Math Core

When two groups overlap, adding their sizes counts the overlap twice. Venn diagrams make the overlap visible, and the inclusion-exclusion formula fixes the double count. Survey problems ("how many students play both?") and "divisible by $3$ or $5$ " problems both live here.

Two sets

Picture two overlapping circles, one for each group. The lens in the middle is the set of things in both groups.

Two overlapping sets A (left) and B (right) inside a rectangle for everyone. The middle lens is A and B. Outside both circles is neither.

If you add $|A|$ and $|B|$ , everyone in the lens is counted twice. Subtract it once.

Inclusion-exclusion for two sets

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

and

$\text{neither} = \text{total} - |A \cup B|.$

Here $|A \cup B|$ is the number in $A$ or $B$ (or both) and $|A \cap B|$ is the number in both.

Worked example: A survey

In a class of $30$ students, $18$ play soccer, $15$ play basketball and $5$ play neither. How many play both?

The number who play at least one sport is $30 - 5 = 25$ . So $25 = 18 + 15 - \text{both}$ , which gives both $= 33 - 25 = 8$ .

To draw the Venn diagram, fill in from the middle out: $8$ in the lens, $18 - 8 = 10$ soccer only, $15 - 8 = 7$ basketball only, and $5$ outside. Check: $8 + 10 + 7 + 5 = 30$ .

Worked example: Divisible by 2 or 3

How many integers from $1$ to $100$ are divisible by $2$ or $3$ ?

Divisible by $2$ : $50$ . Divisible by $3$ : $33$ . Divisible by both means divisible by $6$ : $16$ . So $50 + 33 - 16 = 67$ .

Three sets

For three overlapping sets, the formula adds the singles, subtracts the pairs, and adds back the triple:

$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|.$

The last term is there because the center region is added $3$ times, then subtracted $3$ times, so it needs to be added back once.

In practice, it is often easier to fill in a three-circle Venn diagram from the center outward, especially when the question asks about "exactly one" or "only" regions.

Worked example: Three languages

Of $40$ students, $20$ take French, $18$ take Spanish and $15$ take Latin. Also $7$ take French and Spanish, $5$ take French and Latin, $6$ take Spanish and Latin, and $2$ take all three. How many take none of these languages? How many take only French?

Union: $20 + 18 + 15 - 7 - 5 - 6 + 2 = 37$ . So $40 - 37 = 3$ take none.

For only French, work from the center. The center has $2$ . French and Spanish only: $7 - 2 = 5$ . French and Latin only: $5 - 2 = 3$ . So only French is $20 - 5 - 3 - 2 = 10$ .

Common mistake

"Both" numbers usually include people in all three groups. If $7$ take French and Spanish, and $2$ take all three, then only $5$ are in the French-and-Spanish-but-not-Latin region. Subtract the center before filling in pair regions.

Tip

For the smallest possible overlap of two groups, push them apart: the overlap is at least $|A| + |B| - \text{total}$ (or $0$ if that's negative). The largest possible overlap is the size of the smaller group.

Practice

Practice 1

In a class of $25$ students, $14$ have a dog, $12$ have a cat and $4$ have both. How many have neither a dog nor a cat?