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Which logical connective is ∪? Complete: x ∈ A ∪ B when …	'or' (inclusive): x ∈ A ∪ B when x ∈ A or x ∈ B, or both.	sets math logic	Math::Sets::The algebra of sets
Which logical connective is ∩?	'and': x ∈ A ∩ B when x ∈ A and x ∈ B.	sets math logic	Math::Sets::The algebra of sets
Which logical connective is the complement A′?	'not': x ∈ A′ when x ∉ A (and x is in the universe U).	sets math logic	Math::Sets::The algebra of sets
Rewrite A ⊆ B as a logical statement about members.	For every x: if x ∈ A then x ∈ B. Subset is if-then applied to every member at once.	sets math logic	Math::Sets::The algebra of sets
Rewrite A ∖ B in logic, and as an intersection.	x ∈ A and x ∉ B; equivalently A ∖ B = A ∩ B′.	sets math logic	Math::Sets::The algebra of sets
Why does a complement only make sense relative to a universal set U?	'Everything not in A' needs an everything, and there is no set of everything (Russell). So A′ = U ∖ A for a U fixed in advance.	sets math definition	Math::Sets::The algebra of sets
What is a universal set?	The set U, fixed before the problem starts, of everything under discussion: every set in the problem is a subset of it, and complements are taken out of it. Not a set of everything; there is none.	sets math definition	Math::Sets::The algebra of sets
Three notations for the complement of A?	A′, Ā (a bar over A) and Aᶜ. Sullivan's Precalculus writes Ā and notes the other two in a footnote; this library writes A′.	sets math definition	Math::Sets::The algebra of sets
Same A, different universal set U: does the complement change?	Yes. A = {1, 2} has complement {3, 4} in U = {1, 2, 3, 4}, {3, 4, 5, 6} in U = {1, …, 6} and ∅ in U = {1, 2}. A complement quoted without its U is an unfinished sentence.	sets math trap	Math::Sets::The algebra of sets
Python has |, &, - and ^ for sets. Why is there no complement operator?	A complement needs a universe and Python cannot guess it. Name U yourself and write U - a.	sets math python	Math::Sets::The algebra of sets
State De Morgan's laws for sets.	(A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′. Taking the complement swaps ∪ and ∩.	sets math law	Math::Sets::The algebra of sets
De Morgan's laws in plain words: what is 'not either', and what is 'not both'?	Not either = neither: not in A and not in B. Not both = at least one is missing: not in A or not in B.	sets math law	Math::Sets::The algebra of sets
Rewrite not (a or b) without the outer not.	not a and not b. Likewise not (a and b) is not a or not b. In SQL, NOT (x = 1 OR y = 2) is x <> 1 AND y <> 2.	sets math law	Math::Sets::The algebra of sets
Expand A ∩ (B ∪ C), and its dual A ∪ (B ∩ C).	A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C), and A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C). Unlike + and × on numbers, each operation distributes over the other.	sets math law	Math::Sets::The algebra of sets
Simplify A ∪ (A ∩ B) and A ∩ (A ∪ B).	Both are A (absorption). A ∩ B already lies inside A, so the union adds nothing; A ∪ B already contains A, so the intersection removes nothing.	sets math law	Math::Sets::The algebra of sets
Simplify A ∪ ∅, A ∩ U, A ∪ U and A ∩ ∅.	A, A, U, ∅. ∅ is the identity for ∪ and U is the identity for ∩; U swallows any union and ∅ swallows any intersection (domination).	sets math law	Math::Sets::The algebra of sets
What are A ∪ A′, A ∩ A′ and (A′)′?	U, ∅ and A. Every x is in A or out of it; none is both; not-not brings you back.	sets math law	Math::Sets::The algebra of sets
Why does checking a law on every subset of a 4-element universe count as a proof, not a sample?	Whether x is on the left or the right side depends only on which of A, B, C contain x. The subsets of a small universe realise every combination of memberships, so the check is the law's complete truth table.	sets math principle	Math::Sets::The algebra of sets
What single structure do the algebra of sets and the logic of and, or, not both form?	A Boolean algebra: the same laws, with ∪, ∩, ′, ∅, U matching or, and, not, false, true.	sets math definition	Math::Sets::The algebra of sets
Is (A ∪ B) ∖ B = A? Give a counterexample if not.	No. A = {1}, B = {1}: (A ∪ B) ∖ B = ∅ ≠ A. Removing B also removes the part of A that overlapped it; difference does not undo union.	sets math trap	Math::Sets::The algebra of sets
From A ∪ B = A ∪ C, can you cancel A and conclude B = C?	No. A = {1}, B = ∅, C = {1}: both unions are {1}. A absorbs whatever B and C differ by, so union has no cancellation law.	sets math trap	Math::Sets::The algebra of sets
Is set difference associative? Commutative?	Neither. With A = B = C = {1}: (A ∖ B) ∖ C = ∅ but A ∖ (B ∖ C) = {1}. And A ∖ B ≠ B ∖ A unless A = B.	sets math trap	Math::Sets::The algebra of sets
How many regions does a Venn diagram of n sets have, and what does each stand for?	2^n, one for each pattern of in-or-out across the n sets. Three sets give 8, counting the region outside all the circles.	sets math venn	Math::Sets::The algebra of sets
In a three-set Venn diagram, which set is the region outside all three circles?	(A ∪ B ∪ C)′, the complement of the union: everything in U that is in none of them.	sets math venn	Math::Sets::The algebra of sets
What does the rectangle in a Venn diagram stand for, and the part of it outside a circle?	The rectangle is the universal set U, and the circles are sets inside it. The part of the rectangle outside a circle is that set's complement.	sets math venn	Math::Sets::The algebra of sets
How does a Venn diagram show A ⊆ B, and A and B disjoint?	A ⊆ B: the circle A drawn inside the circle B. Disjoint (A ∩ B = ∅, no member in common): two circles drawn apart. The usual overlapping circles claim nothing; a region may be empty.	sets math venn	Math::Sets::The algebra of sets
Which three properties make ⊆ an order relation?	Reflexive (A ⊆ A); antisymmetric (A ⊆ B and B ⊆ A give A = B, which is extensionality again); transitive (A ⊆ B and B ⊆ C give A ⊆ C).	sets math order	Math::Sets::The algebra of sets
Why is ⊆ a partial order and not a total one?	Some pairs are incomparable: neither {1} ⊆ {2} nor {2} ⊆ {1}. A total order, like ≤ on numbers, compares every pair.	sets math order	Math::Sets::The algebra of sets
What is a poset, and what is the standard example?	A partially ordered set: a set with a reflexive, antisymmetric, transitive relation that need not compare every pair. The subsets of a set under ⊆ are the standard example.	sets math order	Math::Sets::The algebra of sets
Why does sorted([{3}, {1, 2}, {1}]) give a meaningless order in Python, and what is the fix?	sorted() compares with <, which for sets is proper subset, a partial order; {3} is neither below nor above the others, so the result depends on the input order. Sort by a total key, e.g. key=lambda s: (len(s), sorted(s)).	sets math python	Math::Sets::The algebra of sets
How many subsets does a set with n members have, and why?	2^n: each member is independently in or out. {1, 2, 3, 4} has 16 subsets.	sets math definition	Math::Sets::The algebra of sets
Define the symmetric difference A △ B, two ways.	The members of exactly one of A and B: (A ∖ B) ∪ (B ∖ A), equivalently (A ∪ B) ∖ (A ∩ B). In logic it is exclusive or.	sets math definition	Math::Sets::The algebra of sets
{1, 2, 3, 4} △ {1, 4, 5} = ?	{2, 3, 5}. 1 and 4 are in both and drop out; 2, 3 and 5 are each in exactly one.	sets math example	Math::Sets::The algebra of sets
What notations exist for symmetric difference, and which does MathWorld recommend?	A △ B, A ⊖ B, A ∇ B and A + B. MathWorld recommends ⊖, because each of the others already means something else in mathematics.	sets math definition	Math::Sets::The algebra of sets
Why is symmetric difference sometimes written A + B?	The subsets of U form a group under △: associative, commutative, identity ∅, and A △ A = ∅, so every set is its own inverse. With ∩ as multiplication they form a ring, like bits under XOR and AND.	sets math law	Math::Sets::The algebra of sets
What does a chain A △ B △ C △ … contain?	The elements that occur in an ODD number of the sets, not those in exactly one. Each pair of occurrences cancels because A △ A = ∅. E.g. {0,1,2,3,4} △ {2,3,4} △ {2,5} = {0, 1, 2, 5}: 2 is in all three and survives.	sets math law	Math::Sets::The algebra of sets
When is A △ B = ∅?	Exactly when A = B. So an empty symmetric difference is a test of set equality.	sets math law	Math::Sets::The algebra of sets
How do ∀ and ∃ split over a union A ∪ B?	∀x∈A∪B p ⇔ (∀x∈A p) ∧ (∀x∈B p), and ∃x∈A∪B p ⇔ (∃x∈A p) ∨ (∃x∈B p). 'For all' over a union becomes 'and'; 'there exists' over a union becomes 'or'.	sets math law	Math::Sets::The algebra of sets
If some x in A satisfies p and some x in B satisfies p, must some x in A ∩ B satisfy p?	No. A = {1}, B = {2}, p true on both: the witnesses differ and A ∩ B is empty. Only the converse holds: ∃x∈A∩B p implies ∃x∈A p and ∃x∈B p.	sets math trap	Math::Sets::The algebra of sets
If every x in A satisfies p, or every x in B does, must every x in A ∪ B?	No. A = ∅ makes 'every x in A' true vacuously while p can fail on B. The valid direction is the other way: ∀ over the union implies ∀ over each part.	sets math trap	Math::Sets::The algebra of sets
What are the truth values of ∀x∈∅ p(x) and ∃x∈∅ p(x)?	True and false, for every p: no member can fail the test, and no member can pass it. This vacuous truth is the whole proof that ∅ ⊆ A.	sets math law	Math::Sets::The algebra of sets
How is A = B proved in practice?	Show both inclusions, A ⊆ B and B ⊆ A. Each one is a 'take any x ∈ A and show x ∈ B' argument.	sets math law	Math::Sets::The algebra of sets
