Venn Diagrams

The four regions

Two overlapping circles A and B split the universe into four parts: only A, only B, both, neither. Every two-set question is a sum or a difference of these four numbers. Write them in the picture before doing anything else.

Core formulas

• n(A ∪ B) = n(A) + n(B) − n(A ∩ B). The overlap is subtracted once because it was counted in both circles.
• Only A = n(A) − both; only B = n(B) − both.
• Exactly one = only A + only B = n(A) + n(B) − 2 × both.
• Neither = total − n(A ∪ B).
• Total = only A + only B + both + neither.

Worked example

In a society of 150 families, 90 have a car, 70 have a two-wheeler and 40 have both.

• At least one = 90 + 70 − 40 = 120.
• Neither = 150 − 120 = 30.
• Only car = 90 − 40 = 50; only two-wheeler = 70 − 40 = 30. A common wrong answer is 70: taking only car + only two-wheeler = 80 and then 150 − 80, which forgets the 40 who have both.

Reverse puzzles

Sometimes the question gives the answer regions and asks for a circle total or for the overlap. Use the same four boxes and fill in what you know.

• Given union and the two totals: both = n(A) + n(B) − union.
• If everybody is in at least one set: union = total, so both = n(A) + n(B) − total.
• Ratio clues: "only cricket is twice only football", with only C + only F = 30, gives only F = 10, only C = 20.

Percentages

Treat the whole group as 100 and work in percent. Convert to people only at the end.

• 35% failed maths, 25% failed English, 10% failed both → failed at least one = 50%, so passed both = 50%. If 250 passed both, total = 250 ÷ 0.5 = 500.
• The trap: 100 − 35 − 25 = 40% "passed both" forgets that the 10% who failed both were subtracted twice.

Maximum and minimum overlap

When the overlap is not given, it can move within limits (total N, sets of size a and b):

• Maximum both = the smaller set, min(a, b) — the small circle sits fully inside the big one.
• Minimum both = a + b − N, or 0 if that is negative — the circles overlap only as much as they are forced to.
• Maximum neither = N − max(a, b) — again when the smaller circle sits inside the larger.
• Minimum neither = N − (a + b), or 0 if negative. Example: 60 students, 45 like tea, 35 like coffee → both is at least 45 + 35 − 60 = 20 and at most 35.

Quick revision

• Four boxes: only A, only B, both, neither — fill them first.
• Union = A + B − both; neither = total − union; exactly one = A + B − 2 × both.
• Everyone in at least one set → both = A + B − total.
• Percent questions: work in %, convert at the end; passed both ≠ 100 − fail₁ − fail₂.
• Max both = smaller set; min both = A + B − N (not below 0); max neither = N − larger set.