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\begin{center}\begin{large}\textbf{Math 101 - Problem Set 3\\
Due Tuesday, Sept 26}\end{large}\end{center}



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\begin{enumerate} 

\item Let $f: A \to B$ and $A_0 \subseteq A$ and $B_0 \subseteq B$.

\begin{enumerate}
\item Show that if $f$ is surjective, then $f(f^{-1}(B_0)) = B_0.$
\item Show that if $f$ is injective, then $f^{-1}(f(A_0)) = A_0.$
\end{enumerate}

\item Let $f:A \to B$ and $A_0, A_1 \subseteq A$. Prove the following.

\begin{enumerate}
\item $f(A_0 \cup A_1) = f(A_0) \cup f(A_1)$.
\item $f(A_0 \cap A_1) \subseteq f(A_0) \cap f(A_1)$.
\item If $f$ is injective, equality holds in part (b).
\end{enumerate}

\item Let $f:A \to B$ and $g:B \to C$ be injective functions. Show that $g \circ f$ is injective. 

\item Let $f: X \to Y$ be a surjective function. Define the equivalence relation $\sim$ on $X$ as follows: $$a \sim b \iff f(a) = f(b).$$ Recall the function $g: Y \to X/\sim$ we defined in class as $$g(y)= \{x \in X \mid f(x) = y\}.$$ Show $g$ is injective (in a different way than we proved it in class) by assuming $g(y_1) = g(y_2)$ and concluding $y_1=y_2.$

\item Show that $h:X \to Y$ is surjective if and only if $h^{-1}(Y_0) \neq \emptyset$ for all nonempty $Y_0 \subseteq Y$.

\item Denote the identity function for a set $C$ by $i_C$. Given $f:A \to B$, a function $g:B \to A$ is a \textit{\textbf{left inverse}} for $f$ if $g \circ f = i _A$ and $h:B \to A$ is a \textit{\textbf{right inverse}} for $f$ if $f \circ h = i_B$.

\begin{enumerate}
\item Show that if $f$ has a left inverse, $f$ is injective.
\item Show that if $f$ has a right inverse, $f$ is surjective.
\end{enumerate}

\item 
\begin{enumerate}
\item Give an example of a function that has a left inverse but no right inverse.
\item Give an example of a function that has a right inverse but no left inverse.
\item Is it possible for a function to have more than one left inverse? More than one right inverse? Either give examples or prove that it's not possible.
\end{enumerate}

\item Show that if $f$ has both a left inverse $g$ and a right inverse $h$ then $f$ is bijective and $g=h=f^{-1}$.

\item Define the relation $\sim$ on  $\mathbb{R}^2$  by $$(a,b) \sim (c,d) \iff a-b^2 = c-d^2.$$ Show that this is an equivalence relation and describe the equivalence classes.

\item Let $\mathcal {L}$ be the collection of all straight lines in the plane (i.e. $\mathbb{R}^2$). Find a subset of $\mathcal {L}$ that is a partition of $\mathbb{R}^2$ and describe (mathematically) the corresponding equivalence relation.

\end{enumerate}
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