In set theory, which two words are taken as undefined and described only by axioms?	'Set' and 'is a member of' (∈). Every definition of set in words (a collection, a box) is a picture, not a definition; the axioms say what sets do.	sets math definition	Math::Sets::The definition on trial
State the axiom of extensionality, and say what question it settles.	Two sets with the same members are the same set. It settles when two sets are equal: membership is the only thing a set records.	sets math axiom	Math::Sets::The definition on trial
Why is {2, 5} = {5, 2}?	Same members, so equal by extensionality. Order is not membership: a set does not record which member was written first.	sets math axiom	Math::Sets::The definition on trial
Is {1, 1, 2} a legal way to write a set? Which set is it?	Yes: it is {1, 2}. Being listed twice is the same as being a member once, because a set records only whether each object is in it.	sets math axiom	Math::Sets::The definition on trial
The textbook says a set is a collection of 'distinct objects'. What is that word really doing?	Not forbidding repeats in the list, since {1, 1, 2} is legal. It is reaching for extensionality, the rule for when two sets are equal, which the textbook never states.	sets math trap	Math::Sets::The definition on trial
Why is it correct to say THE empty set rather than AN empty set?	Any two empty sets have the same members (none), so by extensionality they are equal. There is exactly one, written ∅.	sets math theorem	Math::Sets::The definition on trial
What does 'well-defined' mean when applied to a set?	For every object it is settled whether it is a member or not. It does not mean anyone knows the answer.	sets math definition	Math::Sets::The definition on trial
Give a set that is well-defined even though nobody knows whether it is empty.	The odd perfect numbers (odd n equal to the sum of its proper divisors). Any given n can be checked, so membership is settled; whether any exists is an open problem.	sets math example	Math::Sets::The definition on trial
State Russell's paradox.	Let R = {x : x ∉ x}, the set of all sets that are not members of themselves. Then R ∈ R holds exactly when R ∉ R, a contradiction, so no such set exists.	sets math theorem	Math::Sets::The definition on trial
What does Russell's paradox show about the definition 'a set is a well-defined collection'?	That a perfectly sharp rule (x ∉ x) can still fail to determine a set. Being well-defined is not enough; the rule must select from a set you already have.	sets math trap	Math::Sets::The definition on trial
Whose foundational system did Russell's paradox break, and how did he hear of it?	Frege's. Russell found the paradox in 1901 and wrote to Frege in 1902; Frege's system assumed every property defines a set.	sets math history	Math::Sets::The definition on trial
State the axiom of separation (Zermelo, 1908).	For any set A and property P, {x ∈ A : P(x)} is a set. A rule may select members from an existing set, never from 'everything'.	sets math axiom	Math::Sets::The definition on trial
With separation in place, what does Russell's argument prove instead of a contradiction?	For any set U, the set R = {x ∈ U : x ∉ x} is not a member of U. So no set contains every set: there is no universal set.	sets math theorem	Math::Sets::The definition on trial
Which older argument has the same shape as Russell's paradox?	Cantor's diagonal argument: build an object that differs from the y-th item at position y, so it is on no list. Russell's R differs from each set y on the question 'is y in y?'.	sets math connection	Math::Sets::The definition on trial
Is {1} ∈ {1, 2}? Is {1} ⊆ {1, 2}?	No and yes. ∈ asks whether the object {1} is one of the members (they are 1 and 2); ⊆ asks whether every member of {1} is in {1, 2}.	sets math trap	Math::Sets::The definition on trial
Which sets is ∅ a subset of, and why?	Every set, including ∅ itself. A ⊆ B means every member of A is in B, and ∅ has no member that could fail the test.	sets math theorem	Math::Sets::The definition on trial
What is the difference between A ⊆ B and A ⊂ B?	A ⊆ B allows A = B. A ⊂ B (proper subset) means A ⊆ B and A ≠ B: B has at least one member A lacks.	sets math definition	Math::Sets::The definition on trial
Is every set a subset of itself? A proper subset of itself?	Yes: every member of A is in A. No: a proper subset must be missing something.	sets math theorem	Math::Sets::The definition on trial
Define A ∪ B, A ∩ B and A ∖ B in one line each.	A ∪ B: members of A or of B (or both). A ∩ B: members of both. A ∖ B: members of A that are not in B.	sets math definition	Math::Sets::The definition on trial
When are two sets disjoint?	When they have no member in common: A ∩ B = ∅. ∅ is disjoint from every set, including itself.	sets math definition	Math::Sets::The definition on trial
Does 'null set' mean the same thing in every book?	No. In algebra and logic it means ∅. In measure theory and probability it means a set of measure zero, which can be infinite: the Cantor set is a null set.	sets math trap	Math::Sets::The definition on trial
Verdict: is 'a set is a well-defined collection of distinct objects' a good definition?	A useful first description, not a definition. 'Collection' is a synonym; when two sets are equal is left out; and Russell shows a well-defined rule can fail to give a set.	sets math principle	Math::Sets::The definition on trial
What does the 'box' definition (a set is a box with things inside) do better than 'a well-defined collection'?	It is built around ∈ (a ∈ A, c ∉ A), it treats the picture as a picture, and it says a set is a single object, so a set can be a member of another set.	sets math principle	Math::Sets::The definition on trial
What does the 'box' definition of a set leave out?	Any rule for when two sets are equal (extensionality) and any notion of well-defined. It cannot explain why {1, 1} = {1} or why there is only one empty set.	sets math trap	Math::Sets::The definition on trial
Is ∈ transitive? Give the standard counterexample.	No. 1 ∈ {1} and {1} ∈ {{1}}, yet 1 ∉ {{1}}: the only member of {{1}} is {1}. A member of a member is not a member. (⊆ is transitive.)	sets math trap	Math::Sets::The definition on trial
Is ∅ = {∅}?	No. ∅ has no members; {∅} has exactly one member, the empty set. An empty box inside a box is a box with something in it.	sets math theorem	Math::Sets::The definition on trial
Name three ways the 'box' picture of a set misleads.	A box in a box is inside the outer box, but ∈ is not transitive; a thing sits in one box, but one element belongs to many sets; a box can hold two identical marbles, but a set has no copies.	sets math trap	Math::Sets::The definition on trial
What does a, b ∈ A abbreviate?	a ∈ A and b ∈ A.	sets math definition	Math::Sets::The definition on trial
Does Russell's paradox depend on the step 'C is either ordinary or extraordinary'?	No. With P meaning C ∈ C, the definition gives P ↔ ¬P. Assuming P gives ¬P, a contradiction, so ¬P; then ¬P gives P. No case split is used, and the argument is valid even in intuitionistic logic, which rejects excluded middle.	sets math theorem	Math::Sets::The definition on trial
A contradiction refutes its premises together. Which premise does ZF set theory give up to escape Russell?	That C, the set of all ordinary sets, exists. Separation only lets you collect the ordinary members of a set you already have.	sets math axiom	Math::Sets::The definition on trial
In ZF, which sets are 'ordinary' (not members of themselves)?	All of them, by the axiom of foundation. So the set of all ordinary sets would be the set of everything, which does not exist.	sets math theorem	Math::Sets::The definition on trial
Which many-valued logic gives 'C ∈ C' a consistent truth value, and which do not?	Łukasiewicz logic: the value ½, since ½ ↔ ¬½ comes out fully true. Kleene's K3 and Gödel's logics also reject excluded middle, but no value there satisfies P ↔ ¬P.	sets math logic	Math::Sets::The definition on trial
Why does no finite-valued Łukasiewicz logic rescue naive set theory?	Curry's sets C_k = {x : x ∈ x → (x ∈ x → … → false)} with k arrows need the truth value k/(k+1). A logic with finitely many values misses some of these, so some C_k gets no value at all.	sets math logic	Math::Sets::The definition on trial
Name two non-classical escapes from Russell's paradox, and what each gives up.	Paraconsistent logic keeps the contradiction but stops it from proving everything. Contraction-free logic blocks the step 'P gives ¬P, therefore ¬P'. Both are research programmes; mainstream mathematics instead denies that R exists.	sets math logic	Math::Sets::The definition on trial
A relation on a class or set S is what, exactly?	A set (class) of ordered pairs (x, y) with x and y in S, that is, a subset of S × S. x R y means (x, y) ∈ R and nothing more.	sets math definition	Math::Sets::Orderings
Membership relation ∈_C and identity relation Id_C on a class C: define both.	∈_C = {(x, y) ∈ C × C : x ∈ y}. Id_C = {(x, y) ∈ C × C : x = y}. Id_C is an equivalence and a partial order at once.	sets math definition	Math::Sets::Orderings
Domain and image (range) of a relation R on C.	dom R = {x : (x, y) ∈ R for some y}; im R = {y : (x, y) ∈ R for some x}. Many books say range for the image.	sets math definition	Math::Sets::Orderings
Inverse of a relation R.	R⁻¹ = {(x, y) : (y, x) ∈ R}. Every relation has one; it need not be a function even when R is.	sets math definition	Math::Sets::Orderings
Composition T ∘ R of two relations, correctly stated.	T ∘ R = {(x, y) : (x, z) ∈ R and (z, y) ∈ T for some z}: first R, then T. (André's Definition 5.5 drops the x from the condition; this is the intended one.)	sets math definition	Math::Sets::Orderings
