Venn Diagram

📖 Theory

A 20-second problem

In a class of 40 students, 25 like cricket, 20 like football, and 12 like both.

How many like neither?

You could stare at the numbers and guess. Or you could draw two overlapping circles and read the answer off the picture in 20 seconds. That picture is a Venn diagram, and it is about to become your favourite exam tool.

📖 Theory

Two friend circles

Think of your school friends as one circle and your college friends as another. A few people belong to both circles, so the circles overlap. Everyone else in the world sits outside both. That outer space is the rectangle drawn around the circles: the universal set.

📖 Theory

The four regions

Two overlapping circles A and B inside a rectangle split the picture into exactly four regions:

• Only A: inside A, outside B
• Only B: inside B, outside A
• Both: the overlapping middle (A ∩ B)
• Neither: inside the rectangle, outside both circles

Every element of the universal set lives in exactly one of these four places. That is what makes counting work.

💭 Think first

Try the class problem in your head. 40 students total, 25 cricket, 20 football, 12 both.

Start with the overlap: 12 sit in the middle. Now, how many like only cricket, how many like only football, and how many like neither?

📖 Theory

Operations as shading

Each set operation is just a region you shade:

• A ∪ B (union): shade both circles completely, "in A or B or both"
• A ∩ B (intersection): shade only the overlapping middle
• A′ (complement): shade everything outside circle A, right up to the rectangle
• A − B (difference): shade A but rub out the overlap, "A only"

If you can shade it, you understand it. No formula needed.

🧩 Quiz

Which region is A ∩ B?

1. Everything covered by the two circles together
2. Only the overlapping middle part
3. The part of A that does not touch B
4. Everything outside both circles

🧩 Quiz

In the class problem, "students who like football but not cricket" is which region?

1. The whole football circle
2. The overlap of the two circles
3. The football circle minus the overlap
4. Everything outside the cricket circle

📖 Theory

Three sets, eight regions

Three overlapping circles A, B, C create 8 regions: the centre (in all three), three "exactly two" overlaps, three "only one" crescents, and the outside.

The strategy stays the same, just repeated:

1. Fill the centre (A ∩ B ∩ C) first.

2. Fill the three pair overlaps, subtracting the centre.

3. Fill the three "only" regions.

4. Whatever is left over sits outside.

⚠️ Watch out

The two marks students lose

First: forgetting the neither region, everything must add up to the universal set total, so check the sum. Second: writing 25 in the "only cricket" region. 25 is the whole cricket circle; only cricket is 25 minus the overlap. Fill the middle first and this mistake becomes impossible.

📖 Theory

Where you will meet this again

SQL joins in DBMS are Venn diagrams (inner join is the overlap, left join is a circle plus the overlap). Probability questions reuse the same regions with numbers between 0 and 1. Even search filters like "BCA students who know Python but not Java" are A − B in disguise.

✅ Summary

Key takeaways

• A Venn diagram shows sets as overlapping circles inside a rectangle (the universal set).
• Two sets create four regions: only A, only B, both, neither.
• Union shades both circles, intersection shades the overlap, difference shades one circle minus the overlap.
• For counting problems, fill the innermost overlap first and work outward, then check the total.
• Memory hook: fill the middle first.