Let $A,B$ be finite sets. Prove that $|A\cup B|=|A|+|B|-|A\cap B|$ by establishing a bijection from $A\cup B$ to $\{1,2,\ldots,|A|+|B|-|A\cap B|\}$.

Question

Let $A,B$ be finite sets. Prove that $|A\cup B|=|A|+|B|-|A\cap B|$ by establishing a bijection from $A\cup B$ to $\{1,2,\ldots,|A|+|B|-|A\cap B|\}$.

First you prove for disjoint finite sets A,B ( i.e. A∩B=∅) that |A∪B|=|A|+|B| by coming up with a bijection N→A∪B. Then you use the following:

A∪B=(A∖B)∪(B∖A)∪(A∩B) is a disjoint (!) decomposition of $A\cup B$.

A=(A∖B)∪(A∩B) is a disjoint (!) decomposition of A

B=(B∖A)∩(B∩A) is a disjoint (!) decomposition of B

With the first part you can argue with equating |A∪B|=... I'm not sure how I would continue the proof. If I could get some pointers that would be helpful. Thanks