Venn Diagrams
Your class of 30 is arguing. Someone claims "loads of us do both French
and Spanish", someone else swears "hardly anyone does either". How do you settle it without
a headache? You draw two overlapping circles — one for French, one for Spanish — and the fog clears
instantly.
That picture is a Venn diagram: overlapping circles that sort things into groups so you
can see what belongs to both, to either, or to
neither. A tangled "how many do both?" question becomes a simple matter of counting the
right region.
Three regions, three questions
A Venn diagram draws sets as overlapping circles inside a rectangle. The rectangle is the
universal set — every item under consideration. Each circle is one set, say
A and B, and where they overlap holds the items
that belong to both.
-
The overlap is the intersection
A \cap B — items in both A
and B. (Read the \cap as "and".)
-
Everything inside either circle is the union
A \cup B — items in A
or B (or both). (Read the \cup
as "or".)
-
Everything outside A is the complement
A' — items not in A. Outside
both circles is "neither".
Writing a number in each region — how many items land there — turns the picture into a
counting tool: a probability is just the items in a region divided by the total.
The golden rule: always fill in the regions from the inside out — start with
the overlap, then work outward to the circle-only regions, then "neither" last.
For two sets A and B drawn inside a universal set:
-
A \cap B (intersection) — the items in both, the overlap;
-
A \cup B (union) — the items in either circle;
-
A' (complement) — the items not in
A;
-
count the union with n(A \cup B) = n(A) + n(B) - n(A \cap B),
subtracting the overlap once so it isn't counted twice;
-
read a probability of a region as
P(\text{region}) = \frac{\text{items in the region}}{\text{total items}}.
Worked example: fill it from the inside out
In a class of 30: 12 study French,
8 study Spanish, and 5 study both.
How many study French only? Spanish only? Neither? Never start from the "12" — start from the overlap.
- Overlap first. "Both" = 5 goes in the middle.
-
French only = 12 - 5 = 7 (the 5 who do both are already
part of the 12).
- Spanish only = 8 - 5 = 3.
-
At least one = 7 + 5 + 3 = 15, so
neither = 30 - 15 = 15.
Reading probabilities off the diagram
Once every region has a number, a probability is just that region divided by the total,
30. Pick a student at random:
-
P(\text{both}) = P(F \cap S) = \frac{5}{30} = \frac{1}{6} \approx 0.17.
-
P(\text{either}) = P(F \cup S) = \frac{7 + 5 + 3}{30} = \frac{15}{30} = 0.5.
-
P(\text{neither}) = \frac{15}{30} = 0.5.
Notice the union the quick way: n(F \cup S) = 12 + 8 - 5 = 15 — add the two
circles, subtract the overlap once.
Another two-set diagram to read
Here the four regions hold 7, 5,
8 and 4 items — 24
altogether. So n(A \cup B) = 7 + 5 + 8 = 20 and
P(A \cap B) = \tfrac{5}{24} \approx 0.21.
Stretch: three circles at once
With three sets you get seven regions inside the circles (plus "none" outside). The
very centre — where all three overlap — is F \cap B \cap T. It looks busier,
but the rule is identical: fill the innermost region first, then work outward.
In this survey of 30 students choosing sports:
2 play all three, 6 play only football, and
7 play none. Reading the middle:
P(\text{all three}) = \tfrac{2}{30} = \tfrac{1}{15}.
The classic Venn slip: reading a circle's total straight into its "only" region. If
12 do French, 8 do Spanish and
5 do both, the French-only region is
12 - 5 = 7, not 12 — because
those 5 "both" students are already inside the French count.
Forget to subtract the overlap and you count those 5 people twice: your
regions would sum to 12 + 5 + 8 = 25 supposed language-takers when only
15 exist, and every total downstream goes wrong. So always write the overlap
in first, then subtract it as you fill each circle-only region.
They're named after John Venn, an English logician who drew them in
1880 to picture logical statements. He'd have been amazed where they ended up:
Venn diagrams now turn up in computer database searches (the AND / OR / NOT of a Google
query is exactly intersection, union and complement), in biology, in probability, and in a million
internet memes.
And the idea underneath them — set theory, the study of collections and their
intersections and unions — is one of the bedrock foundations that all of modern mathematics is
built on. Two circles you can doodle in a margin, holding up a whole subject.
See it explained