R = {(x, y) : x ∈ y} and T = {(x, y) : x = {y}} on S = {a, b, c, {a}, {a, b}, {{c}}, ∅, {∅}}. What is R ∘ T?	{({a}, {a}), ({a}, {a, b}), ({∅}, {∅})}. The book lists only two pairs and misses ({a}, {a, b}): T sends {a} to a, and R sends a to {a, b} as well as to {a}.	sets math trap	Math::Sets::Orderings
Reflexive, irreflexive: define and contrast.	Reflexive: (x, x) ∈ R for every x in S. Irreflexive: (x, x) ∉ R for every x. A relation can be neither, e.g. {(a, a), (a, b)} on {a, b}.	sets math definition	Math::Sets::Orderings
Symmetric, antisymmetric, asymmetric: define all three.	Symmetric: x R y implies y R x. Antisymmetric: x R y and y R x imply x = y. Asymmetric: x R y implies not y R x. Asymmetric = antisymmetric + irreflexive.	sets math definition	Math::Sets::Orderings
Transitive: define, and give the smallest failure.	x R y and y R z imply x R z. R = {(a, b), (b, c)} fails: the shortcut (a, c) is missing.	sets math definition	Math::Sets::Orderings
Does symmetric + transitive give reflexive?	No. R = {(a, a), (b, b), (d, d), (a, b), (b, a)} on {a, b, c, d} is symmetric and transitive and (c, c) is missing. The argument x R y, y R x so x R x only reaches the x that are related to something.	sets math trap	Math::Sets::Orderings
Is the relation 'x and y are distinct siblings' transitive?	No. x T y and y T x would require x T x, and nobody is their own distinct sibling. 'Siblings or the same person' is an equivalence.	sets math trap	Math::Sets::Orderings
Which properties does the empty relation ∅ on a nonempty S have?	Irreflexive, symmetric, antisymmetric, asymmetric and transitive, all vacuously: there is no pair to fail any of them. Not reflexive.	sets math trap	Math::Sets::Orderings
Equivalence relation: definition and what it does to S.	Reflexive, symmetric and transitive. Its classes S_x = {y : x R y} are nonempty, pairwise disjoint and cover S: they partition S.	sets math definition	Math::Sets::Orderings
Partition of a set S: the three conditions.	A family of subsets of S whose union is S, pairwise disjoint (distinct blocks have empty intersection), and none empty.	sets math definition	Math::Sets::Orderings
Finest and coarsest partition of S.	Finest: {{x} : x ∈ S}, every element alone, from Id_S. Coarsest: {S}, one block, from the relation that relates everything. Note {S}, not S and not {{S}}.	sets math example	Math::Sets::Orderings
Quotient set S/R and equivalence class of x.	The class of x is S_x = [x] = {y : x R y}; the quotient set S/R is the set of all classes, which is the partition induced by R.	sets math definition	Math::Sets::Orderings
Partial order (non-strict): definition and symbol.	Reflexive, antisymmetric, transitive; written a ≤ b. Examples: ⊆ on subsets, divides on positive integers, ≤ on numbers.	sets math definition	Math::Sets::Orderings
What is a poset?	A partially ordered set: a set together with a partial order on it. Pairs of elements may be incomparable, like {1} and {2} under ⊆.	sets math definition	Math::Sets::Orderings
Linear (total) order.	A partial order in which every two elements are comparable: a ≤ b or b ≤ a. ≤ on numbers is one; ⊆ and divides are not.	sets math definition	Math::Sets::Orderings
Strict order: definition, and how it relates to the non-strict one.	Irreflexive, asymmetric, transitive; written a < b. Remove the pairs (x, x) from a partial order to get a strict one; add them back to return. The two kinds match one to one.	sets math definition	Math::Sets::Orderings
Comparable elements; chain; antichain.	a and b are comparable when a ≤ b or b ≤ a. A chain is a subset that is linearly ordered; an antichain is a subset with no two comparable members, like the primes under divides.	sets math definition	Math::Sets::Orderings
Maximal versus maximum element of a poset.	Maximal: nothing is strictly above it. Maximum: it is above every other element. A maximum is maximal; a poset can have several maximal elements and no maximum.	sets math definition	Math::Sets::Orderings
Mortimer's ancestors ordered by 'is a descendant of': minimum, maximal elements, maximum?	Mortimer is the minimum (below everyone). The two spontaneously generated ancestors A and E are each maximal. There is no maximum, because A and E are not comparable.	sets math example	Math::Sets::Orderings
Divisibility on {1, ..., 12}: minimum, maximal elements, a chain, an antichain.	Minimum 1. Maximal: 7, 8, 9, 10, 11, 12 (each divides nothing else in range). Chain: 1, 2, 4, 8. Antichain: the primes 2, 3, 5, 7, 11. No maximum.	sets math example	Math::Sets::Orderings
Lexicographic order on pairs.	(a, b) ≤ (c, d) when a < c, or a = c and b ≤ d: compare the first entries, and only on a tie the second. A partial order (linear if both components are); a dictionary and a two-column sort use it.	sets math definition	Math::Sets::Orderings
How many equivalences, partial orders and linear orders are there on a 3-element set?	5 equivalences (the Bell number B(3)), 19 partial orders, 6 linear orders (3!). The identity relation is counted among both equivalences and partial orders.	sets math example	Math::Sets::Orderings
Logical matrix of a relation: what is it, and which property is which shape?	An n × n grid with a 1 in row x, column y when x R y. Reflexive: full diagonal. Symmetric: equal to its transpose. Equivalence: blocks of 1s along the diagonal, one per class. Transitive: the Boolean product M·M stays inside M. R⁻¹ is the transpose; T ∘ R is the Boolean product.	sets math notation	Math::Sets::Orderings
Partition of an interval, in real analysis.	A finite list of cut points a = x₀ < x₁ < ... < xₙ = b. The Riemann integral is squeezed between the lower and upper sums over the cells. Not a partition of a set: the closed cells share endpoints.	sets math definition	Math::Sets::Orderings
How do the two meanings of partition connect?	A Riemann partition P refines Q when P contains all of Q's cut points; refinement is a partial order on partitions, and any two have a common refinement, their union. The half-open cells of a Riemann partition do form a set partition.	sets math definition	Math::Sets::Orderings
Universal class versus universal set.	André's 𝒰 = {x : x = x} is the class of all elements, a proper class that no theory can make a set. The universal set U of a Venn diagram is a set chosen for one problem. Same root, different levels.	sets math class	Math::Sets::Orderings
Proper class, element, set: André's three kinds of object.	Every object is a class. An element is a class that belongs to some class. A set is a class that is an element. A proper class is a class that is not an element, like {x : x ∉ x} or the class of all sets.	sets math class	Math::Sets::Orderings
Hausdorff's ordered pair.	(a, b) = {{a, ∅}, {b, {∅}}}: ∅ and {∅} tag first and second. It has the pair property, (a, b) = (c, d) iff a = c and b = d, and gives a different set from Kuratowski's {{a}, {a, b}}; either will do.	sets math notation	Math::Sets::Orderings
Why is C × D a set when C and D are? Which axioms?	Every pair (c, d) lies in P(P(C ∪ D)): {c} and {c, d} are subsets of C ∪ D. Union gives C ∪ D, power set twice gives P(P(C ∪ D)), and subsets (separation) cuts C × D out of it.	sets math definition	Math::Sets::Orderings
How do you write the empty set in Python, and why not {}?	set(). The braces {} were taken first by the empty dict, so the empty set has no literal of its own.	sets python creating	Math::Sets::Sets in Python
Predict: set('foo') and {'foo'}.	{'f', 'o'} and {'foo'}. set() takes one iterable and adds each item it yields, characters here, with the repeat dropped; braces add each thing written between them, whole.	sets python creating	Math::Sets::Sets in Python
Why does set('a', 'b', 'c') fail?	TypeError: set() takes at most one argument, an iterable. Write set(['a', 'b', 'c']) or {'a', 'b', 'c'}.	sets python trap	Math::Sets::Sets in Python
What set is {('a', 'b', 'c')}?	A set with ONE member, the tuple ('a', 'b', 'c'). Braces do not unpack what is inside them.	sets python trap	Math::Sets::Sets in Python
Which of these build {'a', 'b', 'c'}: set('abc'), set(['a', 'b', 'c']), set('a', 'b', 'c'), {('a', 'b', 'c')}?	The first two. set('a', 'b', 'c') raises TypeError (three arguments); {('a', 'b', 'c')} is a set with one tuple in it.	sets python creating	Math::Sets::Sets in Python
Predict: {2*x for x in [1, 1, 2, 2, 3, 3]}.	{2, 4, 6}. A set comprehension applies the rule to every item and drops repeats as it builds. It always draws from a source, which is Zermelo's separation in code.	sets python creating	Math::Sets::Sets in Python
What must be true of every member of a Python set, and why?	It must be hashable: hash(x) works and never changes while x is stored. A set is a hash table, and a changed hash would leave the member filed in the wrong slot, lost.	sets python hashing	Math::Sets::Sets in Python
Why does s.add([1, 2, 1]) fail, and what is the fix?	TypeError: unhashable type. A list can change, so it cannot be hashed. Add a frozen copy, s.add(tuple([1, 2, 1])), which gives {(1, 2, 1)}.	sets python hashing	Math::Sets::Sets in Python
Is every tuple hashable?	No. (1, [2]) is unhashable because it contains a list. The rule is hashable, not immutable: a tuple is only as hashable as its contents.	sets python hashing	Math::Sets::Sets in Python
How many members does {True, 1, 1.0, '1', (1,)} have?	3. True == 1 == 1.0 and they hash alike, so they are one member (the first written, True, is the one kept); '1' and (1,) are two more.	sets python hashing	Math::Sets::Sets in Python
