What is the difference between a subset and a proper subset in set theory?

Subset (denoted A ⊆ B): every element of A is also in B. Includes the case where A = B. Every set is a subset of itself: A ⊆ A. Proper subset (denoted A ⊂ B): every element of A is in B, AND A is not equal to B (B has at least one element not in A). A set is NOT a proper subset of itself. Example: Let B = {1, 2, 3}. Subsets include: {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}. All 8 are subsets of B. Proper subsets: all except {1,2,3} itself = 7 proper subsets. Total subsets of a set with n elements = 2^n. Proper subsets = 2^n - 1. Empty set {} is a subset of every set.