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\pagestyle{fancy} \lhead{{\sf Homework 13 $|$ Due April 21 (Tuesday)
}} \rhead{\thepage} \cfoot{{\sf Math 3190 $|$ Introduction to Proof $|$
    Spring 2026 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-4mm}

 %%------------------------------------------------------------------------
\begin{exercise} % HW 13, Problem 1

Let $f\colon X\to Y$ be a function, and $C,D\subseteq Y$. Prove the following identities.
\begin{multicols}{2}
\begin{enumerate}[label=(\alph*)]
    \item $f^{-1}(C\cap D)=f^{-1}(C)\cap f^{-1}(D)$
    \item $f^{-1}(C\cup D)=f^{-1}(C)\cup f^{-1}(D)$
    \item $f^{-1}(C\setminus D)=f^{-1}(C)\setminus f^{-1}(D)$
    \item $f^{-1}(C^c)=(f^{-1}(C))^c$.
\end{enumerate}
\end{multicols}
\end{exercise}

%\begin{solution} 
    % To do. Recall that you need to prove both $\Rightarrow$ and $\Leftarrow$ inmplications separately. 
%\end{solution}

\smallskip

%%------------------------------------------------------------------------

\begin{exercise} % HW 13, Problem 2
Let $f\colon X\to Y$ be a function, and $A\subseteq X$ and $C\subseteq Y$. Prove the following.
\begin{enumerate}[label=(\alph*)]
    \item $f(f^{-1}(C))\subseteq C$, with equality iff $C\subseteq\Image(f)$.
    \item $A\subseteq f^{-1}(f(A))$, with equality iff $f$ is injective.
\end{enumerate}
\end{exercise}

%\begin{solution} 
    % To do. Use the definition; this is very short.
%\end{solution}

\smallskip

%%------------------------------------------------------------------------
\begin{exercise} % HW 13, Problem 3
Let $V$ and $W$ be vector spaces, and $f\colon V\to W$ a linear function. 
\begin{enumerate}[label=(\alph*)]
    \item Prove that if $U$ is a subspace of $V$, then $f(U)$ is a subspace of $W$.
    \item Prove that if $Y$ is a subspace of $W$, then $f^{-1}(Y)$ is a subspace of $V$.
\end{enumerate}
Recall that to prove a subset is a subspace, it suffices to show that it contains the zero vector $\bm{0}$, and is closed under addition and scalar multiplication.
\end{exercise}

%\begin{solution} 
    % To do. You just need to show that $\NS(f)$ is closed under addition and scalar multiplication.
%\end{solution}

\smallskip

%%------------------------------------------------------------------------

%%------------------------------------------------------------------------

\begin{exercise} % HW 13, Problem 4
Let $R$ be the relation on $\R$ defined by 
\[
a\sim b\quad\Longleftrightarrow\quad |a|=|b|.
\]
Prove that this is an equivalence relation, and describe the resulting equivalence classes.
\end{exercise}


%\begin{solution} 
    % To do. You just need to show that $\NS(f)$ is closed under addition and scalar multiplication.
%\end{solution}

\smallskip

%%------------------------------------------------------------------------


\begin{exercise} % HW 13, Problem 5
For each of the following relations on $\R^2$, prove that it is an equivalence relation, and then describe the resulting equivalence classes as precisely as you can.
\begin{enumerate}[label=(\alph*)]
\item The relation $R$, defined by 
    \[
    (x_1,y_1)\sim (x_2,y_2)\quad\Longleftrightarrow\quad y_1=y_2.
    \]
\item The relation $R$, defined by
\[
(x_1,y_1)\sim (x_2,y_2)\quad\Longleftrightarrow\quad x_1^2+y_1^2=x_2^2+y_2^2.
\]
\end{enumerate}

\end{exercise}

%\begin{solution} 
    % To do.
%\end{solution}

\smallskip

%%------------------------------------------------------------------------

\begin{exercise} % HW 13, Problem 6
 Let $X=\mathcal{C}^1(\R)$, the differentiable real-valued functions. Define a relation $\sim$ on $X$ where $f(x)\sim g(x)$ if and only if $f(x)$ and $g(x)$ differ by a constant. 
\begin{enumerate}[label=(\alph*)]
    \item Prove that $\sim$ is an equivalence relation. 
    \item Show that $f(x)$ and $g(x)$ are in the same equivalence class if and only if $f'(x)=g'(x)$.
\end{enumerate}
\end{exercise}

%%------------------------------------------------------------------------

\end{document}