Why does s1.add(s2) fail when s2 is a set, and what did you probably mean?	add puts in ONE member, and a set is unhashable, so TypeError. To merge: s1.update(s2) or s1 |= s2. To store s2 as a single member: s1.add(frozenset(s2)).	sets python trap	Math::Sets::Sets in Python
Three ways to get the members of s2 into s1?	s1.update(s2) and s1 |= s2 change s1 in place; s1 | s2 builds a new set and leaves both alone. update() accepts any iterable, |= and | need a set on the right.	sets python operations	Math::Sets::Sets in Python
How do you store a whole set as one member of another set?	Freeze it first: s1.add(frozenset(s2)). A frozenset cannot change, so it is hashable.	sets python frozenset	Math::Sets::Sets in Python
Is + defined for sets?	No: {1} + {2} raises TypeError. + concatenates sequences; the set operation is union, written |. Note that - does work: it is set difference.	sets python trap	Math::Sets::Sets in Python
Give the operator and the method for union, intersection, difference and symmetric difference.	| union(), & intersection(), - difference(), ^ symmetric_difference(). Same result on two sets; the methods also accept any iterable.	sets python operations	Math::Sets::Sets in Python
Give the operators for subset, proper subset, superset and proper superset, and the method names.	<=, <, >=, >. Methods: issubset() for <= and issuperset() for >=; proper subset and proper superset have no method.	sets python operations	Math::Sets::Sets in Python
Which set test has a method but no operator?	isdisjoint(). The nearest operator form is not (a & b), which builds the intersection first.	sets python operations	Math::Sets::Sets in Python
Predict, with a = {1, 2}: a | (3, 4) and a.union((3, 4)).	a | (3, 4) raises TypeError (unsupported operand types set and tuple); a.union((3, 4)) returns {1, 2, 3, 4}. Operators need sets on both sides; methods accept any iterable.	sets python operators-vs-methods	Math::Sets::Sets in Python
X = {1, 3, 5}. Does X & [1, 5, 9] work? Does X.intersection([1, 5, 9])?	The operator raises TypeError; the method returns {1, 5}. A method converts its argument to a set; an operator does not.	sets python operators-vs-methods	Math::Sets::Sets in Python
a.union(b, c) works. Does a.symmetric_difference(b, c)?	No: symmetric_difference takes exactly one argument (TypeError). union, intersection and difference take several. For more sets chain the operator: a ^ b ^ c.	sets python operators-vs-methods	Math::Sets::Sets in Python
Can you call s.__contains__(x) directly?	Yes: it is exactly what x in s calls. A note saying it cannot be used is wrong.	sets python trap	Math::Sets::Sets in Python
How is s1 ^ s2 ^ s3 evaluated, and which elements does it keep?	Left to right, as (s1 ^ s2) ^ s3. It keeps the elements that are in an ODD number of the three sets, not those in exactly one.	sets python chaining	Math::Sets::Sets in Python
Predict: {0, 1, 2, 3, 4} ^ {2, 3, 4} ^ {2, 5}.	{0, 1, 2, 5}. Count occurrences: 0, 1 and 5 once; 2 three times (odd, so it stays); 3 and 4 twice (even, so they cancel).	sets python chaining	Math::Sets::Sets in Python
What are the in-place forms of the set operators?	|=, &=, -=, ^= change the left set in place: s |= {11, 22} adds two members. Each needs a set on the right; update() is the iterable-friendly form of |=.	sets python operations	Math::Sets::Sets in Python
How do you test whether two sets are equal?	a == b. Only membership counts: {1, 9} == {9, 1} is True, whatever order they were built in.	sets python equality	Math::Sets::Sets in Python
Why is list(a) == list(b) not a test for set equality?	Equal sets can iterate in different orders: list({1, 9}) is [1, 9] but list({9, 1}) is [9, 1], so the lists differ while the sets are equal.	sets python equality	Math::Sets::Sets in Python
Why is a.difference(b) == set() not a test for set equality?	It only shows a is a subset of b. {1}.difference({1, 2}) is empty, yet {1} != {1, 2}. Use a == b, or check that a ^ b is empty.	sets python equality	Math::Sets::Sets in Python
Predict: set().issubset([]).	True. The empty set is a subset of everything, so issubset() of an empty iterable is True for an empty set; it is False only for a non-empty one.	sets python trap	Math::Sets::Sets in Python
What does if (s1 <= s2, s3): actually test?	A tuple, (s1 <= s2, s3), which is non-empty and so always true: the branch always runs. Write s1 <= s2 and s1 <= s3.	sets python trap	Math::Sets::Sets in Python
remove(x) versus discard(x) when x is not in the set?	remove raises KeyError; discard does nothing. Both remove x when it is present.	sets python removing	Math::Sets::Sets in Python
What does s.pop() do, and what happens on an empty set?	Removes and returns an arbitrary member, since a set has no first one. On an empty set it raises KeyError.	sets python removing	Math::Sets::Sets in Python
fs is a frozenset. What does fs.add(3) do, and how do you get a frozenset with 3 added?	AttributeError: frozenset has no add. Build a new one, fs.union([3]) or fs | {3}; fs itself is unchanged.	sets python frozenset	Math::Sets::Sets in Python
Predict the types of {1} | frozenset({2}) and frozenset({2}) | {1}.	set, then frozenset. When a set and a frozenset are combined, the LEFT operand decides the type of the result.	sets python frozenset	Math::Sets::Sets in Python
How do you remove duplicates from a list and keep first-seen order?	list(dict.fromkeys(L)): a dict's keys are a set that remembers insertion order. list(set(L)) may reorder.	sets python dedupe	Math::Sets::Sets in Python
Why does list(set([1, 2, 1, 3])) usually come out sorted, and is that reliable?	Luck, not a rule: a small int hashes to itself and lands in slot n mod table size. list(set([10, 3, 8])) is [8, 10, 3], and a set of strings changes order between runs.	sets python dedupe	Math::Sets::Sets in Python
How do you sort a set?	sorted(s) returns a sorted list. A set itself has no order and cannot be sorted in place.	sets python sorting	Math::Sets::Sets in Python
How do you sort a set that mixes ints and strings?	sorted(s, key=lambda x: (isinstance(x, str), x)). Plain sorted() raises TypeError (int against str); the key puts numbers first, since False < True, and never compares across types.	sets python sorting	Math::Sets::Sets in Python
Why keep a set of visited nodes when walking a graph?	A graph with cycles would loop forever without it, and a set answers 'seen already?' with one hash lookup instead of scanning a list.	sets python uses	Math::Sets::Sets in Python
With key sets customers and invoiced, which set operations give the keys of an inner join, an outer join and a left anti-join?	customers & invoiced, customers | invoiced, customers - invoiced. A join keeps the rows whose keys survive the set operation.	sets python uses	Math::Sets::Sets in Python
What does B >= A mean for sets, in English and in Polish?	B is a superset of A: every member of A is in B. B jest nadzbiorem A.	sets python operations	Math::Sets::Sets in Python
s1 = {'a', 'b', 'c'}, s2 = {'c', 'd', 'e'}. Predict s1 | s2 and s1 & s2.	{'a', 'b', 'c', 'd', 'e'} and {'c'}. Union takes every member of either; intersection only the shared 'c'.	sets python kata	Math::Sets::Sets in Python
Predict: set((1, 2, 3)) & set((2, 3, 4)) & set((3, 4, 5)).	{3}: the only number in all three.	sets python kata	Math::Sets::Sets in Python
Is set2 < = set1 (with a space) a subset test?	No: SyntaxError. <= is a single token; the space splits it into < and =.	sets python trap	Math::Sets::Sets in Python
What does d1.keys() & d2.keys() return?	A set of the keys both dicts have. A keys view behaves like a set: |, &, -, ^ and <= all work, and it accepts any iterable on the other side.	sets python dict-views	Math::Sets::Sets in Python
Do dict.items() and dict.values() views support set operations?	items() does, when the values are hashable, and it matches only equal key-value pairs. values() does not (TypeError): values can repeat and need not be hashable.	sets python dict-views	Math::Sets::Sets in Python
Predict: s = {1, 2}; s.update({1: 'a', 5: 'e'}); s.	{1, 2, 5}. Iterating a dict yields its keys, so update adds 1 and 5 and ignores the values.	sets python dict-views	Math::Sets::Sets in Python
'Remove duplicates' hides two decisions. Which?	Which copy represents the group (first seen, last seen, arbitrary), and what order the survivors come in (first-seen, sorted, hash order). Every idiom answers both, silently.	sets python dedupe	Math::Sets::Sets in Python
words = ['Pear', 'fig', 'pear']. Predict list({w.lower(): w for w in words}.values()).	['pear', 'fig']: the LAST spelling at the FIRST spelling's position. A dict keeps a key's first position but overwrites its value on every repeat.	sets python dedupe	Math::Sets::Sets in Python
How do you dedupe by a key and keep the FIRST copy of each group?	Loop with d.setdefault(key(x), x): setdefault never overwrites an existing key. Then list(d.values()).	sets python dedupe	Math::Sets::Sets in Python
What does [k for k, _ in groupby(items)] remove?	Only ADJACENT repeats: [3, 1, 3] stays [3, 1, 3]. Sort first. It is the same rule as Rust's dedup and ABAP's DELETE ADJACENT DUPLICATES.	sets python dedupe	Math::Sets::Sets in Python
Predict: list(dict.fromkeys([1.0, 1, True])).	[1.0]. The three are equal and hash alike, so they are one key, and the first to arrive is kept with its own type.	sets python dedupe	Math::Sets::Sets in Python
How do you remove duplicate rows from a list of lists?	Lists are unhashable, so hash a frozen copy: list(dict.fromkeys(map(tuple, rows))). The fallback, if r not in out: out.append(r), works on anything but is quadratic.	sets python dedupe	Math::Sets::Sets in Python
