Suppose you have two infinite sets A and B, and you are told that there exists a pairing in which each...

in which each element from A is associated with exactly one element from B and no two elements of A are paired up with the same element of B. In this pairing, however, there are elements of B that are not paired up with elements of A. Does this imply that the cardinality of A and the cardinality of B are not equal? What id the two sets were finite, does this change the answer? Explain.
I have no idea what to do. Thank you!

The answer to your first question is, No, not necessarily. The two sets may have the same cardinality, or B may have a larger cardinality than A. Example 1, suppose both A and B are the set of positive integers, and f is the function f(n) = 2n. Clearly, each element of A is associated with exactly one element of B, and no two elements of A are paired with the same element of B. But none of the odd integers in B are paired, but clearly A and B have the same cardinality. Example 2, suppose A is again the set of positive integers, and B is the set of all real numbers; let f(n) = n for each n in A. Again each element in A is uniquely paired with one element in B, and cno two elements in A are paired with the same number in B. But, for example no element of A is paired with 1/2, and clearly the cardinality of A is smaller than that of B.

If both A and B are finite, then B must necessarily have a larger cardinality. For example, Let A be
{1,2,3,4,5}, B = {1,2,3,4,5,6} and f(n) = n for each n in A.