Question

Are the following set of ordered pair of a function? 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} 

Answer 1

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

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

Therefore, f is the function.

Injection test:

As, y can be mother of two or more persons

Therefore, f is not injective.

Surjection test:

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

So, f is surjective.

Therefore, f is surjective function.

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

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

Therefore, g is not a function.