Operations on Sets

Counting two overlapping groups

When two groups $A$ and $B$ overlap, simply adding their sizes double-counts whoever is in both. The inclusion-exclusion principle corrects for that:

|A \cup B| = |A| + |B| - |A \cap B|

Worked example

In a class of 60 students, 25 play cricket and 20 play tennis, and 10 play both. How many play neither?

Solution: First find how many play at least one sport:

|\text{cricket} \cup \text{tennis}| = 25 + 20 - 10 = 35

Out of 60 students total, the rest play neither:

60 - 35 = 25