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.
Detailed notes
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.
Quick revision
- 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.
Types of questions asked
Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.
Type 1: Nested chaincommon2 practice Q
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.
Example: Sparrows, Birds, Animals.
Every sparrow is a bird, every bird an animal → three nested circles.
Type 2: Subsets inside a bigger setvery common3 practice Q
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.
Example: Pens, Pencils, Stationery.
Both are stationery; nothing is both a pen and a pencil → two separate circles inside a third.
Type 3: Partial overlapscommon3 practice Q
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.
Example: Teachers, Women, Singers.
Every pair can share members → three circles, each pair intersecting.
Type 4: Mixed: inside plus crossingoccasional4 practice Q
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.
Example: Fathers, Men, Doctors.
Fathers ⊂ Men; a doctor can be a father or a man → one inside another, third intersecting both.
Shortcut tricks
⚡ Test one pair at a 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.
Example: 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.
One wrong pair eliminates an option — work pair by pair until one option survives.
Where students lose marks
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.
Practice sets — 15 questions
Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.
Topic test · 10 questions
Suggested time 7 min · wrong answers go to your mistake notebook automatically.