The relation R={(x,y)∈A×B:y=x+1}, where A={1,2,3} and B={2,3,4,5}, is of which type?

Correct answer: A function from A to B

To determine whether R is a function from A to B, every element of the domain A must be associated with exactly one element of B. Apply y=x+1: for x=1, y=2; for x=2, y=3; and for x=3, y=4. All three outputs belong to B={2,3,4,5}, and each input in A has exactly one output. Thus R={(1,2),(2,3),(3,4)} is a function from A to B. It is not empty, so option D is false. It is not option A because no input has missing or multiple outputs. The relation need not be described only in the reverse direction; although some elements of B may have no preimage, that does not prevent it from being a function from A to B. Hence option B is correct.