Venn diagram

A Venn diagram shows two groups as overlapping circles.
Where the circles overlap sits the group of items that belong to both, and where they do not overlap sit the items in only one group.
Anything outside both circles belongs to neither.

The key formula is for the number in at least one group, written $A \cup B$ ("A union B").
You add the two groups, but the overlap got counted in both, so you subtract it once: $A \cup B = A + B - (A \cap B)$, where $A \cap B$ ("A intersection B") is the overlap.
For example, if $25$ people study French, $27$ study Spanish, and $15$ study both, then $25 + 27 - 15 = 37$ study at least one.

The very same groups can be organized as a 2×2 table.
The rows are "A" and "not A", the columns are "B" and "not B", and each of the four inner cells counts the people in that combination.
A "Total" row and column are added around the edge.

The table's power is that the totals must agree.
Reading across, the "A" row and the "not A" row add up to the whole group: $A + \text{not } A = \text{total}$.
Reading down, the "B" column and the "not B" column add up to the same total: $B + \text{not } B = \text{total}$.
So if a cell is missing, you can fill it in by making a row or column reach its total.

Here is a full question shown both ways.
In a class of $200$ students, $120$ play an instrument, $100$ play a sport, and $40$ play both.
As overlapping circles, $120 - 40 = 80$ play only an instrument, $40$ play both, $100 - 40 = 60$ play only a sport, and the other $200 - 180 = 20$ play neither.
As a 2×2 table, those same four numbers fill the grid, and the instrument row adds to $120$, the sport column adds to $100$, and everything adds to $200$.

Out of $90$ people who own a bike, $25$ also own a scooter. How many own only a bike?
The overlap "both" is $25$, and the rest of the bike group owns only a bike.
So $90 - 25 = 65$ own only a bike.

In a class, $18$ students study French and $15$ study Spanish, and $6$ study both. How many study at least one of the two languages?
Use $A \cup B = A + B - (A \cap B)$: add the two groups and subtract the overlap, $18 + 15 - 6$.
So $27$ students study at least one language.

A survey of $50$ people is shown in the table below.

What is the missing "not B" value in the "A" row?
(A) $12$
(B) $22$
The two cells in the "A" row must add up to the row total of $32$.
Since the "B" cell is $20$, the "not B" cell is $32 - 20 = 12$.
So (A) is correct.