Venn Diagrams
🔒 Log in to trackChoosing the correct diagram
🔒 Log in to trackFor 'Which diagram represents Doctors, Men, Writers?' style questions, decide each pair with two tests:
- Necessity test — is every X definitionally a Y? (Every mother is a female.) → circle of X drawn inside Y.
- Impossibility test — can X and Y never share a member? (A square and a triangle.) → separate circles.
- Otherwise — overlap possible but neither forced (doctors and writers) → intersecting circles.
Same-type categories usually intersect; category-of-category pairs (squares/rectangles/polygons) nest; mutually exclusive definitions separate.
What the question asks
You get three words (Doctors, Men, Writers; Even numbers, Odd numbers, Prime numbers) and must choose the circle picture that shows how the groups are related. There are no numbers — only relationships.
Two tests for every pair
Look at the three pairs one at a time: (1, 2), (2, 3), (1, 3).
- Inside test. Is every member of X always a member of Y? Every sparrow is a bird → the Sparrows circle is drawn inside Birds.
- Apart test. Can X and Y never share a member? No number is both even and odd → the two circles are drawn separate.
- If neither test holds, some members can be in both and some cannot → the circles intersect.
Judge by definition, not by what is usual. A doctor can be a woman and a woman can be a doctor, so Doctors and Women intersect.
The shapes that appear
| Relationship | Picture | Example |
|---|---|---|
| X ⊂ Y ⊂ Z | three nested circles | Sparrows, Birds, Animals |
| X, Y separate, both inside Z | two separate circles inside a third | Pens, Pencils, Stationery |
| X, Y intersect, both inside Z | two intersecting circles inside a third | Even numbers, Multiples of 3, Integers |
| X ⊂ Y, Z apart from both | one inside another, third separate | Whales, Mammals, Fish |
| X ⊂ Y, Z crosses both | one inside another, third intersecting both | Fathers, Men, Doctors |
| X ⊂ Y, Z crosses only Y | third cuts the outer circle only | Natural numbers, Integers, Negative numbers |
| X, Y intersect, Z inside their overlap | third inside the common part | Multiples of 5, Multiples of 7, Multiples of 35 |
| X, Y separate, Z crosses both | two separate circles joined by a third | Even numbers, Odd numbers, Prime numbers |
| all pairs intersect | three intersecting circles | Teachers, Women, Singers |
| two intersect, third apart | two intersecting circles and a separate one | Women, Engineers, Rivers |
| no links | three separate circles | Pens, Rivers, Tigers |
Tricky cases
- Number sets are exact: check a sample. 2 is even and prime, 3 is odd and prime, 9 is odd and not prime — so Primes cross both Even and Odd while Even and Odd never meet.
- "Crosses only the outer circle." Negative numbers include −1 (an integer) and −½ (not an integer), and no negative number is a natural number. So the Negative circle cuts Integers but stays away from the Naturals circle inside it.
- Inside their overlap. Every multiple of 35 is a multiple of both 5 and 7, so it sits inside the common part of those two circles.
- Gender words. Fathers ⊂ Men; Mothers ⊂ Women; Men and Women are separate.
Method
- Write the three pairs.
- Mark each pair inside, apart or cross.
- Match the three marks to the picture — do not jump to a picture from the first pair alone.
Question types you will see
Each type: how to recognise it, the method step by step, and one question to try.
Nested chain
Each word is a type of the next: sparrow → bird → animal.
Check that every X is a Y and every Y is a Z.
Draw the circles one inside the next.
Why it works: subset of a subset is a subset.
Sparrows, Birds, Animals.
Show solutionHide solution
Every sparrow is a bird, every bird an animal → three nested circles.
Subsets inside a bigger set
Two words are kinds of the third word.
Put both smaller groups inside the big circle.
Decide whether the two smaller groups can share members: never → separate; sometimes → intersecting.
If one group is inside another and the third is unrelated, draw the third apart.
Why it works: the relation between the two small groups is separate from their relation to the big one.
Pens, Pencils, Stationery.
Show solutionHide solution
Both are stationery; nothing is both a pen and a pencil → two separate circles inside a third.
Partial overlaps
No word is a kind of another; some pairs can share members, some cannot.
Test each pair: can share → intersect; never → apart.
All three share → three intersecting circles.
Two share and the third is unrelated → two intersecting circles plus a separate one.
Why it works: with no subsets, only the cross/apart choice matters for each pair.
Teachers, Women, Singers.
Show solutionHide solution
Every pair can share members → three circles, each pair intersecting.
Mixed: inside plus crossing
One word is a kind of another and the third cuts across, or one sits inside the overlap of two.
Draw the subset pair first.
Test the third word against the inner and the outer circle separately.
If the third word is a subset of two overlapping sets, place it inside their common part.
Why it works: the third circle's relation to the inner circle can differ from its relation to the outer one.
Fathers, Men, Doctors.
Show solutionHide solution
Fathers ⊂ Men; a doctor can be a father or a man → one inside another, third intersecting both.
Shortcuts that save time
Never judge the whole picture at once. Run necessity/impossibility on each pair; the correct option is the only one agreeing on every pair.
Q. Mothers, Females, Engineers.
Sol. Mothers ⊂ Females (necessity). Engineers ∩ Females possible (female engineers) and Engineers ⊄ Females (male engineers) → intersecting. Match the option reflecting both facts.
Show solutionHide solution
One wrong pair eliminates an option — work pair by pair until one option survives.
Mistakes to avoid
Where most students lose marks on this subtopic.
Drawing engineers inside females because most examples in mind are female.
Making mutually-overlappable categories fully separate.
Missing a third category's relation to the nested pair.
Quick revision
Read this the night before the exam.
For each pair: always both → inside; never both → apart; else → cross.
Judge by definition, not by what is common.
Test number sets with small examples (2, 3, 9, −1, −½).
A third circle can cross both, only the outer, or sit inside an overlap — check each pair separately.
Decide all three pairs before picking the picture.
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 7 min · wrong answers go to your mistake notebook automatically.