Overlapping sets: Venn diagrams and double-matrix tables

Two sets

When some people belong to two groups, adding the group sizes counts the people in both groups twice. Subtract them once, then add anyone in neither group:

$$\text{Total} = A + B - \text{Both} + \text{Neither}$$

A typical two-set question gives four of these five numbers and asks for the fifth. A Venn diagram makes the parts visible: "only $A$" is $A - \text{Both}$, "only $B$" is $B - \text{Both}$, and "exactly one of the two" is the sum of those, $A + B - 2 \times \text{Both}$. Read carefully which region the question asks about: "speaks French" includes the people who speak both languages, "speaks only French" does not.

The double-matrix table

When everyone is sorted by two yes-or-no attributes (men or women, and on campus or off campus), a two-by-two table with totals is faster and safer than a diagram:

	Attribute 2: yes	Attribute 2: no	Total
Attribute 1: yes
Attribute 1: no
Total			grand total

Every row and every column adds up, so each new number lets you fill in another. You usually need only four well-placed numbers to complete all nine cells. The table works just as well with percents: if no total is given, call the grand total 100.

Be careful with "of". "30% of the employees are remote" is a percent of the grand total; "30% of the sales staff are remote" is a percent of one row. Convert row percents into counts of the whole before you put them in the table.

Three sets

With three groups $A$, $B$ and $C$, the clearest bookkeeping counts how many groups each person belongs to. Let $E_1$, $E_2$ and $E_3$ be the numbers of people in exactly one, exactly two and all three groups. Then

$$A + B + C = E_1 + 2E_2 + 3E_3$$

because a person in two groups is counted in two of the group totals, and a person in three groups in all three. The number of people in at least one group is $E_1 + E_2 + E_3$. Subtracting one equation from the other gives the formula usually quoted:

$$\text{Total} = A + B + C - E_2 - 2E_3 + \text{Neither}$$

Some questions instead give pairwise overlaps such as "in both $A$ and $B$", which include the people in all three. With those, the formula is

$$\text{Total} = A + B + C - (AB + AC + BC) + ABC + \text{Neither}$$

The difference between "exactly two" and "both $A$ and $B$" is the most common source of error in three-set questions, so name which one you have before choosing a formula.

Greatest and least overlap

Sometimes the overlap is not fixed and the question asks how large or small it can be. In a group of 60 where 42 drink tea and 35 drink coffee:

• Greatest overlap: the smaller group sits entirely inside the larger, so at most 35 drink both.
• Least overlap: squeeze the groups apart. They can cover at most all 60 people, so at least $42 + 35 - 60 = 17$ drink both.

If the question also fixes how many are in neither group, the people in at least one group replace the 60 in that calculation.

Checking an answer

Fill in every region and add them up. If the regions do not add to the total, or a region comes out negative, a number went in the wrong place.