If A = {1,2,3,4,5}, B = {1,2,3,4,5}, and R = {(x,y): x + y is odd}, why is R not a function?

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

For a function from A to B, each input x in A must be paired with exactly one output y in B. The relation includes a pair whenever x + y is odd. Take x = 1. Since 1 + 2 = 3 is odd and 1 + 4 = 5 is also odd, both (1,2) and (1,4) belong to R. Thus the same input 1 has at least two images, 2 and 4, which violates the uniqueness condition. In fact, every odd x pairs with all even elements of B, and every even x pairs with all odd elements, so the relation generally has multiple outputs for each input. Option B is false because x = 2 has odd images such as 1, 3 and 5. Options C and D do not affect the function definition. Therefore option A gives the correct reason.