Proof of the Inclusion-Exclusion Principle

Question

Prove the inlcusion-exclusion principle using the fact that for three pairwise disjoint sets $X$, $Y$, $Z$:

$$|X\cup{Y}\cup{Z|}=|X|+|Y|+|Z|$$

I tried setting $X\cup{Y}=A$, but arrive at $|X\cup{Y|}=|X|+|Y|$ due to the sets being disjoint.

Answer 1

Guide:

• Prove that $A$ and $Z$ are disjoint.

• If you can show that then, we have \begin{align} |X \cup Y \cup Z|&= |A \cup Z|\\ &=|A|+|Z| \end{align}

Answer 2

Your setting $A=X\cup Y$ is a good first step.

Now, use this definition in the original expression. Since $X\cup Y\cup Z = (X\cup Y)\cup Z = A\cup Z$, you now have

$$|X\cup Y\cup Z| = |A\cup Z|$$

From here, you can use the inclusion-exclusion principle to get $$|A\cup Z|=|A|+|Z| - |A\cap Z|$$

Now:

1. Write out what $A$ is
2. Think about what $|A\cap Z|$ is (think about disjointedness)