set

A set is a collection of objects, called its elements or members, regarded as a single whole. The only thing that matters about a set is which objects belong to it: order is irrelevant and repetition counts for nothing, so {1, 2, 3}, {3, 1, 2}, and {1, 1, 2, 3} all denote the same set. We write x ∈ A for “x is an element of A” and x ∉ A for “x is not”.

Sets are specified either by listing, {2, 4, 6}, or by a defining property, {x ∈ R : x > 0}, read “the set of real x such that x > 0”. Two sets are equal precisely when they have the same elements — this principle of extensionality is the foundation of set proofs: to show A = B you show each is a subset of the other. The empty set, written ∅, has no elements at all and is a subset of every set.

Sets are the universal language of modern analysis: domains, ranges, intervals, neighborhoods, open and closed sets, and sigma-algebras are all sets, and operations like union, intersection, and complement let us combine them. A genuine caveat: naive “set of all sets satisfying P” constructions can be inconsistent (Russell's paradox), which is why rigorous mathematics rests on axiomatic set theory; for ordinary analysis, however, the naive picture causes no trouble.