# Thread: Proving the Cardinality of 2 finite sets

1. ## Proving the Cardinality of 2 finite sets

Hi all! This is my first post, and my partner and I have struggled to try to understand the following optional/suggested problem:

Let A and B be finite sets. Prove that |A - B| = |A u B| - |B|.
In words: "Prove that the absolute value (or cardinality) of A minus B is equal to the absolute value of the union of A and B, minus the absolute value of B.
On top of this we also are asked: "What does this tell us about |A|, |A - B| + |A n B| (n being the intersection of A and B).

I have tried to set each side of the equation be equal to something, such as letting L =
|A - B| and R = |A u B| - |B|, and showing that some arbitrary value (x) is in both L and R. For example we have: let L = |A - B| and R = |A u B| - |B|. If XeB (if x is an element of B), it is still true that XeAnB (X is an element of the intersection of A and B), because XeA.

However, I am stuck in how this pertains to the absolute value/cardinality and how to prove this. Any help/hints would be greatly appreciated!

(Also, sorry for the weird change of font).

2. ## Re: Proving the Cardinality of 2 finite sets

$A \cup B =(A \cap B^c) \cup B$

$B \cap (A \cap B^c) = \emptyset$

$A-B = A \cap B^c$

so

$|A \cup B| = |A-B| + |B|$

$|A-B| = |A \cup B| - |B|$