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) 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)}

Answer 1

(i) Given y = x + 2

When x = 1 ; y = 1 + 2 = 3

When x = 2 ; y = 2 + 2 = 4

When x = 3 ; y = 3 + 2 = 5

When x = 4 ; y = 4 + 2 = 6

When x = 5 ; y = 5 + 2 = 7

R₁ = {1,3), (2,4), (3, 5), (4, 6), (5,7)}

R₁ is not a function ; 2 and 4 has no image in Y.

(ii) R₂ is a function.

Every element of X has unique image in Y.

1 and 2 have same image 1

3 and 4 have same image 3

It is not one – one function … (1)

7 and 9 has no pre image in X

It is not an onto function … (2)

From (1) and (2) we know that, it is neither one – one nor onto function.

(iii) R₃ is not a function.

1 has two images 1 and 3

3 has two images 5 and 7

(iv) Every element of X has unique image in Y. 

Range = Co-domain

R is a function.

It is an one-one and onto function (or) bijective
function