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)}.

We have,

X = {1, 2, 3} and Y = {4, 5}

∴ X × Y = {(1, 4),(1, 5),(2, 4),(2, 5),(3, 4),(3, 5)}

(i) f = {(1, 4), (1, 5), (2, 4), (3, 5)}

Now, f(1) = 4 and f(1) = 5

We observe that one element of domain maps to two distinct values.

i.e., ‘1’ has no unique image.

Thus, f is not a function.

(ii) g = {(1, 4), (2, 4), (3, 4)}

Now, g(1) = 4, g(2) = 4, g(3) = 4

We observe that each distinct element of domain has unique image.

Thus, g is a function.

(iii) h = {(1,4), (2, 5), (3, 5)}

Now, h(1) = 4, h(2) = 5, h(3) = 5

We observe that each distinct element of domain has unique image.

Thus, h is a function.

(iv) k = {(1,4), (2, 5)}.

Now, k(1) = 4 and k(2) = 5

We observe that ‘3’ does not have any image under the mapping.

Thus, k is not a function.