Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective:
Question:

Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective:

(i) $\{(x, y): x$ is a person, $y$ is the mother of $x\}$

(ii) $\{(a, b): a$ is a person, $b$ is an ancestor of $a\}$

[NCERT EXEMPLAR]

Solution:

(i) f = {(xy) : x is a person, y is the mother of x}

As, for each element x in domain set, there is a unique related element y in co-domain set.

So, f is the function.

Injection test:

As, y can be mother of two or more persons

So, f is not injective.

Surjection test:

For every mother y defined by (xy), there exists a person x for whom y is mother.

So, f is surjective.

Therefore, f is surjective function.

(ii) g = {(ab) : a is a person, b is an ancestor of a}

Since, the ordered map $(a, b)$ does not map ‘ $a$ ‘ – a person to a living person.

So, $g$ is not a function.