Let X = {1, 2, 3} and Y = {4, 5}. Find whether the following subsets of X × Y are functions from X to Y or not.
(i) f = {(1, 4), (1, 5), (2, 4), (3, 5)} (ii) g = {(1, 4), (2, 4), (3, 4)}
(iii) h = {(1,4), (2, 5), (3, 5)} (iv) k = {(1,4), (2, 5)}.
Given, X = {1, 2, 3} and Y = {4, 5}
So, X × Y = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)}
(i) f = {(1, 4), (1, 5), (2, 4), (3, 5)}
f is not a function as f(1) = 4 and f(1) = 5
Hence, pre-image ‘1’ has not unique image.
(ii) g = {(1, 4), (2, 4), (3, 4)}
It’s seen clearly that g is a function in which each element of the domain has unique image.
(iii) h = {(1,4), (2, 5), (3, 5)}
It’s seen clearly that h is a function as each pre-image with a unique image.
And, function h is many-one as h(2) = h(3) = 5
(iv) k = {(1, 4),(2, 5)}
Function k is not a function as ‘3’ has not any image under the mapping.