How does ABAP remove duplicates from an internal table?	SORT by the fields, then DELETE ADJACENT DUPLICATES COMPARING the same fields (adjacent only; keeps the first of each run). Or INSERT the rows into a table WITH UNIQUE KEY, where the first wins and later duplicates set sy-subrc = 4.	sets python dedupe	Math::Sets::Sets in Python
How does Rust remove duplicates from a Vec?	sort() then dedup() gives sorted order and removes adjacent repeats. Filtering with a HashSet, where seen.insert(x) returns true only the first time, keeps first-seen order.	sets python dedupe	Math::Sets::Sets in Python
Why does Python need a second set type, frozenset?	A set can change after it is stored, so its hash could change and the container would lose it; Python therefore refuses to hash a set. frozenset drops every mutating method, and so it can be a set member or a dict key.	sets python frozenset	Math::Sets::Sets in Python
Predict: {1, 2} in {frozenset({1, 2})}.	True, even though {1, 2} is unhashable: set membership (and remove, discard) treat a set argument as if it were frozen.	sets python frozenset	Math::Sets::Sets in Python
Predict: {frozenset({1, 2}): 'x'}[{1, 2}].	TypeError: unhashable type. A dict lookup does not freeze the key for you; write d[frozenset({1, 2})].	sets python frozenset	Math::Sets::Sets in Python
t = frozenset({1}); u = t; t |= {2}. What are t and u now?	t = frozenset({1, 2}), u = frozenset({1}). A frozenset has no in-place |=, so Python builds a new frozenset and rebinds t; u still names the old one.	sets python frozenset	Math::Sets::Sets in Python
What does f.copy() return for a frozenset f?	f itself: f.copy() is f is True. Nothing can change it, so a copy would protect nothing.	sets python frozenset	Math::Sets::Sets in Python
Why does Rust have no frozenset?	Immutability belongs to the binding, not the type: let s is frozen, let mut s is not. And the borrow checker never hands out a &mut to a stored member. The exception is interior mutability: BTreeSet<Cell<i32>> compiles, and Clippy's mutable_key_type lint warns.	sets python frozenset	Math::Sets::Sets in Python
Can a Rust HashSet contain HashSets?	No: HashSet does not implement Hash, because equal sets can iterate in different orders, so insert fails with E0599. Use BTreeSet as the element type; it hashes its members in sorted order.	sets python frozenset	Math::Sets::Sets in Python
How is a set of sets built in ABAP, which has no frozenset?	Freeze each inner set into a key by hand, e.g. sort it and join it into a string with a separator, and store that in a HASHED TABLE WITH UNIQUE KEY.	sets python frozenset	Math::Sets::Sets in Python
l1 = ['a', 'a', 'b', 'c', 'd'], l2 = ['a', 'b', 'c', 'f']. Why is set(l1) ^ set(l2) equal to {'d', 'f'} while the Counter version (c1 - c2 | c2 - c1) gives a, d, f?	A set drops the copies first, so 'a' is in both and cancels. Counter is a multiset: l1 has two 'a' and l2 one, so one 'a' is left over. Counter has no ^, and - binds tighter than |, so the line is (c1 - c2) | (c2 - c1).	sets python dedupe	Math::Sets::Sets in Python
Predict: bool(set()) and bool({0}).	False and True. An empty set is falsy; any non-empty set is truthy, whatever its members.	sets python trap	Math::Sets::Sets in Python
Math Vault, relational symbols: what does x ∈ A mean, and what is the example?	Set membership: the element x is in the set A. Example: i ∈ ℂ.	membership	Math::Sets::Symbols (Math Vault)
Math Vault, relational symbols: what does x ∉ A mean, and what is the example?	Set non-membership: x is not in A. Example: π ∉ ℚ.	membership	Math::Sets::Symbols (Math Vault)
Math Vault, relational symbols: what does A ∋ x mean, and what is the example?	Set inclusion read from the set's side: A includes the element x. A ∋ x ⟺ x ∈ A. The sign is a mirrored ∈, not the quantifier ∃.	membership trap	Math::Sets::Symbols (Math Vault)
Math Vault, relational symbols: what does A ∌ x mean, and what is the example?	Set non-inclusion: A does not include the element x. Example: {a} ∌ {a}, since the set {a} is not one of its own members.	membership	Math::Sets::Symbols (Math Vault)
Math Vault, relational symbols: what does A = B mean, and what is the example?	Set equality: A and B have the same members (extensionality). Example: ∅ = {}.	relation	Math::Sets::Symbols (Math Vault)
Math Vault, relational symbols: what does A ⊆ B mean, and what is the example?	Subset: every member of A is a member of B, equality allowed. Example: ℕ ⊆ ℚ.	relation	Math::Sets::Symbols (Math Vault)
Math Vault, relational symbols: what does A ⊂ B mean, and what is the example?	Proper subset, in Math Vault's convention: A ⊂ B ⟺ A ⊆ B and A ≠ B. Jech and Halmos use ⊂ for the plain subset; check the book.	relation trap	Math::Sets::Symbols (Math Vault)
Math Vault, relational symbols: what does A ⊄ B mean, and what is the example?	Non-subset: A is not a subset of B, so there is an x ∈ A with x ∉ B. One witness is enough.	relation	Math::Sets::Symbols (Math Vault)
Math Vault, relational symbols: what does A ⊇ B mean, and what is the example?	Superset: every member of B is a member of A. Example: {5, 8, 1} ⊇ {1, 8}.	relation	Math::Sets::Symbols (Math Vault)
Math Vault, relational symbols: what does A ⊃ B mean, and what is the example?	Proper superset: A ⊃ B ⟺ B ⊂ A, so A ⊇ B and A ≠ B.	relation	Math::Sets::Symbols (Math Vault)
Math Vault, relational symbols: what does A ⊉ B mean, and what is the example?	Non-superset: A is not a superset of B. Example: ℕ ⊉ ℝ.	relation	Math::Sets::Symbols (Math Vault)
Math Vault, cardinality symbols: what is |A|?	The cardinality of A, its size by matching. Example: |ℕ| = |ℤ|, since the integers can be listed.	cardinality	Math::Sets::Symbols (Math Vault)
Math Vault, cardinality symbols: what is ℵ₀, ℵ₁, ℵ₂, …?	The aleph numbers: the infinite cardinals in order of size. ℵ₀ = |ℕ|. Under the continuum hypothesis, ℵ₁ = 2^(ℵ₀).	cardinality	Math::Sets::Symbols (Math Vault)
Math Vault, cardinality symbols: what is ℶ₀, ℶ₁, ℶ₂, …?	The beth numbers, by repeated power set: ℶ₀ = |ℕ|, ℶ₁ = |𝒫(ℕ)|, ℶ₂ = |𝒫(𝒫(ℕ))|. ℶ₁ = ℵ₁ is the continuum hypothesis.	cardinality	Math::Sets::Symbols (Math Vault)
Math Vault, cardinality symbols: what is ℷ (gimel)?	The gimel function, ℷ(κ) = κ^(cf κ). By König's theorem ℷ(κ) > κ for every infinite cardinal κ.	cardinality	Math::Sets::Symbols (Math Vault)
Math Vault, cardinality symbols: what is 𝔠?	The cardinality of the continuum: |ℝ| = 𝔠 = |𝒫(ℕ)| = 2^(ℵ₀) = ℶ₁.	cardinality	Math::Sets::Symbols (Math Vault)
Math Vault, cardinality symbols: what is ω, ω₁, ω₂, …?	The omega ordinals: ω = {0, 1, 2, …} is the first infinite ordinal, and ω + 1 = ω ∪ {ω} its successor; ω₁ is the first uncountable ordinal.	ordinal	Math::Sets::Symbols (Math Vault)
Math Vault, cardinality symbols: what is Ω?	In Math Vault and Cantor, the first uncountable ordinal: Ω = ω₁. In Halbeisen, Ω is the class of all ordinals. Two conventions.	ordinal trap	Math::Sets::Symbols (Math Vault)
Math Vault, cardinality symbols: what is ε₀, ε₁, …?	The epsilon ordinals, the fixed points of α ↦ ω^α: ε₀ = ω^(ω^(ω^…)), the least ordinal with ω^ε = ε.	ordinal	Math::Sets::Symbols (Math Vault)
Math Vault, cardinality symbols: what is S(α)?	The successor of the ordinal α, α ∪ {α}, also written α + 1. Addition is defined by recursion: α + S(β) = S(α + β).	ordinal	Math::Sets::Symbols (Math Vault)
Math Vault, cardinality symbols: what is κ⁺?	The successor cardinal of κ, the least cardinal greater than κ. ℵ_α⁺ = ℵ_(α+1).	cardinality	Math::Sets::Symbols (Math Vault)
Which sign says that the set A includes the element x, read from the set's side?	A ∋ x, the mirrored membership sign.	membership	Math::Sets::Symbols (Math Vault)
Which sign says that A is a subset of B and A ≠ B, and what is its trap?	A ⊂ B in Math Vault, Cunningham and this library; in Jech and Halmos ⊂ allows equality, so a book may mean ⊆ by it.	relation trap	Math::Sets::Symbols (Math Vault)
What is the difference between the aleph and the beth numbers?	ℵ_α lists all infinite cardinals in order of size; ℶ_α is reached by repeated power sets from |ℕ|. ℵ₀ = ℶ₀ always; ℵ₁ = ℶ₁ is the continuum hypothesis, undecidable in ZFC.	cardinality	Math::Sets::Symbols (Math Vault)
Which three symbols name the same size: that of the real line?	𝔠, 2^(ℵ₀) and ℶ₁, all equal to |ℝ| = |𝒫(ℕ)|.	cardinality	Math::Sets::Symbols (Math Vault)
What does a witness for A ⊄ B look like?	An element x with x ∈ A and x ∉ B. One such x proves the non-subset claim.	relation	Math::Sets::Symbols (Math Vault)
In an unbracketed set expression, which operation is done first, and what is the full order?	Complement first, then intersection, then union; equals go left to right. So A ∪ B ∩ C means A ∪ (B ∩ C). Difference has no agreed place: Python reads it above ∩, Isabelle, Pascal and SQL level with ∪, and most textbooks bracket it always.	sets math precedence	Math::Sets::Reading a set expression
Why is the order of set operations complement, intersection, union?	It copies logic, where not binds tighter than and, and and tighter than or: ′ is not, ∩ is and, ∪ is or. Difference is 'and not', two connectives, so logic gives it no place of its own.	sets math precedence	Math::Sets::Reading a set expression
How does Python group A | B & C?	A | (B & C): & binds tighter than |, the same as ∩ before ∪.	sets math python	Math::Sets::Reading a set expression
List Python's set operators from tightest to loosest.	- then & then ^ then |. Inherited from C's bitwise operators; it matches the shared textbook order for ∩ and ∪, and puts - (difference) above & by Python's own choice.	sets math python	Math::Sets::Reading a set expression
How is A ∖ B ∖ C read, and does it matter?	(A ∖ B) ∖ C, left to right. It matters: A ∖ (B ∖ C) is a different set, because difference is not associative.	sets math precedence	Math::Sets::Reading a set expression
