Set X has x members and set Y has y members. Set Z consists of all members that are in ?

Question

Set X has x members and set Y has y members. Set Z consists of all members that are in either set X or set Y with the exception of the k common members (k is greater than 0). Represent the number of members in set Z.

The answer is x+y-2k, but I want to know how to get to this answer.

Answer 1

x + y counts the k elements that are in both X and Y each twice.  But Z excludes all of the elements that are in both X and Y, and excludes no other elements, so x + y - 2k counts Z.

Answer 2

If X and Y have k common elements, X has x elements, and Y has y elements, then X has x-k elements that are not in Y and Y has y-k elements that are not in X. But Z consists of exactly these two groups of elements, so Z has (x-k) + (y-k) = x + y - 2k elements.