Question:
Let $f$ be the subset of $Z \times Z$ defined by $f=\{(a b, a+b): a, b \in Z\}$. Is $f$ a function from $Z$ to $Z$ : justify your answer.
Solution:
The relation $f$ is defined as $f=\{(a b, a+b): a, b \in Z\}$
We know that a relation f from a set A to a set B is said to be a function if every element of set A has unique images in set B.
Since $2,6,-2,-6 \in Z(2 \times 6,2+6),(-2 \times-6,-2+(-6)) \in f$
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.