With U = {1, 2, 3, 4, 5}, evaluate {5} ∪ {1, 2}′ ∩ {2, 3} by the precedence rule, and with the other grouping.	By the rule ∩ goes first: {5} ∪ ({3, 4, 5} ∩ {2, 3}) = {5} ∪ {3} = {3, 5}. Grouped as ({5} ∪ {3, 4, 5}) ∩ {2, 3} it is {3}. Two answers from one line of symbols, so bracket when in doubt.	sets math precedence	Math::Sets::Reading a set expression
What is wrong with writing x ∈ A ∩ x ∈ B?	∩ needs a set on each side, but 'x ∈ A' is a statement, true or false. Write x ∈ A ∧ x ∈ B, or x ∈ A ∩ B.	sets math trap	Math::Sets::Reading a set expression
What is wrong with writing x ∈ A ∧ B?	∧ needs a statement on each side, but B is a set. Write x ∈ A ∧ x ∈ B.	sets math trap	Math::Sets::Reading a set expression
What does Python do with x in A & x in B?	& binds tighter than in, so it reads x in (A & x) in B, a chained comparison that tries to compute A & 2 and raises TypeError.	sets math python	Math::Sets::Reading a set expression
What does x in A and B return in Python, and why is it dangerous?	If x is in A it returns the set B itself, which is truthy even when x is not in B, so an if takes the wrong branch and nothing complains. Write x in A and x in B.	sets math python	Math::Sets::Reading a set expression
Using Kuratowski pairs, (a, b) = {{a}, {a, b}}, what is (1, 2) ∩ (2, 1)?	{{1, 2}}. (1, 2) = {{1}, {1, 2}} and (2, 1) = {{2}, {1, 2}} share only the member {1, 2}.	sets math notation	Math::Sets::Reading a set expression
Read as open intervals, what is (1, 2) ∪ (2, 1)?	(1, 2). The interval (2, 1) is the set of t with 2 < t < 1, which is empty. Same symbols as the ordered pair, a different set: the context decides.	sets math notation	Math::Sets::Reading a set expression
Where do real systems put set difference ∖ in the order of operations?	Python: above ∩ (- before &). Lean 4: level with ∩ at 70, above ∪ at 65, and non-associative. Isabelle/HOL, Pascal, the SQL standard, Z: level with ∪, below ∩. Most textbooks: no level, always bracketed.	sets math precedence	Math::Sets::Reading a set expression
Which unbracketed expressions change value with the convention for ∖, and which does not?	A ∪ B ∖ C and A ∖ B ∩ C change: Python reads A ∪ (B ∖ C) and (A ∖ B) ∩ C, Isabelle (A ∪ B) ∖ C and A ∖ (B ∩ C). A ∩ B ∖ C does not, because A ∩ (B ∖ C) = (A ∩ B) ∖ C is a law.	sets math precedence	Math::Sets::Reading a set expression
Why does A ∖ B ∪ C ∩ D′ not test which convention for ∖ a reader uses?	Its ∖ is leftmost and its ∪ is lowest in every system, so all read it as (A ∖ B) ∪ (C ∩ D′). To tell conventions apart, put ∖ beside a ∪ or ∩ that could take it either way: A ∪ C ∖ B.	sets math precedence	Math::Sets::Reading a set expression
Read the symbol: ∈  ∉  ∋  ∌	is a member of, is not, contains. Explained under: glossary entry: membership.	symbol	Math::Sets::Symbol index
Which symbol says: is a member of, is not, contains?	∈  ∉  ∋  ∌ (explained under: glossary entry: membership)	symbol reverse	Math::Sets::Symbol index
Read the symbol: {a, b, c}	the set whose members are a, b, c. Explained under: what is a set.	symbol	Math::Sets::Symbol index
Which symbol says: the set whose members are a, b, c?	{a, b, c} (explained under: what is a set)	symbol reverse	Math::Sets::Symbol index
Read the symbol: {x ∈ A : P(x)}, {x | P(x)}	the set of x such that P(x). Explained under: glossary entry: set builder notation.	symbol	Math::Sets::Symbol index
Which symbol says: the set of x such that P(x)?	{x ∈ A : P(x)}, {x | P(x)} (explained under: glossary entry: set builder notation)	symbol reverse	Math::Sets::Symbol index
Read the symbol: ∅, { }	the empty set. Explained under: glossary entry: empty set.	symbol	Math::Sets::Symbol index
Which symbol says: the empty set?	∅, { } (explained under: glossary entry: empty set)	symbol reverse	Math::Sets::Symbol index
Read the symbol: U, 𝒰, E, Ω	the universal set (Ω also the class of ordinals, or ω₁). Explained under: glossary entry: universal set.	symbol	Math::Sets::Symbol index
Which symbol says: the universal set (Ω also the class of ordinals, or ω₁)?	U, 𝒰, E, Ω (explained under: glossary entry: universal set)	symbol reverse	Math::Sets::Symbol index
Read the symbol: ⊆  ⊂  ⊊  ⊇  ⊃  ⊄  ⊉	subset, proper subset (two conventions), superset, and their negations. Explained under: glossary entry: symbols.	symbol	Math::Sets::Symbol index
Which symbol says: subset, proper subset (two conventions), superset, and their negations?	⊆  ⊂  ⊊  ⊇  ⊃  ⊄  ⊉ (explained under: glossary entry: symbols)	symbol reverse	Math::Sets::Symbol index
Read the symbol: A = B	the same members, by extensionality. Explained under: glossary entry: extensionality.	symbol	Math::Sets::Symbol index
Which symbol says: the same members, by extensionality?	A = B (explained under: glossary entry: extensionality)	symbol reverse	Math::Sets::Symbol index
Read the symbol: ∪  ∩	union, intersection; ∪𝓕 and ∩𝓕 over a family. Explained under: algebra of sets.	symbol	Math::Sets::Symbol index
Which symbol says: union, intersection; ∪𝓕 and ∩𝓕 over a family?	∪  ∩ (explained under: algebra of sets)	symbol reverse	Math::Sets::Symbol index
Read the symbol: ∖  −	difference, A ∖ B = {x ∈ A : x ∉ B}. Explained under: algebra of sets.	symbol	Math::Sets::Symbol index
Which symbol says: difference, A ∖ B = {x ∈ A : x ∉ B}?	∖  − (explained under: algebra of sets)	symbol reverse	Math::Sets::Symbol index
Read the symbol: A′  Aᶜ  Ā  U − A	complement within the universal set. Explained under: glossary entry: complement.	symbol	Math::Sets::Symbol index
Which symbol says: complement within the universal set?	A′  Aᶜ  Ā  U − A (explained under: glossary entry: complement)	symbol reverse	Math::Sets::Symbol index
Read the symbol: △  ⊕	symmetric difference. Explained under: glossary entry: symmetric difference.	symbol	Math::Sets::Symbol index
Which symbol says: symmetric difference?	△  ⊕ (explained under: glossary entry: symmetric difference)	symbol reverse	Math::Sets::Symbol index
Read the symbol: A × B	Cartesian product, the set of ordered pairs. Explained under: glossary entry: cartesian product.	symbol	Math::Sets::Symbol index
Which symbol says: Cartesian product, the set of ordered pairs?	A × B (explained under: glossary entry: cartesian product)	symbol reverse	Math::Sets::Symbol index
Read the symbol: (a, b), ⟨a, b⟩	ordered pair. Explained under: glossary entry: ordered pair.	symbol	Math::Sets::Symbol index
Which symbol says: ordered pair?	(a, b), ⟨a, b⟩ (explained under: glossary entry: ordered pair)	symbol reverse	Math::Sets::Symbol index
Read the symbol: (x₁, y₁), (x₂, y₂); p = (p₁, p₂), q = (q₁, q₂)	a point by its coordinates: Sullivan numbers the points and letters the axes, Wikipedia and linear algebra letter the points and number the axes, so x₂ is q₁. Explained under: distance in n dimensions.	symbol	Math::Sets::Symbol index
Which symbol says: a point by its coordinates: Sullivan numbers the points and letters the axes, Wikipedia and linear algebra letter the points and number the axes, so x₂ is q₁?	(x₁, y₁), (x₂, y₂); p = (p₁, p₂), q = (q₁, q₂) (explained under: distance in n dimensions)	symbol reverse	Math::Sets::Symbol index
Read the symbol: 𝒫(A), 2ᴬ	power set. Explained under: glossary entry: power set.	symbol	Math::Sets::Symbol index
Which symbol says: power set?	𝒫(A), 2ᴬ (explained under: glossary entry: power set)	symbol reverse	Math::Sets::Symbol index
Read the symbol: |A|, n(A), #A, card(A)	cardinality. Explained under: glossary entry: cardinality.	symbol	Math::Sets::Symbol index
Which symbol says: cardinality?	|A|, n(A), #A, card(A) (explained under: glossary entry: cardinality)	symbol reverse	Math::Sets::Symbol index
Read the symbol: ℵ₀, ℵ₁, ℶ₁, 𝔠, κ⁺, ℷ	infinite cardinals, the continuum, successor, gimel. Explained under: glossary entry: aleph beth and the continuum c.	symbol	Math::Sets::Symbol index
Which symbol says: infinite cardinals, the continuum, successor, gimel?	ℵ₀, ℵ₁, ℶ₁, 𝔠, κ⁺, ℷ (explained under: glossary entry: aleph beth and the continuum c)	symbol reverse	Math::Sets::Symbol index
Read the symbol: ω, ω₁, ε₀, α + 1, S(α)	ordinals and successors. Explained under: glossary entry: ordinal.	symbol	Math::Sets::Symbol index
Which symbol says: ordinals and successors?	ω, ω₁, ε₀, α + 1, S(α) (explained under: glossary entry: ordinal)	symbol reverse	Math::Sets::Symbol index
Read the symbol: ℕ  ℤ  ℚ  ℝ  ℂ	the number sets. Explained under: glossary entry: number sets n z q r c.	symbol	Math::Sets::Symbol index
Which symbol says: the number sets?	ℕ  ℤ  ℚ  ℝ  ℂ (explained under: glossary entry: number sets n z q r c)	symbol reverse	Math::Sets::Symbol index
Read the symbol: x R y, (x, y) ∈ R, dom R, ran R	a relation and its domain and range. Explained under: glossary entry: relation.	symbol	Math::Sets::Symbol index
Which symbol says: a relation and its domain and range?	x R y, (x, y) ∈ R, dom R, ran R (explained under: glossary entry: relation)	symbol reverse	Math::Sets::Symbol index
Read the symbol: f: A → B, x ↦ f(x), f[A], f⁻¹, g ∘ f	function, image of a set, inverse, composition. Explained under: relations and functions.	symbol	Math::Sets::Symbol index
Which symbol says: function, image of a set, inverse, composition?	f: A → B, x ↦ f(x), f[A], f⁻¹, g ∘ f (explained under: relations and functions)	symbol reverse	Math::Sets::Symbol index
Read the symbol: [x], A/~	equivalence class, quotient set. Explained under: glossary entry: equivalence relation.	symbol	Math::Sets::Symbol index
Which symbol says: equivalence class, quotient set?	[x], A/~ (explained under: glossary entry: equivalence relation)	symbol reverse	Math::Sets::Symbol index
