# Section 5.1

Remember the definitions: Definition 5.3.1 Let $A$ and $B$ be non-empty sets, let $R$ be a relation from $A$ to $B$, and let $x\in A$. The relation class of $x$ with respect to $R$, denoted $R[x]$, is the set defined by $R[x]=\{y\in B|xRy\}$. Definition 2.2.1 Let $a$ and $b$ be integers. The number $a$ divides the number $b$ if there is some integer $q$ such that $aq=b$. If $a$ divides $b$, we write $a|b$, and we say that $a$ is a factor of $b$, and that $b$ is divisible by $a$.

## 5.1.1

* Let $aSb\Leftrightarrow a=|b|$, for all $a,b\in \mathbb {Z}$. Then $S[3]=\{b\in \mathbb {Z} :3Sb\}$, but by the definition of the relation $S$ is $S[3]=\{b\in \mathbb {Z} :3=|b|\}$ and the only elements that satisfy this property are $3$ and $-3$, since $3=|3|=|-3|$ and therefore $S[3]=\{-3,3\}$. Analogously, we have to: $S[-3]=\{b\in \mathbb {Z} :-3Sb\}=\{b\in \mathbb {Z} :-3=|b|\}=\emptyset$. $S[6]=\{b\in \mathbb {Z} :6Sb\}=\{b\in \mathbb {Z} :6=|b|\}=\{-6,6\}$.
* Let $aDb\Leftrightarrow a|b$, for all $a,b\in \mathbb {Z}$. Then $D[3]=\{b\in \mathbb {Z} :3Db\}=\{b\in \mathbb {Z} :\exists k\in \mathbb {Z} {\text{ and }}b=3k\}=\{...,-6,-3,-1,0,1,3,6,...\}$. $D[-3]=\{b\in \mathbb {Z} :-3Db\}=\{b\in \mathbb {Z} :\exists k\in \mathbb {Z} {\text{ and }}b=-3k\}=\{...,-6,-3,-1,0,1,3,6,...\}$. $D[6]=\{b\in \mathbb {Z} :6Db\}=\{b\in \mathbb {Z} :\exists k\in \mathbb {Z} {\text{ and }}b=6k\}=\{...,-12,-6,0,6,12,...\}$.
* Let $aTb\Leftrightarrow b|a$, for all $a,b\in \mathbb {Z}$. Then $T[3]=\{b\in \mathbb {Z} :3Tb\}=\{b\in \mathbb {Z} :b|3\}=\{b\in \mathbb {Z} :\exists k\in \mathbb {Z} {\text{ and }}3=bk\}=\{-3,-1,1,3\}$. $T[-3]=\{b\in \mathbb {Z} :-3Tb\}=\{b\in \mathbb {Z} :b|-3\}=\{b\in \mathbb {Z} :\exists k\in \mathbb {Z} {\text{ and }}-3=bk\}=\{-3,-1,1,3\}$. $T[6]=\{b\in \mathbb {Z} :6Tb\}=\{b\in \mathbb {Z} :b|6\}=\{b\in \mathbb {Z} :\exists k\in \mathbb {Z} {\text{ and }}6=bk\}=\{-6,-3,2,-1,1,2,3,6\}$.
* Let $aQb\Leftrightarrow a+b=7$, for all $a,b\in \mathbb {Z}$. Then $Q[3]=\{b\in \mathbb {Z} :3Qb\}=\{b\in \mathbb {Z} :3+b=7\}=\{4\}$. $Q[-3]=\{b\in \mathbb {Z} :-3Qb\}=\{b\in \mathbb {Z} :-3+b=7\}=\{10\}$. $Q[6]=\{b\in \mathbb {Z} :6Qb\}=\{b\in \mathbb {Z} :6+b=7\}=\{1\}$.

## 5.1.2

* Let $S$ be the relation defined by $(x,y)S(z,w)\Leftrightarrow y=3w{\text{ for all }}(x,y),(z,w)\in \mathbb {R} ^{2}$. $S[(0,0)]=\{(z,w)\in \mathbb {R} ^{2}:(0,0)S(z,w)\}=\{(z,0)\}$. Because $y=3w\Rightarrow 0=3w$, and therefore $w=0$. The geometric description of the relation class are: the $x$-axis. $S[(3,4)]=\{(z,w)\in \mathbb {R} ^{2}:(3,4)S(z,w)\}=\{(z,{\dfrac {4}{3}})\}$. Because $y=3w\Rightarrow 4=3w$, and therefore $w={\dfrac {4}{3}}$. The geometric description of the relation class are: the the line whose equation is $y={\dfrac {4}{3}}$.
* Let $T$ be the relation defined by $(x,y)T(z,w)\Leftrightarrow x^{2}+3y^{2}=7z^{2}+w^{2}{\text{, for all }}(x,y),(z,w)\in \mathbb {R} ^{2}$. $T[(0,0)]=\{(z,w)\in \mathbb {R} ^{2}:(0,0)T(z,w)\}=(0,0).Because0^{2}+3\cdot 0^{2}=7z^{2}+w^{2}\Rightarrow Z=0andw=o$. $T[(3,4)]=\{(z,w)\in \mathbb {R} ^{2}:(3,4)T(z,w)\}=\{(z,{\sqrt {57-7z^{2}}})\}$. Because $3^{2}+3\cdot 4^{2}=7z^{2}+w^{2}\Rightarrow w={\sqrt {57-7z^{2}}}$. The geometric description of the relation class are the graph of $y={\sqrt {57-7z^{2}}}$.
* Let $Z$ be the relation defined by $(x,y)T(z,w)\Leftrightarrow x=z\vee y=w{\text{, for all }}(x,y),(z,w)\in \mathbb {R} ^{2}$. $Z[(0,0)]=\{(z,w)\in \mathbb {R} ^{2}:(0,0)Z(z,w)\}=\{(0,w)\vee (z,0)\}$. $Z[(3,4)]=\{(z,w)\in \mathbb {R} ^{2}:(3,4)Z(z,w)\}=\{(3,w)\vee (z,4)\}$.

## 5.1.3

Let $A=\{1,2,3\}$. Each of the following subsets of $A\times A$ defines a relation on $A$. Is each relation reflexive, symmetric and/or transitive?
* $M=\{(3,3),(2,2),(1,2),(2,1)\}$. is symmetric only
* $N=\{(1,1),(2,2),(3,31),(1,2)\}$. is reflexive only