Question

Let X and Y be sets where [X] = 15 and \Y= 10. Let f:X + Y be a function. (a) If f is surjective then what is f(x))?
(b) Is it possible for f to be injective? Be sure to explain your answer.

Answer 1

Answer:

Let $f: X\rightarrow Y$ be a function.

a) If f is surjective then each element of Y has a preimage in X, this means that $f(X)=Y$.

Since Y has less elements that X, then f can not be injective because each element of X must have a image, then by the Pigeonhole principle at least one element of Y has more that one preimage.