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#deck:Math::Sets::Symbols
How is A β a read, and which symbol is it often confused with? A contains a: the membership sign mirrored, the same fact as a β A from the set's side. Not β, which means 'there exists'. notation trap
What does a β A mean? a is not a member of A. notation
Which symbols are used for the universal set? U, π°, E or Ξ©, fixed in advance. Not ΞΎ, which a circulating chart invented. notation
A β B versus A β B: what is the trap? β allows equality in every book. β means proper subset in Cunningham and this library, but allows equality in Jech and Halmos. Check the book's convention. trap
What does A β B say, as a witness? There is an x β A with x β B. notation
What is {x β 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
Name four spellings of the complement of A. Aβ², AαΆ, Δ and U β A, all within a universal set U. notation
Two spellings of the symmetric difference? A β³ B and A β B: the members of exactly one of the two sets. notation
Two spellings of the power set of A? π«(A) and 2ᴬ: the set of all subsets of A. notation
Four spellings of the cardinality of A? |A|, n(A), #A and card(A). notation
What are β΅β, βΆβ and π ? β΅β = |β|; βΆβ = |π«(β)| = |β| = π , the continuum. The continuum hypothesis says β΅β = βΆβ. definition
What is Ο + 1 as a set? Ο βͺ {Ο}: the successor of the first infinite ordinal. definition
What do β, β€, β, β, β 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
What is the difference between βx, β!x and βx? There exists an x; there is exactly one x; there is no x (the same as Β¬βx). notation
What do β’ and β¨ mean, and how do they differ? A β’ B: B is provable from A (proofs). A β¨ B: every model of A satisfies B (truth). GΓΆdel's completeness theorem makes them agree in first-order logic. definition
Is A β£ B the sign for 'does not yield'? No. β¬ is 'does not prove'. β£ is β’ read right to left, or in category theory 'is left adjoint to'; Halbeisen uses it as an end-of-proof mark. trap
What do β΄ and β΅ mean? β΄ therefore, β΅ because: three dots pointing up or down, shorthand of school proofs. notation
What does β or β‘ at the end of a proof mean? End of proof, Halmos's tombstone, standing for QED, quod erat demonstrandum. notation
How are := and :βΊ read? Is defined as: := for a term, :βΊ for a statement. Halbeisen uses both. notation
What does β mean? Does not imply: A β B says A can hold while B fails. notation
Which symbol does Cori and Lascar use for equality inside the formal language? β, where every other book writes =. notation
What does ΞΊ β (Ξ»)βΏ_r say? However the n-element subsets of a set of size ΞΊ are split into r pieces, some subset of size Ξ» has all its n-element subsets in one piece. Ramsey's theorem is Ο β (Ο)βΏ_r. definition