Read the symbol: ≤, <, ≺ on a poset	partial order, strict order. Explained under: glossary entry: partial order.	symbol	Math::Sets::Symbol index
Which symbol says: partial order, strict order?	≤, <, ≺ on a poset (explained under: glossary entry: partial order)	symbol reverse	Math::Sets::Symbol index
Read the symbol: ≈, ~, ≼, |A| ≤ |B|	equinumerous, injects into. Explained under: glossary entry: symbols.	symbol	Math::Sets::Symbol index
Which symbol says: equinumerous, injects into?	≈, ~, ≼, |A| ≤ |B| (explained under: glossary entry: symbols)	symbol reverse	Math::Sets::Symbol index
Read the symbol: κ → (λ)ⁿ_r	partition relation. Explained under: glossary entry: partition relation arrow notation.	symbol	Math::Sets::Symbol index
Which symbol says: partition relation?	κ → (λ)ⁿ_r (explained under: glossary entry: partition relation arrow notation)	symbol reverse	Math::Sets::Symbol index
Read the symbol: ∀  ∃  ∃!  ∄	for all, there exists, exactly one, none. Explained under: glossary entry: quantifier.	symbol	Math::Sets::Symbol index
Which symbol says: for all, there exists, exactly one, none?	∀  ∃  ∃!  ∄ (explained under: glossary entry: quantifier)	symbol reverse	Math::Sets::Symbol index
Read the symbol: ¬  ∧  ∨  ⇒  →  ⇔  ↔  ⇏	not, and, or, implies, iff, does not imply. Explained under: glossary entry: connectives.	symbol	Math::Sets::Symbol index
Which symbol says: not, and, or, implies, iff, does not imply?	¬  ∧  ∨  ⇒  →  ⇔  ↔  ⇏ (explained under: glossary entry: connectives)	symbol reverse	Math::Sets::Symbol index
Read the symbol: ⊢  ⊬  ⊨  ⊣	proves, does not prove, models; the reversed turnstile. Explained under: glossary entry: turnstile.	symbol	Math::Sets::Symbol index
Which symbol says: proves, does not prove, models; the reversed turnstile?	⊢  ⊬  ⊨  ⊣ (explained under: glossary entry: turnstile)	symbol reverse	Math::Sets::Symbol index
Read the symbol: ∴  ∵	therefore, because. Explained under: glossary entry: therefore and because.	symbol	Math::Sets::Symbol index
Which symbol says: therefore, because?	∴  ∵ (explained under: glossary entry: therefore and because)	symbol reverse	Math::Sets::Symbol index
Read the symbol: ∎  ■  □	end of proof. Explained under: glossary entry: end of proof qed.	symbol	Math::Sets::Symbol index
Which symbol says: end of proof?	∎  ■  □ (explained under: glossary entry: end of proof qed)	symbol reverse	Math::Sets::Symbol index
Read the symbol: :=  :⟺  ≝	is defined as (a term, a statement). Explained under: README.md#three-books-three-notations.	symbol	Math::Sets::Symbol index
Which symbol says: is defined as (a term, a statement)?	:=  :⟺  ≝ (explained under: README.md#three-books-three-notations)	symbol reverse	Math::Sets::Symbol index
Read the symbol: ≃	equality inside Cori and Lascar's formal language. Explained under: reading a formula.	symbol	Math::Sets::Symbol index
Which symbol says: equality inside Cori and Lascar's formal language?	≃ (explained under: reading a formula)	symbol reverse	Math::Sets::Symbol index
How is A ∋ a read, and which symbol is it often confused with?	A contains a: the membership sign mirrored, the same fact as a ∈ A from the set's side. Not ∃, which means 'there exists'.	notation trap	Math::Sets::Symbols
What does a ∉ A mean?	a is not a member of A.	notation	Math::Sets::Symbols
Which symbols are used for the universal set?	U, 𝒰, E or Ω, fixed in advance. Not ξ, which a circulating chart invented.	notation	Math::Sets::Symbols
A ⊆ B versus A ⊂ B: what is the trap?	⊆ allows equality in every book. ⊂ means proper subset in Cunningham and this library, but allows equality in Jech and Halmos. Check the book's convention.	trap	Math::Sets::Symbols
What does A ⊄ B say, as a witness?	There is an x ∈ A with x ∉ B.	notation	Math::Sets::Symbols
What is {x ∈ A : P(x)}, and what else is the colon written as?	The set of members of A that satisfy P; the colon is 'such that' and a bar | means the same.	notation	Math::Sets::Symbols
Name four spellings of the complement of A.	A′, Aᶜ, Ā and U − A, all within a universal set U.	notation	Math::Sets::Symbols
Two spellings of the symmetric difference?	A △ B and A ⊕ B: the members of exactly one of the two sets.	notation	Math::Sets::Symbols
Two spellings of the power set of A?	𝒫(A) and 2ᴬ: the set of all subsets of A.	notation	Math::Sets::Symbols
Four spellings of the cardinality of A?	|A|, n(A), #A and card(A).	notation	Math::Sets::Symbols
What are ℵ₀, ℶ₁ and 𝔠?	ℵ₀ = |ℕ|; ℶ₁ = |𝒫(ℕ)| = |ℝ| = 𝔠, the continuum. The continuum hypothesis says ℵ₁ = ℶ₁.	definition	Math::Sets::Symbols
What is ω + 1 as a set?	ω ∪ {ω}: the successor of the first infinite ordinal.	definition	Math::Sets::Symbols
What do ℕ, ℤ, ℚ, ℝ, ℂ stand for, and where does Z come from?	Natural numbers, integers (German Zahlen), rationals (quotients), reals, complex numbers; each is a subset of the next.	notation	Math::Sets::Symbols
What is the difference between ∃x, ∃!x and ∄x?	There exists an x; there is exactly one x; there is no x (the same as ¬∃x).	notation	Math::Sets::Symbols
What do ⊢ and ⊨ mean, and how do they differ?	A ⊢ B: B is provable from A (proofs). A ⊨ B: every model of A satisfies B (truth). Gödel's completeness theorem makes them agree in first-order logic.	definition	Math::Sets::Symbols
Is A ⊣ B the sign for 'does not yield'?	No. ⊬ is 'does not prove'. ⊣ is ⊢ read right to left, or in category theory 'is left adjoint to'; Halbeisen uses it as an end-of-proof mark.	trap	Math::Sets::Symbols
What do ∴ and ∵ mean?	∴ therefore, ∵ because: three dots pointing up or down, shorthand of school proofs.	notation	Math::Sets::Symbols
What does ∎ or □ at the end of a proof mean?	End of proof, Halmos's tombstone, standing for QED, quod erat demonstrandum.	notation	Math::Sets::Symbols
How are := and :⟺ read?	Is defined as: := for a term, :⟺ for a statement. Halbeisen uses both.	notation	Math::Sets::Symbols
What does ⇏ mean?	Does not imply: A ⇏ B says A can hold while B fails.	notation	Math::Sets::Symbols
Which symbol does Cori and Lascar use for equality inside the formal language?	≃, where every other book writes =.	notation	Math::Sets::Symbols
What does κ → (λ)ⁿ_r say?	However the n-element subsets of a set of size κ are split into r pieces, some subset of size λ has all its n-element subsets in one piece. Ramsey's theorem is ω → (ω)ⁿ_r.	definition	Math::Sets::Symbols
What is a set, in the school definition, and what does 'well-defined' add?	A well-defined collection of objects: for any object it is settled whether it belongs. 'The three best students' is not a set; 'the students in the quiz' is.	sets math definition	Math::Sets::Set katas
Roster form versus set-builder form: give both for the vowels.	Roster lists the members: {a, e, i, o, u}. Set-builder states the property: {x : x is a vowel of the English alphabet}.	sets math notation	Math::Sets::Set katas
Can every set be written in roster form?	No. The real numbers between 2 and 5 have no first member and no next member, so they can only be given by a property: {x : x is real and 2 < x < 5}.	sets math trap	Math::Sets::Set katas
Does the order of listing or a repeated member change a set?	No. {1, 2, 3} = {3, 1, 2}, and the set of letters of BANANA is {B, A, N}, three members. A set is only its members.	sets math definition	Math::Sets::Set katas
When is a set finite, and when infinite?	Finite when its members can be counted off by a natural number n(A) = m; otherwise infinite. The digits in the decimal expansion of √2 form a finite set (at most ten), though the expansion never ends.	sets math definition	Math::Sets::Set katas
What is the empty set, and name two sets that are empty in disguise.	The set with no members, written ∅ or { }. Primes that are composite; rational numbers that are irrational; irrationals whose decimals terminate.	sets math definition	Math::Sets::Set katas
Is the set of even primes empty?	No: it is {2}. One member is not none.	sets math trap	Math::Sets::Set katas
Define A ⊆ B and A ⊂ B (proper subset).	A ⊆ B when every member of A is a member of B. A ⊂ B when also A ≠ B. Every set is a subset of itself; ∅ is a subset of every set.	sets math definition	Math::Sets::Set katas
1 ∈ {1}?  {2} ∈ {2}?  {2} ∈ {{2}}?  ∅ ∈ {1, 2, 3}?  ∅ ⊆ {1, 2, 3}?	True, false, true, false, true. ∈ asks whether the thing is a member; ⊆ asks whether each of its members is.	sets math trap	Math::Sets::Set katas
What is the cardinality of {a, {a}}, and of {∅, {∅}, {∅, {∅}}}?	2 and 3. A member that is itself a set counts as one member; braces inside braces do not open.	sets math trap	Math::Sets::Set katas
What is the power set P(A), and how many members does it have?	The set of all subsets of A. If n(A) = p then n(P(A)) = 2^p, and A has 2^p − 1 proper subsets.	sets math definition	Math::Sets::Set katas
P(∅) = ?  P({∅}) = ?	P(∅) = {∅}, one member. P({∅}) = {∅, {∅}}, two members. The power set of the empty set is not empty.	sets math trap	Math::Sets::Set katas
n(A) = 10: how many subsets? n(A) = 100?	2^10 = 1024, and 2^100 ≈ 1.27 × 10^30.	sets math counting	Math::Sets::Set katas
What is the universal set U, and give two examples.	The set everything under discussion is drawn from, fixed in advance: all students of the school when sets of students are discussed; the real numbers when naturals, rationals and irrationals are. The rectangle of a Venn diagram.	sets math definition	Math::Sets::Set katas
Define A ∪ B, A ∩ B and A − B in set-builder form.	A ∪ B = {x : x ∈ A or x ∈ B}; A ∩ B = {x : x ∈ A and x ∈ B}; A − B = {x : x ∈ A and x ∉ B}.	sets math definition	Math::Sets::Set katas
When are A and B disjoint?	When A ∩ B = ∅: no member in common. Two circles drawn apart in a Venn diagram.	sets math definition	Math::Sets::Set katas
