Let A = {1,2,3,4}, B = {1,5,9,11,15,16} and f = {(1,5), (2,9), (3,1), (4,5), (2,11)}

Are the following true?

(i) f is a relation from A to B

(ii) f is a function from A to B.

Justify your answer in each case.

Given: A = {1,2,3,4}, B = {1,5,9,11,15,16} and f = {(1,5), (2,9), (3,1), (4,5), (2,11)}

Now, A × B = {(1,1), (1,5), (1,9), (1,11), (1,15), (1,16), (2,1), (2,5), (2,9), (2,11), (2,15), (2,16), (3,1), (3,5), (3,9), (3,11), (3,15), (3,16), (4,1), (4,5), (4,9), (4,11), (4,15), (4,16).

(i) f is a relation from A to B.

f = {(1,5), (2,9), (3,1), (4,5), (2,11)}.

A relation from a non empty set A to a non empty set B is a subset of the Cartesian product A × B.

And we can see f is a subset of A × B.

Hence f is a relation from A to B statement is true.

(ii) f is a function from A to B.

f = {(1,5), (2,9), (3,1), (4,5), (2,11)}.

As we observe that same first element i.e. 2 corresponds to two different images that is 9 and 11. Thus f is not a function from A to B.