Problem 0: Complete the Canvas quiz "PSet 8 - Canvas".

Problem (10 points): Draw Venn or Euler diagrams to disprove each of the following.

• For any sets $A, B, C$, if $A \cup C$ = $B \cup C$ then $A = B$.
• For any sets $A, B, C$, if $A \not\subseteq B$ and $B \not\subseteq C$, then $A \not\subseteq C$.

Problem (20 points): Prove each of the following using an element argument.

• For any sets $A$, $B$, and $C$, $(A - C) \cap (B - C) = (A \cap B) - C$.
• For that for any sets $A$, $B$, and $C$, if $A \subseteq B$ and $A \subseteq C$, then $A \subseteq B \cap C$.

Problem (30 points): For each of the following statements, prove that the statement is true or provide a counterexample that shows that it is false.

• If $X$ and $Y$ are any sets, $f:X \rightarrow Y$, and $A, B \subseteq X$, then $f(A \cup B) = f(A) \cup f(B)$.
• If $X$ and $Y$ are any sets, $f:X \rightarrow Y$, and $A, B \subseteq X$, then $f(A \cap B) = f(A) \cap f(B)$.
• If $X, Y, Z$ are any sets and $f:X \rightarrow Y$ and $g:Y \rightarrow Z$ are functions so that $g \circ f$ is one-to-one, then $g$ is one-to-one.
• If $X$ is any set and $f:X \rightarrow X$, $g:X \rightarrow X$, and $h:X \rightarrow X$ are functions so that $h$ is one-to-one and $h \circ f = h \circ g$, then $f = g$.