1. Prove the following identities for union and intersection. Use words to explain your argument (rather than only using symbols like $\forall, \exists, \Leftrightarrow$, etc.). (a) For any sets $A, B, C, A \cup(B \cup C)=(A \cup B) \cup C$. (b) For any sets $A, B, C, A \cap(B \backslash C)=(A \cap B) \backslash(A \cap C)$. (c) For any sets $A, B, C,(A \backslash C) \cap(B \backslash C)=(A \cap B) \backslash C$. (d) For any sets $A, B,(A \backslash B) \cup B=A$ if and only if $B \subseteq A$.

Step 1 ：The question is asking to prove some identities related to set theory. These identities are about union, intersection and difference of sets. We can prove these identities by showing that for any element, it belongs to the left hand side if and only if it belongs to the right hand side.

Step 2 ：Let's start with the first identity: For any sets \(A, B, C, A \cup(B \cup C)=(A \cup B) \cup C\).

Step 3 ：An element belongs to the left hand side \(A \cup(B \cup C)\) if it belongs to A or if it belongs to the union of B and C. Similarly, an element belongs to the right hand side \((A \cup B) \cup C\) if it belongs to the union of A and B or if it belongs to C.

Step 4 ：Since the union operation is associative, these two conditions are equivalent for any element and any sets A, B, and C. Therefore, the identity is proven.

Step 5 ：\(\boxed{\text{The identity } A \cup(B \cup C)=(A \cup B) \cup C \text{ is proven to be true for all sets A, B, and C.}}\)