If A = {1, 2, 3, 4}, B = {1, 2, 3, 4}, and R = {(x, y): x − y is odd}, why is R not a function?

Correct answer: Because x = 1 has images 2 and 4

The governing definition says that a function from A to B must assign exactly one element of B to every element of A. In this relation, (x,y) is included whenever x−y is odd. Consider x=1: y=2 gives 1−2=−1, which is odd, and y=4 gives 1−4=−3, which is also odd. Therefore x=1 has at least two different images, 2 and 4. This violates the uniqueness requirement, so R is not a function and option A is correct. Option B is false because x=2 is related to y=1 and y=3. Option C is false because B is not empty, while option D is irrelevant: equal domain and codomain sets do not guarantee that a relation is a function.