Venn Diagrams

Some classmates joined the football club, some the basketball club, and a few joined both. The rosters live on two sheets of paper, and counting who got double-listed — or listed nowhere — turns into a mess. Draw the two rosters as two circles and everything becomes clear at once. That is a Venn diagram: circles that overlap so sets can line up in order. Collecting data starts with a questionnaire (see surveys and sampling); this lesson is about untangling the data once it is back.

Intersection: the overlap

Lay two circles on top of each other and the shared region in the middle is the intersection — the members that belong to both groups. Students who like football and basketball stand exactly there.

Say set $A$ has 20 people, set $B$ has 14, and the intersection holds 6. Then the part belonging to $A$ alone is $20 - 6 = 14$ people — subtract the 6 first so nobody gets counted twice.

Union: both circles as one

The whole area covered by the two circles is the union — everyone in at least one group, possibly both. Adding the group sizes directly would count the intersection twice, so the intersection must be subtracted back out:

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

With the numbers above: $20 + 14 - 6 = 28$ people. That subtracted 6 is exactly the double-counted crowd, counted once at last.

Complement: the world outside the circle

Inside the circle live the members; outside it live stories too. The complement is everything "not in the set". Out of 40 students, 26 like mathematics — so the complement, the ones who do not, numbers $40 - 26 = 14$ . Surveys most often overlook precisely the people outside the circle.

Worked example: two clubs, one picture

Football and basketball

In a class of 40, 25 students like football, 18 like basketball, and 10 like both. How many like only football, only basketball, and neither?

Football only: $25 - 10 = 15$ ; basketball only: $18 - 10 = 8$ ; the middle intersection holds 10. Adding the three regions, $15 + 10 + 8 = 33$ students like at least one sport, so $40 - 33 = 7$ like neither. One circle diagram answers four questions in a single sweep.

What about three circles?

Swap in three circles — can swim, can ride a bike, can skate — and the inside of the picture splits into 7 regions: 3 belonging to exactly one group, 3 to exactly two, and 1 to all three, since $3 + 3 + 1 = 7$ . Every extra circle roughly doubles the regions, so three-set problems must be counted region by region. This region-counting habit is the foundation for counting outcomes in probability.

Check yourself

Quick quiz

1. Set A has 20 people, set B has 14, and the intersection has 6. How big is the union?
2. In a class of 30, 18 students like football, 15 like basketball and 8 like both. How many like neither?
3. Two sets have an empty intersection. What does that mean?