Question

Let $f$ be the subset of $Z\times Z$ defined by $f=\left\{\right(ab,a+b):a,b\in Z\}$. Is $f$ a function from $Z$ to $Z$ : justify your answer

Answer 1

The relation $f$ is defined as
$f=\left\{\right(ab,a+b):a,b\in Z\}$
We know that a relation $f$ from set A to set B is said to be a function if every element of set A has unique images in set B.
Since $2,6\in Z$,

$\Rightarrow (2\times 6,2+6)\in f$

Again since, $-2,-6\in Z$,

$\Rightarrow (-2\times -6,-2+(-6))\in f$

i.e., $(12,8),(12,-8)\in f$
It can be seen that the same first element i.e., $12$ corresponds to two different images i.e., $8$ and $-8.$
Thus relation $f$ is not a function