MathsISC Class 11 · Sets

Set Operations, De Morgan's Laws and Survey Problems with Venn Diagrams

Updated 17 October 2026 · Fact-checked

For finite sets, n(A ∪ B) = n(A) + n(B) − n(A ∩ B) and, for three sets, n(A ∪ B ∪ C) = Σn(A) − Σn(A ∩ B) + n(A ∩ B ∩ C). De Morgan's laws are (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′. Survey problems are solved by filling a Venn diagram from the innermost region outward.

Understand Set Operations, De Morgan's Laws and Venn Diagram Problems

Operations. $A \cup B$ = elements in A or B (or both); $A \cap B$ = elements in both; $A - B$ = elements of A not in B; $A^{'}$ = elements of U not in A; the symmetric difference $A Δ B = (A - B) \cup (B - A)$ . Sets are disjoint if $A \cap B = \emptyset$ .

Laws. Commutative, associative, distributive: $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ and $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ . De Morgan: $(A \cup B)^{'} = A^{'} \cap B^{'}$ and $(A \cap B)^{'} = A^{'} \cup B^{'}$ . Also $A - B = A \cap B^{'}$ and $A \cup A^{'} = U$ , $A \cap A^{'} = \emptyset$ .

Counting formulae. $n (A \cup B) = n (A) + n (B) - n (A \cap B)$ . $n (A - B) = n (A) - n (A \cap B)$ . $n (A^{'}) = n (U) - n (A)$ . For three sets: $n (A \cup B \cup C) = n (A) + n (B) + n (C) - n (A \cap B) - n (B \cap C) - n (A \cap C) + n (A \cap B \cap C)$ .

Venn diagram method. (1) Draw the circles for the sets inside a rectangle for U. (2) Start with the most restricted region, the triple intersection. (3) Fill the two-set overlaps by subtracting the triple intersection. (4) Fill the "only" regions by subtracting. (5) The region outside all circles is $n (U) - n (A \cup B \cup C)$ . Check that the regions add up to n(U).

Checking a law numerically. To verify De Morgan's law for given sets, find the left side and the right side separately and compare them element by element.

Set Operations, De Morgan's Laws and Venn Diagram Problems: formulas to remember

Two sets: $$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$; Subtract the overlap once.
Three sets: $$n(A \cup B \cup C) = Σ n(A) - Σ n(A \cap B) + n(A \cap B \cap C)$$; Inclusion-exclusion.
De Morgan: $$(A \cup B)' = A' \cap B'$ | $(A \cap B)' = A' \cup B'$$; Complement swaps ∪ and ∩.
Difference: $$A - B = A \cap B'$ | $n(A - B) = n(A) - n(A \cap B)$$; Elements only in A.

Set Operations, De Morgan's Laws and Venn Diagram Problems in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

A language survey

Suppose a survey of 100 people finds that 60 speak Hindi, 45 speak English and 20 speak both.

Find the number who speak only Hindi, only English, and neither.

Try it first, then show the solutionSolution

Only Hindi $= 60 - 20 = 40$ ; only English $= 45 - 20 = 25$ .
At least one language: $60 + 45 - 20 = 85$ .
Neither: $100 - 85 = 15$ .

Answer: Only Hindi 40; only English 25; neither 15.

Memory hook: Subtract the overlap before counting the "only" regions.

Tricks to remember Set Operations, De Morgan's Laws and Venn Diagram Problems

Short memory aids. Say each one aloud once, then use it on a question the same day.

Complement flips
Break the bar and change the sign: ∪ becomes ∩.
Use it when: De Morgan's laws.
Inside out
Fill a three-set Venn diagram from the centre outward.
Use it when: Survey problems.

Set Operations, De Morgan's Laws and Venn Diagram Problems: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Counting the union · 1 mark · MCQBoard level

If n(A) = 12, n(B) = 9 and n(A ∩ B) = 4, then n(A ∪ B) is:
(a) 25
(b) 21
(c) 17
(d) 13

Try it first, then show the solutionSolution

$12 + 9 - 4 = 17$ .

