Moderate# Properties of Composition of Functions

ABSALG-LM9MJY

Let $S$, $T$, $U$ be three (non-empty) sets.

Let $f:S\rightarrow T$ be a function with domain $S$ and codomain $T$, and let $g:T\rightarrow U$ a function with domain $T$ and codomain $U$.

The *composition of $g$ with $f$*, denoted $g\circ f$, is a third function $h=g\circ f:S\rightarrow U$ with domain $S$ and codomain $U$, defined by $h(s)=(g\circ f)(s)=g(f(s))$, and also often called the *composite function* (when $f$ and $g$ are understood).

That is, we **first** apply $f$ to $s$ to yield $f(s)$, and **then** apply $g$ to $t=f(s)$ to get $g(t)=g(f(s))$.

Notice that it is **not** necessary for $f$ to have image all of $T$ for $g\circ f$ to be defined.

You may know the saying

Vedi Napoli e poi muori!,

meaning

See Naples and die!.

The intention is not that Naples is so bad it will kill you, but that it's so beautiful you would not want to be anywhere else for the rest of your life. Oh, and yes, this usually refers to Naples, Italy, not Naples, Florida (also a nice place, though). For people who love Venice, Italy, this saying is often recycled as *See Venice and die!*

Consider the set $S=\{$ sees NI, sees NF, sees VI, sees VF, sees VLOS, sees VLOU$\}$ where $NI$ is Naples, Italy; $NF$ is Naples, Florida; $VI$ is Venice, Italy; $VF$ is Venice, Florida; $VLOS$ is Venice, Los Angeles; and $VLOU$ is Venice, Louisiana.

Let $M=\{$ marries in NI, marries in VI$\}$, and

$W=\{$writes a diary, writes a book, writes a movie$\}$

$D=\{$ dies in NF, dies in VF, dies in VLOS, dies in VLOU $\}$.

For any two of the above sets, $A$ and $B$, and any function $f:A\rightarrow B$, let $f(a)=b$ mean in words *$a$ then $b$*, where *then* implies **$a$ precedes $b$ in time**.

For example, if $f:S\rightarrow T$ has:

$$f({\rm sees\,NF})={\rm marries \, in \,NI}$$

this means *sees Naples, Florida*, **then** *marries in Naples, Italy*.

Which of the following is **TRUE**?