?

Abstract Algebra

Free Version

Upgrade subject to access all content

Easy

Map, Relation, Correspondence, Function

ABSALG-GTNKP4

Let $S$ and $T$ be two (non-empty) sets.

Informally, a function from $S$ to $T$ is a rule that associates to each element of $S$ just one element of $T$. The set $S$ is often called the domain of the function. A function $f$ from $S$ to $T$ is often written $f:S\rightarrow T$ and the notation $f(s)=t$ means $f$ assigns the element $t\in T$ to the element $s\in S$. For this, we also write $f:s\mapsto t$. Sometimes we call a function a map or mapping, although the word map or mapping may also mean a correspondence or relation, described below.

Another definition of a function is more like associating to a function its graph. We define a function $f$ from $S$ to $T$ as a subset $R$ of the Cartesian product $S\times T$ such that $(s,t_1)$ and $(s,t_2)$ are in $R$ if and only if $t_1=t_2$, so $t=t_1=t_2$ is the value $f(s)=t$.

There is a generalization of a function that is important both in abstract mathematics and in its applications. This is the notion of correspondence or relation​. A correspondence or relation $\gamma$ between $S$ and $T$ is a subset $R$ of $S\times T$, with no further restriction. The domain of $\gamma$ is, therefore, the set of $a\in S$ for which there exists a $b\in T$ with $(a,b)\in R$.

$S=\{1, i\}$ and $T=\{e^{i\theta}\mid -\pi<\theta\le \pi\}$ be the unit circle (here $i=\sqrt{-1}$).

Which of the following subsets $R$ of $S\times T$ defines a function from $S$ to $T$ with domain $S$? ​

Select ALL that apply.

A

$R=\{(s,t)\mid s\in S\;{\rm and}\; t\in T\; {\rm with} \; t^4=s\}$

B

$R=\{(s,t)\mid s\in S\;{\rm and}\; t\in T\; {\rm with} \; t=s^4\}$

C

$R=\{(s,t)\mid s\in S\;{\rm and}\; t\in T\; {\rm with} \; t^3=s^4, \;{\rm and}\; (-\sqrt{-1})(2t+1)>0\}$

D

$R=\{(s,t)\mid s\in S\;{\rm and}\; t\in T\; {\rm with} \; t=\sqrt{2}/(1-s)\}$

E

$R=\{(s,t)\mid s\in S\;{\rm and}\; t\in T\; {\rm with} \; |t|=|s|\}$