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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.	definition
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}.	notation
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}.	trap
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.	definition
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.	definition
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.	definition
Is the set of even primes empty?	No: it is {2}. One member is not none.	trap
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.	definition
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.	trap
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.	trap
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.	definition
P(∅) = ?  P({∅}) = ?	P(∅) = {∅}, one member. P({∅}) = {∅, {∅}}, two members. The power set of the empty set is not empty.	trap
n(A) = 10: how many subsets? n(A) = 100?	2^10 = 1024, and 2^100 ≈ 1.27 × 10^30.	counting
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.	definition
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}.	definition
When are A and B disjoint?	When A ∩ B = ∅: no member in common. Two circles drawn apart in a Venn diagram.	definition
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.	definition
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.	formula
A − B = ?  (as an intersection)	A − B = A ∩ B′: difference is intersection with the complement.	formula
n(A ∪ B) = ?	n(A) + n(B) − n(A ∩ B). The overlap was counted twice and is subtracted once (inclusion–exclusion).	formula
n(A − B) = ?  (only in A)	n(A) − n(A ∩ B).	formula
n(neither in A nor in B) = ?	n(A′ ∩ B′) = n((A ∪ B)′) = n(U) − n(A ∪ B).	formula
n(in exactly one of A, B) = ?	n((A − B) ∪ (B − A)) = n(A) + n(B) − 2 n(A ∩ B).	formula
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.	formula
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.	example
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.	example
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.	example
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.	example
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.	example
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.	notation
Is 'x ∈ A ∩ x ∈ B' a meaningful expression?	No. ∩ joins sets, not statements. Write x ∈ A ∩ B, or 'x ∈ A and x ∈ B'.	trap
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.	trap
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.	definition
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.	trap
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.	trap
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}.	example
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.	trap
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}.	example
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}.	example
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 &.	notation
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}.	trap
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.	trap
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.	trap
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.	trap