College level, not hard but need specific steps.
#1. The distinction between âŠ† and âˆˆ. Recall that for sets A and B, A âŠ† B means that A is a subset of B. Recall also that x âˆˆ C means that x is an element of C. Also, y /âˆˆ D means that y is not an element of D. Let âˆ… denote the empty set. (
1a) We have already proved that âˆ… âŠ† A for any set A. However, âˆ… âˆˆ âˆ… / ; explain why.
(1b) Explain why âˆ… âˆˆ P(X) for any set X, where P(X) is the power set of X.
(1c) Let U be the universal set. Suppose that A and B are sets. Explain why A âˆ’ B âˆˆ P(A), where A âˆ’ B is the set difference of A and B. (1d) It is possible for x, A, and B to be such that x âˆˆ A and A âˆˆ B. Given an example. (1e) It is possible that both A âˆˆ B and A âŠ† B. Given an example.