Define the complement A′, and say what it needs.	A′ = U − A = {x : x ∈ U and x ∉ A}. It needs a universal set U fixed first; the same A has different complements under different U.	sets math definition	Math::Sets::Set katas
State De Morgan's laws for sets.	(A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′. Not either is neither; not both is at least one missing.	sets math formula	Math::Sets::Set katas
A − B = ?  (as an intersection)	A − B = A ∩ B′: difference is intersection with the complement.	sets math formula	Math::Sets::Set katas
n(A ∪ B) = ?	n(A) + n(B) − n(A ∩ B). The overlap was counted twice and is subtracted once (inclusion–exclusion).	sets math formula	Math::Sets::Set katas
n(A − B) = ?  (only in A)	n(A) − n(A ∩ B).	sets math formula	Math::Sets::Set katas
n(neither in A nor in B) = ?	n(A′ ∩ B′) = n((A ∪ B)′) = n(U) − n(A ∪ B).	sets math formula	Math::Sets::Set katas
n(in exactly one of A, B) = ?	n((A − B) ∪ (B − A)) = n(A) + n(B) − 2 n(A ∩ B).	sets math formula	Math::Sets::Set katas
n(A ∪ B ∪ C) = ?	n(A) + n(B) + n(C) − [n(A ∩ B) + n(B ∩ C) + n(C ∩ A)] + n(A ∩ B ∩ C). The triple overlap was added three times and subtracted three times, so it is added back once.	sets math formula	Math::Sets::Set katas
40 students, 22 play badminton, 11 play both badminton and table tennis, 16 play neither. How many play table tennis but not badminton?	2. n(B ∪ T) = 40 − 16 = 24; n(T) = 24 − 22 + 11 = 13; only table tennis = 13 − 11 = 2.	sets math example	Math::Sets::Set katas
120 students, 92 can teach a language, 46 can teach maths, every student teaches something. How many teach both? Exactly one?	Both: 92 + 46 − 120 = 18. Exactly one: 92 + 46 − 2 · 18 = 102.	sets math example	Math::Sets::Set katas
100 students, 70 like physics, 60 like chemistry. Least possible number liking both? Greatest possible number liking neither?	Both ≥ 70 + 60 − 100 = 30, because n(P ∪ C) ≤ 100. Neither ≤ 100 − 70 = 30, when every chemistry student also likes physics.	sets math example	Math::Sets::Set katas
56 students got certificates in at least one of three activities; 17, 28 and 25 got each; 4 got all three. How many got exactly two?	6. The pair overlaps sum to 17 + 28 + 25 + 4 − 56 = 18; each of the 4 triple students is in three pair counts, so exactly two = 18 − 3 · 4.	sets math example	Math::Sets::Set katas
100 students: 60 English, 50 German, 35 Spanish; 40 E and G, 30 G and S, 25 E and S; 25 all three. At least two languages? At most one? None?	At least two: (40 + 30 + 25) − 2 · 25 = 45. At most one: 55. None: 100 − (145 − 95 + 25) = 25.	sets math example	Math::Sets::Set katas
What does a Venn diagram show, and what are the rectangle and the regions?	Sets as circles inside a rectangle that is U; n circles make 2^n regions, one per pattern of membership, and every set operation is a choice of regions. Four sets cannot be drawn with circles.	sets math notation	Math::Sets::Set katas
Is 'x ∈ A ∩ x ∈ B' a meaningful expression?	No. ∩ joins sets, not statements. Write x ∈ A ∩ B, or 'x ∈ A and x ∈ B'.	sets math trap	Math::Sets::Set katas
Which is the right symbol: {2, 3, 4} … {1, 2, 3, 4, 5}; ∅ … {∅}; {x : x = n²} … {x : x = n³}?	⊂ (proper subset); ⊂, since ∅ is a subset of every set and ∅ ≠ {∅}; ⊄, since 4 = 2² is not a cube.	sets math trap	Math::Sets::Set katas
What is the solution set of an equation or inequality?	The set of every value that makes it true: set-builder notation with the equation as the test. {x : x² = 4} = {−2, 2}; {(x, y) : x² + y² = 1} is the unit circle, so a graph is a solution set drawn.	sets math definition	Math::Sets::Set katas
What is the solution set of x² − x + 1 < 0, and why?	∅, the empty set: the discriminant 1 − 4 is negative and the parabola opens up, so it never goes below the axis. 'No solution' is a set, not a failure.	sets math trap	Math::Sets::Set katas
When are the x-intercepts of a quadratic in the solution set of its inequality?	For ≤ 0 and ≥ 0, since the quadratic is 0 there; never for < 0 and > 0.	sets math trap	Math::Sets::Set katas
In Python, with A = {1, 2, 3}, B = {3, 4, 5}, C = {5, 6, 7}, what is A | B & C, and why?	{1, 2, 3, 5}: & binds tighter than |, so it is A | (B & C) = A | {5}, the same as ∩ before ∪. The other reading, (A | B) & C, would give {5}.	sets math example	Math::Sets::Set katas
In Python, with A = {1, 2, 3, 4}, B = {3, 4, 5}, C = {4, 5, 6}, what is A - B & C, and why?	∅: - binds tighter than &, so it is (A - B) & C = {1, 2} & {4, 5, 6}. Read as A - (B & C) it would be {1, 2, 3}. Python inherits the place of - from C; textbooks give ∖ no place and bracket it.	sets math trap	Math::Sets::Set katas
In Python, with A = {1, 2}, B = {2, 3}, C = {2, 4}, what is A | B ^ A & C, and in what order?	{1, 2, 3}: & first, A & C = {2}; then ^, B ^ {2} = {3}; then |, A | {3}. The ^ of symmetric difference sits between & and |, and only this bracketing of the five gives {1, 2, 3}.	sets math example	Math::Sets::Set katas
In Python, with A = {1, 2}, B = {2, 3}, C = {1, 3}, D = {3, 4}, E = {1, 4}, what is A | B ^ C - D & E, step by step?	{1, 2, 3}: - first, C - D = {1}; then &, {1} & E = {1}; then ^, B ^ {1} = {1, 2, 3}; then |, A | {1, 2, 3}. The ladder is - then & then ^ then |, and only this bracketing of the fourteen gives {1, 2, 3}.	sets math example	Math::Sets::Set katas
What is Python's order for the four set operators, and where do comparisons go?	Tightest first: - then & then ^ then |, left to right within a level; <=, == and in come after all four, so A - B <= C is (A - B) <= C. The ladder is C's, but C puts comparisons before &.	sets math notation	Math::Sets::Set katas
Why does the kata A | B ^ A & C with A = {1, 2}, B = {2, 3}, C = {3, 4} not test where ^ sits?	A & C = ∅ and B ^ ∅ = B, so the ^ step does nothing: 3 of 5 bracketings give {1, 2, 3}, among them ^ above & and ^ below |. With C = {2, 4} only Python's reading gives {1, 2, 3}.	sets math trap	Math::Sets::Set katas
Why can A | B ^ C - A & D never show whether ^ sits below |?	(C - A) & D is disjoint from A, and (A | B) ^ X = A | (B ^ X) whenever A and X are disjoint, on all 65 536 quadruples of subsets of {1, 2, 3, 4}. Replace the second A by a fresh set E.	sets math trap	Math::Sets::Set katas
Which of -, &, ^ and | needs brackets within its own level?	Only -: (A - B) - C differs from A - (B - C) on 2 800 of 4 096 triples of subsets of {1, 2, 3, 4}, while ^, & and | are associative, so A ^ B ^ C is the same set either way.	sets math trap	Math::Sets::Set katas
What does an assert on the final set of a precedence kata prove?	Only that some reading gives that set. A kata tests a step when every rival bracketing gives a different set, so try them all: two for three operands, fourteen for five.	sets math trap	Math::Sets::Set katas
Define a set in four rules.	'Set' and 'member of' are undefined. (1) Extensionality: two sets with the same members are the same set. (2) A set is one object, so it can be a member of another set. (3) Separation: a rule picks the members of a set you already have, {x ∈ A : P(x)}, never {x : P(x)}. (4) There is an empty set, and by (1) only one.	sets math definition	Math::Sets::What is a set
Everything about sets is defined through one relation. Which, and how does the program check that nothing is lost?	Membership, ∈. Equality is 'each member of either is in the other' and subset is 'every member of A is in B'; written in 'in' alone, they agree with Python's == and <= on all 64 pairs of subsets of {0, 1, 2}.	sets math definition	Math::Sets::What is a set
What does a set record, and what does it forget?	It records which objects are its members. It forgets the order they were named in, how many times, and the rule that picked them: {2, 5} = {5, 2}, {1, 1, 2} = {1, 2}, and two different rules can give one set.	sets math axiom	Math::Sets::What is a set
Why is 'a set is a collection of things' not a definition, even though it is a fair description?	'Collection' is a synonym: it names the idea without reducing it to anything simpler. Something has to be the first undefined word, and for sets it is 'set' and 'member of'; the axioms say what they do, not what they are.	sets math trap	Math::Sets::What is a set
Which rule of the definition makes 'THE empty set' a correct phrase, and why?	Extensionality. Two rules that select nothing have the same members, none, so they select the same set. The program finds one distinct empty set among {n in 0..9 : n² < 0} and {w in [set] : len(w) > 5}.	sets math axiom	Math::Sets::What is a set
How many members does {∅} have, and which rule of the definition is at work?	One. A set is a single object, so ∅ can be a member of another set; {∅} is a box holding an empty box, and ∅ ≠ {∅}.	sets math trap	Math::Sets::What is a set
How many levels down does ∈ look?	Exactly one. 1 ∈ {1} and {1} ∈ {{1}}, but 1 ∉ {{1}}: a member of a member is not a member. ⊆ is the relation that chains.	sets math trap	Math::Sets::What is a set
Run Russell's rule under separation on A = {∅, {∅}, {{∅}}}. What is R = {x ∈ A : x ∉ x}, and where is it not?	R is a harmless set (here R = A, since no set is a member of itself), and R ∉ A: if R were a member y of A, 'y ∈ y' would get opposite answers from A's membership and from R's rule. So no set has every set as a member.	sets math theorem	Math::Sets::What is a set
Why does 'a set is one object' matter beyond ∅ versus {∅}?	Because a set can then be a member of a set: functions become sets of pairs, numbers become sets, and Russell's question 'is this set a member of itself?' can be asked. It is what makes set theory a foundation and not a notation.	sets math connection	Math::Sets::What is a set