Venn Diagrams
🔒 Log in to trackTwo-circle counting
🔒 Log in to trackWith two sets there are exactly four regions: only A, only B, both, neither. Every two-circle question is a sum or difference of these:
- Union (at least one) = only A + only B + both.
- Only A = |A| − both; Only B = |B| − both.
- Neither = grand total − union.
Exam data typically gives |A|, |B| and the overlap (or hides one of them behind the union). Rearranging the inclusion-exclusion identity recovers any missing region in one step.
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.
Question types you will see
Each type: how to recognise it, the method step by step, and one question to try.
Union and neither
Two totals, their overlap and the group size; asks at least one, neither or only one set.
Union = A + B − both.
Neither = total − union.
Only A = A − both.
Why it works: the overlap is counted twice in A + B, so it is removed once.
150 families: 90 car, 70 two-wheeler, 40 both. How many have neither?
Show solutionHide solution
Union = 90 + 70 − 40 = 120; neither = 150 − 120 = 30.
Reverse two-set puzzle
Some region values are given and a circle total, the overlap or the group size is asked; may include a ratio clue.
Draw the four boxes: only A, only B, both, neither.
Fill the known boxes.
Solve for the missing box from the total or the ratio.
Why it works: the four boxes always add to the total.
30 like mango, 15 like only guava, 8 like neither. Group size?
Show solutionHide solution
Mango (30, includes both) + only guava 15 + neither 8 = 53.
Percentage sets
Data given in percent; the answer may be in percent or in people.
Work the whole problem in percent with total 100.
Find the percent of the asked region.
Convert: people = percent × total ÷ 100 (or total = people ÷ percent × 100).
Why it works: percentages obey the same region algebra as counts.
35% failed maths, 25% failed English, 10% failed both; 250 passed both. Total?
Show solutionHide solution
Failed at least one = 50%, passed both = 50% → total = 500.
Maximum and minimum overlap
The overlap is not given; asks the largest or smallest possible both / neither.
Max both = smaller set.
Min both = A + B − N (0 if negative).
Max neither = N − larger set; min neither = N − (A + B) (0 if negative).
Why it works: the smaller circle can sit fully inside the larger, or the circles can be pushed apart until the total forces overlap.
60 students; 45 tea, 35 coffee. Minimum who like both?
Show solutionHide solution
45 + 35 − 60 = 20.
Formula sheet
The intersection is counted twice on the right, so subtract once.
'Only A' is A minus the overlap.
Everyone outside both circles.
Shortcuts that save time
Write the numbers INTO the regions: the lens, the two moons, and outside. Sums become single looks instead of formula recalls.
Q. 60 students; 35 like tea, 30 coffee, 10 both. How many like neither?
Sol. Tea-only 25 + both 10 + coffee-only 20 = 55 inside; 60 − 55 = 5.
Show solutionHide solution
Fill the diagram first; the answer is whatever region is left, counted.
Mistakes to avoid
Where most students lose marks on this subtopic.
Counting the intersection twice when totalling.
Treating 'like tea' (including both) as 'like only tea'.
Forgetting the 'neither' region sits outside both circles.
Quick revision
Read this the night before the exam.
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.
Practice: 15 questions
Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.
Topic test · 10 questions
Suggested time 6 min · wrong answers go to your mistake notebook automatically.