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\pagestyle{fancy} \lhead{{\sf Homework 12 $|$ Due April 14 (Tuesday)
}} \rhead{\thepage} \cfoot{{\sf Math 3190 $|$ Introduction to Proof $|$
    Spring 2026 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-4mm}


\begin{exercise} % HW 12, Problem 1
    Let $S\subseteq G$ be a nonempty subset of a group. Define $S^{-1}=\{s^{-1}\mid s\in S\}$. Define the \emph{subgroup generated by $S$} to be
    \[
    \<S\>=\big\{s_1\cdots s_k\mid s_i\in S\cup S^{-1}\big\}.
    \]

\begin{enumerate}[label=(\alph*)]
    \item Prove that $\<S\>$ is actually a subgroup of $G$.
    \item Prove that
    \[
    \<S\>=\bigcap_{S\subseteq H_\alpha\leq G} H_\alpha. 
    \]
    That is, the subgroup generated by $S$ is the intersection of all subgroups of $G$ that contain $S$.
\end{enumerate}
\end{exercise}

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

\smallskip

%%------------------------------------------------------------------------
\begin{exercise} % HW 12, Problem 2
Let $f\colon X\to Y$ be a function, and $A,B\subseteq X$. Prove the following identities.
\begin{enumerate}[label=(\alph*)]
    \item $f(A\cap B)\subseteq f(A)\cap f(B)$
    \item $f(A\cup B)=f(A)\cup f(B)$
    \item $f(A\setminus B)\subseteq f(A)\setminus f(B)$.
\end{enumerate}
\end{exercise}

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

\smallskip

%%------------------------------------------------------------------------
\begin{exercise} % HW 12, Problem 3
Prove that the following functions are bijective. 
\begin{enumerate}[label=(\alph*)]
\item $f\colon\R\to\R$, where $f(x)=x^3$
\item $f\colon\R\to(0,\infty)$, where $f(x)=e^x$
\item $f\colon(0,\infty)\to(0,1)$, where $f(x)=\frac{1}{x+1}$.
\end{enumerate}
\end{exercise}

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

\smallskip

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

\begin{exercise} % HW 12, Problem 4
Let $f:A\to B$ and $g:B\to C$ be functions.
\begin{enumerate}[label=(\alph*)]
\item Show that if $g\circ f$ is surjective then $g$ is surjective.
\item Show that if $g\circ f$ is injective then $f$ is injective.
\item Give an example in which $g\circ f$ is surjective, but $f$ is not surjective.
\item Give an example in which $g\circ f$ is injective, but $g$ is not injective.
\end{enumerate}
\end{exercise}

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

\smallskip

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

\begin{exercise} % HW 12, Problem 5
Recall that the \emph{nullspace} of a linear map $f\colon V\to W$ between vector spaces is the set of elements that get sent to the zero vector. Similarly, the \emph{kernel} of a homomorphism $\phi\colon G\to H$ between groups is the set of elements that get sent to the identity element:
\[
\NS(f)=\big\{v\in V:f(v)=\0\},\qquad \ker(\phi)=\big\{g\in G:\phi(g)=e_H\}.
\]
\begin{enumerate}[label=(\alph*)]
\item Prove that a linear map $f\colon V\to W$ is injective if and only if $\NS(f)=\{0\}$.
\item Prove that a group homomorphism $f\colon G\to H$ is injective if and only if $\ker(f)=\{e_H\}$.
\end{enumerate}
\end{exercise}

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

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

\end{document}