Venn Diagrams
🔒 Log in to trackWhat a Venn diagram encodes
🔒 Log in to trackA Venn diagram is a picture of set membership: every point inside a circle belongs to that category; overlap regions belong to both. Two habits solve most CGL venn questions:
- Region bookkeeping. Name the 4 regions of a two-circle diagram (only A, only B, both, neither) and the 8 regions of a three-circle one. Every question is a sum or difference of named regions.
- Relation reading. 'Choose the right diagram for X, Y, Z' asks about definitional relations: is one category ALWAYS inside another (nested circles)? Can they NEVER meet (separate circles)? Otherwise (overlap possible but not forced) draw intersecting circles.
Draw the regions and label them with the given numbers before computing — unlabelled diagrams cause double-counting errors.
What a Venn diagram shows
A Venn diagram is a picture of membership. Each circle is a group (people who drink tea, even numbers, doctors). A point inside a circle belongs to that group. A point where two circles overlap belongs to both. The rectangle around the circles is the universal set — everyone being talked about — and the space outside all circles holds those who belong to none.
The diagram is only useful if you read it by regions, not by circles. A region is a piece of the picture that cannot be split further by any circle line.
Counting regions
- Two intersecting circles make 4 regions: only A, only B, both, neither.
- Three mutually intersecting circles make 8 regions: 3 "only" regions, 3 "exactly two" regions, 1 "all three" region and 1 "none" region.
- In general each region is a yes/no answer for every circle, so n circles drawn with every overlap give 2ⁿ regions (including the outside).
- Circle A alone is made of 4 of the 8 regions in a three-circle diagram: only A, A-and-B only, A-and-C only, and all three.
Words → set symbols
| Phrase | Symbol | Regions (three circles) |
|---|---|---|
| in A or B (at least one) | A ∪ B | everything inside A or B |
| in both A and B | A ∩ B | A-B only plus all three |
| in A but not B | A − B | only A plus A-C only |
| in A and B but not C | (A ∩ B) − C | A-B only |
| in exactly one of A, B | (A ∪ B) − (A ∩ B) | only A, only B |
| in none | outside all circles | the "none" region |
The commonest slip is to read "in both A and B" as the single A-and-B-only region. "Both A and B" says nothing about C, so the all-three region is included.
Reading numbers off a diagram
When the numbers are written region by region, every question is just adding the right regions. Suppose a survey of news habits — Newspaper (N), TV (T), Radio (R) — gives: only N 20, only T 25, only R 5, N-T only 15, T-R only 6, N-R only 4, all three 10, none 15 (total 100).
- Total newspaper readers = 20 + 15 + 4 + 10 = 49 (all four N-regions).
- At least two sources = 15 + 6 + 4 + 10 = 35.
- Exactly one source = 20 + 25 + 5 = 50.
- Not TV = total − all T-regions = 100 − 56 = 44.
- At most one source = exactly one + none = 50 + 15 = 65.
Negative phrases
- Neither / none is the outside region.
- Not A is everything outside circle A, including the none region.
- At most one includes those in no circle at all.
- Either A or B but not both is the two "only" regions.
Method that never fails
- Write the region numbers (or letters) in the picture.
- Turn the phrase into a list of regions.
- Add those regions only once each.
Question types you will see
Each type: how to recognise it, the method step by step, and one question to try.
Phrase to set expression
A phrase like 'doctors and singers but not players' with options written in ∩, ∪ and − symbols.
List which regions the phrase covers.
For each option, list the regions it covers.
Pick the option whose regions match exactly.
Why it works: two expressions mean the same only if they cover the same regions.
Which expression shows 'in A and B but not C'?
Show solutionHide solution
The phrase is the A-B-only region = (A ∩ B) − C. A ∩ B alone would also include the all-three region.
Reading region numbers
Numbers are given for each region (only A, A-B only, all three, none) and a question asks for a total.
Turn the question into a list of regions.
Add those region numbers, each once.
Use total − region sum for 'not' questions.
Why it works: regions never overlap, so adding them never double-counts.
Only N 20, N-T only 15, N-R only 4, all three 10. How many read the newspaper?
Show solutionHide solution
All N-regions: 20 + 15 + 4 + 10 = 49.
Counting regions
Asks how many regions a diagram has, or how many regions make up one circle.
Each region is a yes/no choice for every circle.
n circles with all overlaps → 2ⁿ regions including the outside.
Regions inside one circle of three = 2² = 4.
Why it works: every combination of in/out appears exactly once.
How many regions does a diagram of three mutually intersecting circles have (including the outside)?
Show solutionHide solution
2³ = 8.
Negative and 'at most' phrases
Phrases like not A, neither, at most one, either but not both.
'Not A' = total − all regions of A (includes the none region).
'At most one' = exactly one + none.
'Either but not both' = the two only-regions.
Why it works: negative phrases are easiest as complements of positive ones.
Total 100; the four T-regions add to 56. How many do not watch TV?
Show solutionHide solution
100 − 56 = 44.
Formula sheet
The intersection is counted twice on the right, so subtract once.
'Only A' is A minus the overlap.
Shortcuts that save time
Write the numbers INTO the regions: |A∩B| in the lens, only-A and only-B in the moons, neither 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 empty times its count.
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.
Read by regions; two circles = 4 regions, three circles = 8 (including outside).
"Both A and B" includes the all-three region; "A and B only" does not.
A − B keeps the parts of A outside B, whatever C does.
Not A includes the none region; at most one = exactly one + none.
List the regions first, then add each one once.
Practice: 12 questions
Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.
Topic test · 12 questions
Suggested time 8 min · wrong answers go to your mistake notebook automatically.