(a)   if $f$ and $g$ are injective then $g\circ f$ is injective,

(b)   if $f$ and $g$ are surjective then $g\circ f$ is surjective, and

(c)   if $f$ and $g$ are bijective then $g\circ f$ is bijective.

Let $f:S\to T$ be a function and let $U\subseteq S$. The image of $U$ under $f$ is the subset of $T$ given by $$f\left(U\right)=\left\{f\right(u\left)\right|u\in U\}.$$

Let $f:S\to T$ be a function. The image of $f$ $U$ under $f$ is the is the subset of $T$ given by $$\mathrm{im}f=\left\{f\right(s\left)\right|s\in S\}.$$ Note that $\mathrm{im}f=f\left(S\right)$.

Let $f:S\to T$ be a function and let $V\subseteq T$. The inverse image of $V$ under $f$ is the subset of $S$ given by $$f^{-1}\left(V\right)=\{s\in S|f\left(s\right)\in V\}.$$

Let $f:S\to T$ be a function and let $t\in T$. The fiber of $f$ over $t$ is the subset of $S$ given by $$f^{-1}\left(t\right)=\{s\in S|f\left(s\right)=t\}.$$

Let $f:S\to T$ be a function. Show that the set $F=\left\{f^{-1}\right(t\left)\right|t\in T\}$ of fibers of the map $f$ is a partition of $S$.

(a)   Let $f:S\to T$ be a function. Define $$\begin{array}{rcl}f\prime :S& ⟶& imf\\ s& ⟼& f\left(s\right)\end{array}$$ Show that the map $f\prime$ is well defined and surjective.

(b)   Let $f:S\to T$ be a function and let $F=\left\{f^{-1}\right(t\left)\right|t\in \mathrm{im}f\}=\{f^{-1}\left(t\right)|t\in T\}-\varnothing$ be the set of nonempty fibers of the map $f$. Define $$\begin{array}{rrcl}\hat{f}:& F& ⟶& T\\ & f^{-1}\left(t\right)& ⟼& t\end{array}$$ Show that the map $\hat{f}$ is well defined and injective.

(c)   Let $f:S\to T$ be a function and let $F=\left\{f^{-1}\right(t\left)\right|t\in \mathrm{im}f\}=\{f^{-1}\left(t\right)|t\in T\}-\varnothing$ be the set of nonempty fibers of the map $f$. Define $$\begin{array}{rrcl}\hat{f}\prime :& F& ⟶& \mathrm{im}T\\ & f^{-1}\left(t\right)& ⟼& t\end{array}$$ Show that the map $\hat{f}\prime$ is well defined and bijective.

Let $S$ be a set. The power set of $S$, $2^S$, is the set of all subsets of $S$.

Let $S$ be a set and let ${\{0,1\}}^S$ be the set of all functions $f:S\to \{0,1\}$. Given a subset $T\subseteq S$ define a function $f_T:S\to \{0,1\}$ by $$f_T\left(s\right)=\{\begin{array}{ll}0,& \text{if}s\notin T,\\ 1,& \text{if}s\in T.\end{array}$$

Show that $$\begin{array}{rcll}\phi :& 2^S& ⟶& {\{0,1\}}^S\\ & T& ⟼& f_T\end{array}\text{is a bijection.}$$

Let $\circ :S\times S\to S$ be an associtaive operation on a set $S$. An identity for $\circ$ is an element $e\in S$ such that if $s\in S$ then $e\circ s=s\circ e=s$.

Let $e$ be an identity for an associative operation $\circ$ on a set $S$. Let $s\in S$. A left inverse for $s$ is an element $t\in S$ such that $t\circ s=e$. A right inverse for $s$ is an element $t\prime \in S$ such that $s\circ t\prime =e$. An inverse for $s$ is an element $s^{-1}\in S$ such that $s^{-1}\circ s=s\circ s^{-1}=e$.

1. Let $\circ$ be an operation on a set $S$. Show that if $S$ contains an identity for $\circ$ then it is unique.
2. Let $e$ be an identity for an associative operation $\circ$ on a set $S$. Let $s\in S$. Show that if $s$ has an inverse then it is unique.

1. Let $S$ and $T$ be sets and let ${\iota }_S$ and ${\iota }_T$ be the identity maps on $S$ and $T$, respectively. Show that for any function $f:S\to T$, $${\iota }_T\circ f=f,\text{and}f\circ {\iota }_S=f.$$
2. Let $f:S\to T$ be a function. Show that if an inverse function to $f$ exists then it is unique. (Hint: The proof is very similar to the proof in Ex. 5b above.)