Answer: (c) 17

Example 2: Verifying De Morgan's law · 3 marks · Short answerBoard level

Let U = {1, 2, ..., 10}, A = {2, 4, 6, 8, 10} and B = {1, 2, 3, 4, 5}. Find A ∪ B, A ∩ B, A − B, A′ and verify that (A ∪ B)′ = A′ ∩ B′.

Try it first, then show the solutionSolution

A ∪ B = {1, 2, 3, 4, 5, 6, 8, 10}; A ∩ B = {2, 4}; A − B = {6, 8, 10}; A′ = {1, 3, 5, 7, 9}. (1.5 marks)
(A ∪ B)′ = {7, 9}. (0.75 mark)
B′ = {6, 7, 8, 9, 10}, so A′ ∩ B′ = {7, 9}, equal to (A ∪ B)′. (0.75 mark)

Answer: (A ∪ B)′ = A′ ∩ B′ = {7, 9}.

Example 3: Two-set survey · 4 marks · Short answerHarder board level

In a class of 60 students, 35 like cricket, 30 like football and 10 like neither. Find the number who like both, only cricket and only football.

Try it first, then show the solutionSolution

Like at least one: $60 - 10 = 50$ . (1 mark)
Both: $35 + 30 - 50 = 15$ . (1 mark)
Only cricket: $35 - 15 = 20$ . (1 mark)
Only football: $30 - 15 = 15$ . (1 mark)

Answer: Both 15; only cricket 20; only football 15.

Example 4: Three-set survey · 4 marks · Short answerHarder board level

In a group of 100 students, 40 study maths (M), 35 physics (P) and 30 chemistry (C); 15 study M and P, 10 study P and C, 12 study M and C, and 5 study all three. Find the number who study none, only maths and exactly two subjects.

Try it first, then show the solutionSolution

$n (M \cup P \cup C) = 40 + 35 + 30 - 15 - 10 - 12 + 5 = 73$ ; none: $100 - 73 = 27$ . (1.5 marks)
Only M: $40 - (15 - 5) - (12 - 5) - 5 = 40 - 10 - 7 - 5 = 18$ . (1.5 marks)
Exactly two: $(15 - 5) + (10 - 5) + (12 - 5) = 10 + 5 + 7 = 22$ . (1 mark)

Answer: None 27; only maths 18; exactly two subjects 22.

Example 5: Finding unknown overlaps · 5 marks · Long answerHOTS / competency

n(U) = 60, n(A) = 35, n(B) = 30 and n(A′ ∩ B′) = 10. Find n(A ∩ B), n(A − B) and n(B − A). By De Morgan's law, express A′ ∩ B′ in another form and state the number of elements in n(A′ ∪ B′) if n(A ∩ B) is as found.

Try it first, then show the solutionSolution

$A' \cap B' = (A \cup B)'$ so $n (A \cup B) = 60 - 10 = 50$ . (1.5 marks)
$n (A \cap B) = 35 + 30 - 50 = 15$ . (1 mark)
$n (A - B) = 35 - 15 = 20$ ; $n (B - A) = 30 - 15 = 15$ . (1 mark)
$A' \cup B' = (A \cap B)'$ (De Morgan), so $n (A' \cup B') = 60 - 15 = 45$ . (1.5 marks)

Answer: n(A ∩ B) = 15; n(A − B) = 20; n(B − A) = 15; n(A′ ∪ B′) = 45.

Where students lose marks in Set Operations, De Morgan's Laws and Venn Diagram Problems

The slips examiners see most often, and what to write instead.

Mistake: Subtracting the triple overlap wrongly
Using n(A ∩ B) instead of n(A ∩ B) − n(A ∩ B ∩ C) for "exactly two".
Do this: Exactly two = pair overlap minus the triple intersection.
Usually costs: 2 marks.
Mistake: Forgetting to subtract the neither group
Using n(U) for the union.
Do this: Union = n(U) − n(neither).
Usually costs: 1 mark.
Mistake: Not flipping the sign in De Morgan's law
Writing (A ∪ B)′ = A′ ∪ B′.
Do this: The complement turns ∪ into ∩.
Usually costs: 1 to 2 marks.

