## Functions

 Let $S$, $T$ and $U$ be sets and let $f:S\to T$ and $g:T\to U$ be functions. Show that (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 $f(U)= {f(u) | u∈U}.$ Let $f:S\to T$ be a function. The image of $f$ $U$ under $f$ is the is the subset of $T$ given by $imf= {f(s) | s∈S}.$ 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 $f-1(V) = {s∈S | f(s)∈V}.$ 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 $f-1(t) = {s∈S | f(s)=t}.$ Let $f:S\to T$ be a function. Show that the set $F=\left\{{f}^{-1}\left(t\right)\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}t\in T\right\}$ of fibers of the map $f$ is a partition of $S$. (a)   Let $f:S\to T$ be a function. Define $f′:S ⟶ imf s ⟼ f(s)$ Show that the map $f\prime$ is well defined and surjective. (b)   Let $f:S\to T$ be a function and let $F=\left\{{f}^{-1}\left(t\right)\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}t\in \mathrm{im}f\right\}=\left\{{f}^{-1}\left(t\right)\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}t\in T\right\}-\varnothing$ be the set of nonempty fibers of the map $f$. Define $f^: F ⟶ T f-1(t) ⟼ t$ Show that the map $\stackrel{^}{f}$ is well defined and injective. (c)   Let $f:S\to T$ be a function and let $F=\left\{{f}^{-1}\left(t\right)\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}t\in \mathrm{im}f\right\}=\left\{{f}^{-1}\left(t\right)\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}t\in T\right\}-\varnothing$ be the set of nonempty fibers of the map $f$. Define $f^′: F ⟶ imT f-1(t) ⟼ t$ Show that the map $\stackrel{^}{f}\prime$ 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 ${\left\{0,1\right\}}^{S}$ be the set of all functions $f:S\to \left\{0,1\right\}$. Given a subset $T\subseteq S$ define a function ${f}_{T}:S\to \left\{0,1\right\}$ by $fT(s) = { 0, ifs∉T, 1, ifs∈T.$ Show that $φ: 2S ⟶ {0,1}S T ⟼ fT is a bijection.$ Let $\circ :S×S\to S$ 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 ${s}^{-1}\circ s=s\circ {s}^{-1}=e$. Let $\circ$ be an operation on a set $S$. Show that if $S$ contains an identity for $\circ$ then it is unique. 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. 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 $f:S\to T$, $ιT∘f=f, and f∘ιS=f.$ 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.)

## Notes and References

These notes are written to highlight the analogy between groups and group actions, rings and modules, and fields and vector spaces.

## References

[Ram] A. Ram, Notes in abstract algebra, University of Wisconsin, Madison 1993-1994.

[Bou] N. Bourbaki, Algèbre, Chapitre 9: Formes sesquilinéaires et formes quadratiques, Actualités Sci. Ind. no. 1272 Hermann, Paris, 1959, 211 pp. MR0107661.

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.