Sets

A = B, A and B are equal sets, A and B are equivalent sets

A = {1, 2, 3, 4, 5}. The set B contains all natural numbers less than 6, so B = {1, 2, 3, 4, 5}. The two sets have identical elements, so they are equal. Equal sets always have the same number of elements, so they are also equivalent.

{2, 5, 6, 8, 11}

First find B ∩ C. The common elements are {6, 8}. Now take the union with A: {2, 5, 8, 11} ∪ {6, 8} = {2, 5, 6, 8, 11}.

{2, 5, 6, 8, 11}

A ∪ B = {2, 4, 5, 6, 8, 11} and A ∪ C = {2, 5, 6, 7, 8, 11}. Their intersection is the common elements {2, 5, 6, 8, 11}.

8 subsets, 7 proper subsets

For a set with n elements, number of subsets = 2^n and number of proper subsets = 2^n - 1. Here n = 3, so subsets = 2^3 = 8 and proper subsets = 8 - 1 = 7.