Question

Examine each of the following relations given below and state in each case, giving reasons whether it is a function or not?
$\left(i\right)R=\left\{\right(2,1),(3,1),(4,2\left)\right\}$
$\left(ii\right)R=\{2,2),(2,4),(3,3),(4,4)\}$
$\left(iii\right)R=\{1,2),(2,3),(3,4),(4,5),(5,6),(6,7)\}$

Answer 1

$\left(i\right)$ Checking whether $R$ is a fuction or not
$R=\left\{\right(2,1),(3,1),(4,2\left)\right\}$
Since every element of domain has one and only one image
Hence, this relation is a function.

$\left(ii\right)$ Checking whether $R$ is a fuction or not
$R=\{2,2),(2,4),(3,3),(4,4)\}$
Since, the same first elements $2$ corresponds to two different images $2$ and $4.$ Hence, this relation is not a function.

$\left(iii\right)$ Checking whether relation $R$ is a function or not
$R=\left\{\right(1,2),(2,3),(3,4),(4,5),(5,6),(6,7\left)\right\}$
Since every element of domain has one and only one image
Hence, this relation is a function.