Question

Let $A=\{1,2,3,4,5,6\}.$ Define a relation $R$ from $A$ to $A$ by $R=\left\{\right(x,y):y=x+1\}$

$\left(i\right)$ Depict this relation using an arrow diagram
$\left(ii\right)$ Write down the domain, codomain, and range of $R$.

Answer 1

$\left(i\right)$ Given:
$A=\{1,2,3,4,5,6\},R$ is a relation from $A$ to $A$
$R=\left\{\right(x,y):y=x+1\}$

Make relation from $A$ to $A$
Put $x=1,2,3,4,5,$ we get
$y=2,3,4,5,6$
$\therefore R=\left\{\right(1,2),(2,3),(3,4),(4,5),(5,6\left)\right\}$

Drawing diagram
The relation using an arrow diagram is$\left(ii\right)$ Given:
$A=\{1,2,3,4,5,6\},R$ is a relation from $A$ to $A$
$R=\left\{\right(x,y):y=x+1\}$

Make relation from $A$ to $A$
Put $x=1,2,3,4,5,$ we get
$y=2,3,4,5,6$
$\therefore R=\left\{\right(1,2),(2,3),(3,4),(4,5),(5,6\left)\right\}$

Finding domain of $R$
$\because R=\left\{\right(1,2),(2,3),(3,4),(4,5),(5,6\left)\right\}$
The domain $R$ is the set of all first elements of ordered pairs in relation $R$.
Hence, domain of $R=\{1,2,3,4,5\}$

Finding Co-domain of $R$
The whole set $A$ is the co-domain of $R$
Hence, co-domain of $R=A=\{1,2,3,4,5,6\}$

Finding range of $R$
The range of $R$ is the set of second elements of ordered pairs in relation $R$.
$R=\left\{\right(1,2),(2,3),(3,4),(4,5),(5,6\left)\right\}$

Hence, Range of $R$ is $\{2,3,4,5,6\}$