8 min readConcept sheet2-mark items; 5-mark problems; 10-mark 'types with a survey problem'

Venn diagrams

Venn diagrams show how groups overlap with circles. Pick the diagram that fits the relationship, and use n(A ∪ B) = n(A) + n(B) − n(A ∩ B) for counts.

Two groupsn(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Three groupsAdd the three, subtract the three pairs, add back the centre

NeitherTotal − n(A ∪ B)

Only A (two groups)n(A) − n(A ∩ B)

Choose the right diagram

Relationship	Example	Diagram
One group inside another	Vitamin, Nutrient	small circle inside a big circle
Three nested groups	Mango, Fruit, Food	three circles, one inside another
Partly overlapping	Women, Doctors	two overlapping circles
Completely separate	Iron, Wood	two separate circles
Two inside one, separate from each other	Rice, Wheat, Cereal	two separate circles inside a big circle
Two inside one, overlapping	Mothers, Teachers, Women	two overlapping circles inside a big circle
One inside another, third separate	Dietitians, Professionals, Pulses	small circle inside a big one, third circle apart
Three partly overlapping	Students, Singers, Dancers	three overlapping circles

How to fill a Venn diagram (start from the middle)

1
Draw the circles inside a box; the box stands for the total.
2
Write the 'all three' value in the centre first.
3
Fill each 'exactly two' region: the pair value minus the centre.
4
Fill each 'only one' region: the circle total minus everything else inside that circle.
5
Add all the regions inside the circles and subtract from the total to get 'none'.
6
Check that each circle adds up to its given total.

Worked example 1: choosing the diagram

Problem

Which diagram best shows the relationship between Calcium, Mineral and Phosphorus?

Working

Step 1: test each pair: calcium is a mineral, and phosphorus is a mineral.
Step 2: calcium is not phosphorus; they are different minerals.
Step 3: so both small circles lie inside the 'mineral' circle and do not touch each other.
Check: every true statement about the groups is shown, and nothing false is shown.

Answer

Two separate circles inside one large circle.

Worked example 2: two groups

Problem

In a hostel of 120 students, 70 drink tea, 55 drink coffee and 25 drink both. How many drink neither?

Working

Standard method (formula): n(tea ∪ coffee) = 70 + 55 − 25 = 100, so neither = 120 − 100 = 20.
Shortcut (fill from the middle): both = 25; tea only = 70 − 25 = 45; coffee only = 55 − 25 = 30. Inside the circles = 45 + 25 + 30 = 100, so outside = 20.
Check: 45 + 25 + 30 + 20 = 120.

Answer

20 students drink neither (75 drink only one of the two).

Worked example 3: three groups

Problem

In a canteen survey of 60 students, 30 like idli, 25 like dosa and 20 like pongal. 10 like idli and dosa, 8 like dosa and pongal, 6 like idli and pongal, and 3 like all three. How many like none of them?

Working

Standard method (formula): union = 30 + 25 + 20 − 10 − 8 − 6 + 3 = 54, so none = 60 − 54 = 6.
Shortcut (fill from the middle): centre 3. Exactly two: idli-dosa 10 − 3 = 7, dosa-pongal 8 − 3 = 5, idli-pongal 6 − 3 = 3.
Only one: idli 30 − 7 − 3 − 3 = 17; dosa 25 − 7 − 5 − 3 = 10; pongal 20 − 5 − 3 − 3 = 9. Inside = 17 + 10 + 9 + 7 + 5 + 3 + 3 = 54, so none = 6.
Check: idli 17 + 7 + 3 + 3 = 30, dosa 10 + 7 + 5 + 3 = 25, pongal 9 + 5 + 3 + 3 = 20; and 54 + 6 = 60.

Answer

6 students like none of the three.

How exams ask this

SSC, railway and state PSC papers ask 'which diagram best represents' three words, and count questions on two or three groups; bank papers use Venn diagrams to test syllogisms. Placement tests give survey data and ask for 'only one', 'exactly two' or 'none'.

Exam angle

2 marks: “Draw a Venn diagram to show the relationship between Fruit, Mango and Iron.” The mango circle sits inside the fruit circle; the iron circle stands apart from both.

5 marks: “In a hostel of 120 students, 70 drink tea, 55 drink coffee and 25 drink both. How many drink neither, and how many drink only one of the two?” Union = 100, so 20 drink neither. Only tea 45 + only coffee 30 = 75 drink only one.

10 marks: “What is a Venn diagram? Explain the types of relationships it shows, with examples, and solve a three-group survey problem.” Define; draw the common types with examples; give the two- and three-group formulas; solve the idli-dosa-pongal survey, filling from the centre (none = 6).

Remember it

Fill from the middle out: all three → exactly two → only one → none. Union = add − overlaps (+ centre for three groups).