Exam tips

Draw and label every region of the Venn diagram.
Check that the regions add up to n(U).
Write the formula before substituting.
Verify a law with a small example when asked.

Practice questions on Set Operations, De Morgan's Laws and Venn Diagram Problems

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
(A ∩ B)′ equals:
(a) A′ ∩ B′
(b) A′ ∪ B′
(c) A ∪ B′
(d) A′ ∩ B
Hint
De Morgan's second law.
Answer
(b) A′ ∪ B′
Q2 · 2 marks · Short answerBoard level
If n(A) = 20, n(B) = 15 and n(A ∪ B) = 30, find n(A ∩ B).
Hint
Rearrange the formula.
Answer
$n (A \cap B) = 20 + 15 - 30 = 5$ .
Q3 · 3 marks · Short answerHarder board level
In a town 70% read newspaper X and 50% read newspaper Y; 30% read both. What percentage read neither?
Hint
Union first.
Answer
Read at least one: $70 + 50 - 30 = 90$ ; neither $= 10$ .
Q4 · 3 marks · Short answerHarder board level
Prove with a Venn diagram or elements that A − (B ∪ C) = (A − B) ∩ (A − C) for A = {1, 2, 3, 4, 5, 6}, B = {2, 4, 6} and C = {3, 6}.
Hint
Evaluate both sides.
Answer
B ∪ C = {2, 3, 4, 6}; A − (B ∪ C) = {1, 5}. A − B = {1, 3, 5}, A − C = {1, 2, 4, 5}, and their intersection is {1, 5}. Both sides are {1, 5}.
Q5 · 4 marks · Short answerHarder board level
Of 120 people, 65 take tea, 55 take coffee and 40 take milk; 25 take tea and coffee, 15 take coffee and milk, 20 take tea and milk, and 8 take all three. How many take none of the three?
Hint
Inclusion-exclusion.
Answer
$n (T \cup C \cup M) = 65 + 55 + 40 - 25 - 15 - 20 + 8 = 108$ . None $= 120 - 108 = 12$ .
Q6 · 4 marks · Case-basedHOTS / competency
Case study: Suppose a college of 200 students offers three optional activities: sports (S), music (M) and drama (D). 90 take S, 70 take M, 60 take D; 30 take S and M, 20 take M and D, 25 take S and D; 10 take all three.
(i) Find how many take at least one. (1 mark)
(ii) Find how many take none. (1 mark)
(iii) Find how many take exactly one. (1 mark)
(iv) Find how many take exactly two. (1 mark)
Hint
Draw the three-circle diagram.
Answer
(i) $90 + 70 + 60 - 30 - 20 - 25 + 10 = 155$ . (ii) $200 - 155 = 45$ . (iii) Only S: $90 - 20 - 15 - 10 = 45$ where 20 = (30 − 10), 15 = (25 − 10); only M: $70 - 20 - 10 - 10 = 30$ ; only D: $60 - 15 - 10 - 10 = 25$ ; total 100. (iv) $(30 - 10) + (20 - 10) + (25 - 10) = 20 + 10 + 15 = 45$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.n(A ∪ B)?Show answer
$n (A) + n (B) - n (A \cap B)$ .
2.(A ∩ B)′?Show answer
A′ ∪ B′.
3.A − B in terms of complement?Show answer
A ∩ B′.
4.n(A′)?Show answer
n(U) − n(A).

Set Operations, De Morgan's Laws and Venn Diagram Problems: frequently asked questions

Why do we subtract n(A ∩ B) in the union formula?

The elements in the overlap are counted once in n(A) and once in n(B), so they must be subtracted once.

Are A − B and B − A equal?

No, in general. A − B = B − A only when A = B, and then both differences are the empty set. For A ≠ B they are different, and they are disjoint.

What is the symmetric difference?

The elements in exactly one of the two sets: (A − B) ∪ (B − A).