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 $Z=\{m \in \mathbb{Z} \mid m> 0\}$.