mathematical reasoning hw only one question

College level, not hard but need specific steps.

#1. The distinction between ⊆ and ∈. Recall that for sets A and B, A ⊆ B means that A is a subset of B. Recall also that x ∈ C means that x is an element of C. Also, y /∈ D means that y is not an element of D. Let ∅ denote the empty set. (

1a) We have already proved that ∅ ⊆ A for any set A. However, ∅ ∈ ∅ / ; explain why.

(1b) Explain why ∅ ∈ 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 − B ∈ P(A), where A − B is the set difference of A and B. (1d) It is possible for x, A, and B to be such that x ∈ A and A ∈ B. Given an example. (1e) It is possible that both A ∈ B and A ⊆ B. Given an example.