Inclusion-Exclusion (Subtraction) Principle

The Inclusion-Exclusion Principle helps count elements correctly when sets overlap.

For two sets the following is always true, \( |A \cup B| = |A| + |B| - |A \cap B| \).

Examples:

• If $30$ students are taking CISC 2210, $25$ are taking CISC 3115, and $10$ are taking both, then $30 + 25 - 10 = 45$ students take either of these $2$ classes.
• If $15$ people like pizza, $12$ like burgers, and $5$ like both, then $15 + 12 - 5 = 22$ people like pizza or burgers.

We subtract the overlap because it was counted twice.

With three or more sets, more complicated versions of Inclusion-Exclusion are used.