subset

A set A is a subset of a set B, written A ⊆ B, when every element of A is also an element of B. Picture B as a region and A as a smaller region sitting entirely inside it. Crucially the two may coincide: every set is a subset of itself, and the empty set is a subset of absolutely everything (vacuously, since it has no element that could fail to be in B).

Spelled out as an implication, A ⊆ B means “for all x, if x ∈ A then x ∈ B”. This is why the standard recipe for proving a subset relation is: take an arbitrary x ∈ A, and show x ∈ B. And it gives the universal method for set equality: A = B if and only if both A ⊆ B and B ⊆ A, the celebrated double-inclusion argument.

When A ⊆ B but A ≠ B we call A a proper subset, sometimes written A ⊊ B. A warning about notation: some authors use ⊂ for proper subset and others use it loosely to mean ⊆, so read each text's conventions. The collection of all subsets of B is itself a set, the power set, and it is always strictly larger than B — a fact that ultimately produces an unending hierarchy of infinities.