How do I calculate the number of elements in set Z using set theory?

[courtrigrad]

Hello all

Set X has x elements and Sset Y has y elements and Set Z consists of all elements are are in either set X or set Y with the exception of the k common elements. I know that the answer is

x+ y - 2k, however how would I get this? Should I just use a practical example?

 

[matt grime]

No, proof by example isn't a proof.

X sym diff Y = (X \ Y)u(Y\X)

the union is of disjoint sets so the card of the lhs is the sum of the cards on the rhs. now just find card (X\Y) hint X\Y = X\(XnY)

 

[HallsofIvy]

You are told that there are k elements in both X and Y. That means there are x-k elements in x that are NOT in y and y- k elements that are in Y but NOT in X. Z will contain (x- k)+ (y- k)= x+ y- 2k elements.