How can I prove that $|A| + |B| = |A\cup B| + |A\cap B|$?

Question

If $A$ and $B$ are sets then $|A| + |B| = |A\cup B| + |A\cap B|$.

Answer 1

Hint: for disjoint sets $A$ and $B$, $|A \cup B| = |A| + |B|$.

Now, $A \cup B = A \cup ( B \setminus A )$ (notice that the two are disjoint), and $B = ( B \setminus A ) \cup ( B \cap A )$.

Answer 2

Hint: try to prove that $|A\cup B|-|A|=|B|-|A\cap B|$.

What is the relation between the sets $(A\cup B)\setminus A$ and $B\setminus(A\cap B)$?

Answer 3

You can also view this formula as $|A \cup B| = |A| + |B| - |A \cap B|$ which might make the reasoning for this clearer. Basically, the cardinality of the union is the sum of the cardinality of each set, but elements in both sets (the cardinality of the intersection) are double-counted. So subtracting $|A \cap B|$ gives you the actual cardinality of the union.