Question

If X= { 1, 2, 3, 4, 5 }, Y= { 1, 3, 5, 7, 9 } determine which of the following relations from X to Y are functions? Give reason for your answer. If it is a function, state its type.
(i) $R_1$={(x, y)| y=x+2, x $ϵ$ X, y $ϵ$ Y}
(ii) $R_2$ = { (1, 1), (2, 1), (3, 3), (4, 3), (5, 5) }
(iii) $R_3$ = { (1, 1), (1, 3), (3, 5), (3, 7), (5, 7) }
(iv) $R_4$ = { (1, 3), (2, 5), (4, 7), (5, 9), (3, 1) }

Answer 1

$\left(i\right)$ Every element of $X$ does not have an image in $Y$ thus $R_1$ is not a function.

$\left(ii\right)R_2$ is a function. Since two inputs have same output $\therefore$ it is many-one.

In addition $domain\ne codomain$ therefore it is not onto.

$\left(iii\right)$ Every element of $X$ does not have an image in $Y$ thus $R_3$ is not a function.

$\left(iv\right)R_4$ is a function. Since $domain=codomain$ it is onto.

Also no two elements has same image $\therefore$ it is one-one.