Answer
Use a Venn diagram
Work Step by Step
We have to prove that
$$\begin{align*}
|A\cup B\cup C|&=|A|+|B|+|C|-|A\cap B|-|B\cap C|\\
&-|C\cap A|+|A\cap B\cap C|.
\end{align*}$$
We will use a Venn diagram in which we draw each of the three sets $A$, $B$ and $C$. For a better understanding we note each of the subsets in the drawing by the letters $D,E,F,G,H,I,J$ as follows:
$$\begin{align*}
A\cap B&=G\cup J\\
B\cup C&=I\cup J\\
C\cap A&=H\cup J\\
A\cap B\cap C&=J.
\end{align*}$$
The number of elements of $A\cup B\cup C$ is:
$$|A\cup B\cup C|=|D|+|E|+|F|+|G|+|H|+|I|+|J|.$$
Now we calculate the number of elements of the sets on the right side:
$$\begin{align*}
&|A|+|B|+|C|-|A\cap B|-|B\cap C|-|C\cap A|+|A\cap B\cap C|\\
&=(|D|+|G|+|H|+|J|)+(|E|+|G|+|I|+|J|)\\
&+(|F|+|I|+|H|+|J|)-(|G|+|J|)-(|I|+|J|)-(|H|+|J|)\\
&+|J|\\
&=|D|+|G|+|H|+|J|+|E|+|G|+|I|+|J|\\
&+|F|+|I|+|H|+|J|-|G|-|J|-|I|-|J|-|H|-|J|\\
&+|J|\\
&=|D|+|E|+|F|+|G|+|H|+|I|+|J|.
\end{align*}$$
Since we got the same expression in both cases it means the identity if proved.
