Set Theory glossary

Clear, one-line definitions of the Set Theory terms used across the OgbonLab textbooks. Each entry links to the interactive sections where the idea is taught.

33 terms

axiom of choice: Every collection of nonempty sets admits a choice function picking one element from each; independent of ZF and equivalent to Zorn's Lemma.; See: The Axiom of Choice and Beyond, The Axiom of Choice and Its Equivalents
banach-tarski paradox: Using the Axiom of Choice, a solid ball in ℝ³ can be partitioned into finitely many pieces and reassembled into two balls of the original size.
cantor diagonal: Cantor's argument that ℝ is uncountable: assume a countable enumeration exists, then construct a real differing from the n-th listed real in the n-th decimal digit.
cantor-schröder-bernstein: If injections A → B and B → A both exist, then a bijection A → B exists; equivalently, |A| = |B|.; See: The Cantor-Schröder-Bernstein Theorem
cardinality: A measure of the 'size' of a set; two sets have the same cardinality iff there exists a bijection between them.
cartesian product: A × B = {(a, b) : a ∈ A, b ∈ B}, all ordered pairs.; See: Ordered Pairs and Cartesian Products
choice function: Given a collection of nonempty sets, a function selecting one element from each set; its existence is guaranteed by the Axiom of Choice.
closure (of a set): The smallest set containing the original and closed under a given operation, such as the transitive closure of a relation.
complement of a set: The set of elements in the universe not belonging to A; written A^c, A', or A̅.
continuum hypothesis: The statement that no set has cardinality strictly between ℵ₀ and 2^ℵ₀; independent of ZFC by Gödel and Cohen.
countable: A set is countable when its cardinality equals |ℕ| (or it is finite); examples include ℤ, ℚ, and any infinite subset of ℕ.; See: Countable and Uncountable Sets
disjoint sets: Sets with no common elements: A ∩ B = ∅.
equinumerous: Two sets are equinumerous when a bijection exists between them; written A ~ B.; See: Equinumerous Sets
image of a set: Given f: A → B and S ⊆ A, the image f(S) = {f(x) : x ∈ S}.
indexed family: A collection of sets {A_i : i ∈ I} indexed by elements of an index set I.
indexed intersection: ⋂_{i ∈ I} A_i, the set of elements belonging to every A_i in the family.
indexed union: ⋃_{i ∈ I} A_i, the set of elements belonging to at least one A_i in the family.
intersection of sets: A ∩ B = {x : x ∈ A and x ∈ B}; elements common to both sets.
ordered pair: (a, b), a pair where order matters; (a, b) ≠ (b, a) unless a = b.; See: Ordered Pairs and Cartesian Products
ordinal: A transitive set well-ordered by ∈; ordinals generalize the natural numbers and index well-orderings up to isomorphism.
partition: A collection of nonempty, pairwise disjoint subsets whose union is the whole set.
poset: A partially ordered set, a set together with a reflexive, antisymmetric, and transitive order relation.
power set: 𝒫(A), the set of all subsets of A; for finite A, |𝒫(A)| = 2^|A|.
proper class: A collection too large to be a set, such as the class of all sets or all ordinals; cannot be a member of another class in ZFC.
roster notation: Listing the elements of a set explicitly between braces, such as {2, 3, 5, 7}.
set difference: A B = {x : x ∈ A and x ∉ B}; elements in A but not in B.
set-builder notation: Expressing a set as {x : P(x)}, read 'the set of all x such that P(x) holds.'
uncountable: A set whose cardinality exceeds |ℕ|; no bijection with ℕ exists. ℝ and 𝒫(ℕ) are the canonical examples.; See: Countable and Uncountable Sets
union of sets: A ∪ B = {x : x ∈ A or x ∈ B}; elements in either set (or both).
universal set: The reference set U from which all elements under consideration are drawn; complement is taken with respect to U.
well-ordered: A totally ordered set in which every nonempty subset has a least element; ℕ is well-ordered but ℤ and ℚ are not.
well-ordering theorem: Every set admits a well-ordering in which every nonempty subset has a least element; equivalent to the Axiom of Choice.
zorn's lemma: Every nonempty partially ordered set in which every chain has an upper bound contains a maximal element; equivalent to the Axiom of Choice.