Subsets and Equality

Set Theory · Axiom Academy

Learn how sets can contain other sets and when two sets are considered equal. Set A is a subset of set B (written A ⊆ B) if every element of A is also an element of B. A is a proper subset of B (written A ⊂ B) if A ⊆ B but A ≠ B. This means B has at least one element not in A. Two sets are equal if and only if each is a subset of the other: A = B ⟺ (A ⊆ B) ∧ (B ⊆ A). 4 The Empty Set is a Subset of Everything A remarkable fact: ∅ ⊆ A for any set A. The empty set is contained in every set! Let's explore when the subset relation holds and when it doesn't.

This is the written version of the interactive lesson above. See the full Set Theory course.