Problem
Homework 1 Math 105 - Summer 2023 (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 $A,\lceil\}$ one has $(A \backslash B) \cup B=A$ if and only if $B \subseteq A$. (2) Let $f: X \rightarrow Y$ be a function. If $C$ is a subset of $X$ we write \[ f(C):=\{f(x) \mid x \in C\} \] In other words, $f(C)$ is the image of the set $C$ under the function $f$. (a) Prove that $f(A \cap B) \subseteq f(A) \cap f(B)$ for all $A, B \subseteq X$. (b) Give an example of a function $f: X \rightarrow Y$ and subsets $A, B \subseteq X$ for which the containment in part (a) is proper. (3) Let $f: X \rightarrow Y$ be a function. If $Z \subseteq Y$ we define the inverse image of $Z$ under $f$ to be the set \[ f^{-1}(Z):=\{x \in X \mid f(x) \in Z\} . \] (a) Let $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be the function defined by $f(n)=n^{2}-2 n+1$. Find $f^{-1}(Z)$ if $Z=\{m \in \mathbb{Z} \mid m>0\}$.
Solution
Step 1 ：Let's consider an element x. If x is in A ∪ (B ∪ C), then x is in A or x is in (B ∪ C). If x is in (B ∪ C), then x is in B or x is in C. Therefore, if x is in A ∪ (B ∪ C), then x is in A or x is in B or x is in C.
Step 2 ：Similarly, if x is in (A ∪ B) ∪ C, then x is in (A ∪ B) or x is in C. If x is in (A ∪ B), then x is in A or x is in B. Therefore, if x is in (A ∪ B) ∪ C, then x is in A or x is in B or x is in C.
Step 3 ：Since the two conditions are the same, we can conclude that A ∪ (B ∪ C) = (A ∪ B) ∪ C.
Step 4 ：\(\boxed{A \cup(B \cup C)=(A \cup B) \cup C}\) is true for any sets A, B, and C. This is known as the associative law of union.
