Subsets and Proper Subsets

A set A is a subset of another set B if every element of A is also in B. If A and B are given sets, we say: A is a subset of B if for every a ∈ A, a ∈ B.

If A is a subset of B, we denote it as A ⊆ B.

Proper Subset

If A ⊆ B, but B has at least one element not in A, then A is a proper subset of B. We denote this as A ⊂ B.

Example: If A = {1,2} and B = {1,2,3}, then A ⊂ B, because 3 is in B but not in A.

Examples

• If A = {apple, pear}, B = {apple, pear, peach}, then A ⊂ B.
• If C = {red, blue}, D = {red, blue}, then C ⊆ D, but not a proper subset.
• The empty set (∅) is a subset of every set.

Important Notes

• Every set is a subset of itself.
• The empty set is a subset of every set.
• For proper subsets, one always has fewer elements than the other.

Practice Exercise