Given the domain D x y z and the range R 9 10 11 12 which

Given the domain D = {x, y, z} and the range R = {9, 10, 11, 12} which of the following are functions and which are relations?

Then, state which ones are onto.

a) {(x, 9), (y, 11)}                                             c) {(x, 9), (y, 9), (z, 9)}

b) {(y, 9), (x, 11), z, 12),(x, 10)}

Solution

By the definition of function,we know that a function is a rule which maps every element of a set A to a unique element of a set B.

(a) Here, x is mapped to 9, y is mapped to 11 but z is not mapped to any of the elements of the range, so it ruins the definition of function.

Thus, it is not a function but indeed it is a relation.

(b) Here, x is mapped to 9, y is mapped to 9 and z is mapped to 9.

Observe that every element of the domain has a unique image in the range.

So, it is a function.

Further, it is not an onto function as the other elements of the range do not have any pre-image in the domain.

Moreover, it is known that, every function is a relation .

So, the given function is a relation too.

(c) Here, x is mapped to 10 and 11 both, so x does not have a unique image.

Thus, it is not a function but indeed it is a relation.