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) R₁ = {(x, y)| y = x + 2, x ∈ X , y ∈ Y }

(ii) R₂ = {(1, 1), (2, 1), (3, 3), (4, 3), (5, 5)}

(iii) R₃ = {(1, 1), (1, 3), (3, 5), (3, 7), (5, 7)}

(iv) R₄ = {(1, 3), (2, 5), (4, 7), (5, 9), (3, 1)}

Here we are given

X = {1, 2, 3, 4, 5}, Y = { 1, 3, 5, 7, 9 }

(i) R₁ = {(x, y)| y = x + 2, x ∈ X , y ∈ Y }

So using the value we can write the function in set form

R₁ = {(1,3),(3,5),(5,7)}

From the above form it’s clear that as the elements of X doesn’t have a unique image in Y so it’s not a function.

(ii) R₂ = {(1, 1), (2, 1), (3, 3), (4, 3), (5, 5)}

Here R₂ is a function as all the element of X has a distinct image in Y. As 1 and 2 of X is related to 1 of Y and 3, 4 of X is related to 3 of Y so this function is onto function.

(iii) R₃ = {(1, 1), (1, 3), (3, 5), (3, 7), (5, 7)}

Here in this relation 1 of X is related to 1 and 3 of Y which clearly contradict the definition of a function which says that every elements of a domain should have at most one image. So it’s clear that it’s not a function.

(iv) R₄ = {(1, 3), (2, 5), (4, 7), (5, 9), (3, 1)}

The above expression is a onto and one-one function as all the elements of X has a unique image in Y and no two elements of X have same image in Y. these functions are also termed as bijective function.