A set specifies membership without counting duplicate listings.
Key knowledge
A set is a well-defined collection of distinct elements. Two sets are equal when they have the same members. Their union contains members of either set, and their intersection contains members of both.
Conditions and scope
Specify the universe when taking complements.
This is elementary set language, not unrestricted permission to form a set from every possible predicate.
Sources · 3
References
Sources retain their recorded scope. A source link alone does not establish verification of every statement.
2026-10-10 · Checked definitions; no full axiomatic set theory is claimed.
Foundational review
2026-10-10 · Fresh primary-source inspection of the complete compact record, including every statement, equation, quantity and material condition, followed by independent AI editorial review. This check does not certify all linked lessons or every passage of every cited publication.
Sources from other places
Scope or convention differs
Distinct membership, set equality, union and intersection.
Elementary set language does not grant unrestricted comprehension.
Australian Mathematical Sciences Institute and University of Melbourne
A function f:A→B assigns exactly one element of B to each element of A. A is the domain, B the codomain, and the range consists of outputs actually attained.
A probability space consists of an outcome set Ω, a sigma-algebra of events and a probability P. Probabilities are nonnegative, P(Ω)=1, and the probability of a countable union of pairwise disjoint events equals the sum of their probabilities.
ZFC combines extensionality, empty-set, pairing, union, power-set, infinity, foundation and choice axioms with separation and replacement schemas. Separation selects elements from an already given set.
A metric d on X is nonnegative, vanishes exactly when its arguments coincide, is symmetric and satisfies the triangle inequality. Open balls contain points at distance less than a positive radius.
On a measure space, the integral of a nonnegative measurable f is the supremum of integrals of nonnegative simple functions below f. A real measurable f is integrable when ∫|f|dμ<∞; then ∫f=∫f⁺−∫f⁻.
A sigma-algebra contains the whole space and is closed under complements and countable unions. A measure assigns nonnegative extended values, gives the empty set zero and is countably additive on disjoint measurable sets; sigma-finiteness means a countable cover by finite-measure sets.
A DFA consists of finite states Q, finite alphabet Σ, total transition δ:Q×Σ→Q, start q₀ and accepting states F. It reads a finite string once, accepts when the final state lies in F and recognizes a regular language.
Regular expressions built from letters, the empty word and empty language using union, concatenation and star describe exactly the DFA/NFA-recognizable languages. Subset construction replaces an NFA's possible active states by DFA states.
A context-free grammar has finite variables, terminal alphabet, rules A→w with one variable on the left, and a start variable. Its language contains terminal strings derived from the start; S→0S1|ε generates {0ⁿ1ⁿ:n≥0}.
An n-element set has 2ⁿ subsets; its k-element subsets correspond to length-n binary strings with k ones. Their number is C(n,k), with C(n,0)=C(n,n)=1. For n≥2 and 1≤k≤n−1, Pascal’s recurrence is C(n,k)=C(n−1,k−1)+C(n−1,k). C(n,k) is also the coefficient of xᵏyⁿ⁻ᵏ in (x+y)ⁿ.
Disjoint finite alternatives have total count equal to the sum of their counts. Sequential choices with exactly nᵢ options at each stage for every earlier permitted choice have total count ∏nᵢ; in particular |A×B|=|A||B|.