set

A set is a collection of distinct objects regarded as a whole. The objects are called elements or members, and an element either belongs to the set or it does not. Sets are typically denoted by capital letters and written with curly braces, for example {1, 2, 3} or {x ∈ ℕ : x > 0}. Repetitions are ignored, so {1, 1, 2} equals {1, 2}.

Two sets are equal when they contain exactly the same elements. The statement that an object a belongs to a set A is written a ∈ A, while a ∉ A means it does not belong. The universe of discourse U is the ambient collection of objects used to define operations such as complement.

Basic operations include union A ∪ B, intersection A ∩ B, and difference A \ B (elements in A not in B). The subset relation is written A ⊆ B; A is a proper subset of B if A ⊆ B and A ≠ B. The power set P(A) is the set of all subsets of A.

Sets can be finite or infinite. The cardinality |A| counts elements; for finite sets it is a nonnegative integer, while infinite sets may be countable or uncountable, with examples ℕ and ℝ. Standard notable sets include the natural numbers ℕ, integers ℤ, rationals ℚ, real numbers ℝ, and complex numbers ℂ.

Foundations: set theory treats sets as primitive objects in axiomatic systems. Georg Cantor initiated modern set theory, introducing ideas of membership and cardinality. Later axiomatizations, such as Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC), formalize the subject and address paradoxes like Russell's paradox.