MathsICSE Class 8 · Sets

Complement of a Set and Survey Problems with Venn Diagrams (ICSE Class 8)

Updated 17 October 2026 · Fact-checked

The complement $A'$ of a set $A$ is the set of elements of the universal set $U$ that are not in $A$. So $A \cup A' = U$ and $A \cap A' = \varnothing$, and $n(A') = n(U) - n(A)$. In surveys, "neither" means $n(U) - n(A \cup B)$.

The orange area outside both circles is (A ∪ B)′, the 'neither' group; the four regions must add up to n(U).

Illustration drawn for QuestionBuddy.

Understand Complement of a Set and Survey Problems

The CISCE curriculum includes the complement of a set. It only makes sense inside a universal set $U$ , the set of all elements being considered.

Complement. $A^{'} = {x : x \in U, x \in / A}$ . If $U = {1, 2, \dots, 10}$ and $A = {2, 4, 6, 8, 10}$ then $A^{'} = {1, 3, 5, 7, 9}$ .

Facts. $A \cup A^{'} = U$ , $A \cap A^{'} = \emptyset$ , $(A^{'})^{'} = A$ , $U^{'} = \emptyset$ , $\emptyset^{'} = U$ and $n (A^{'}) = n (U) - n (A)$ .

Complement of a union (extension). $(A \cup B)^{'} = A^{'} \cap B^{'}$ : not belonging to $A$ or $B$ means belonging to neither. You can verify it on any example.

Survey problems. A survey of $n (U)$ people about two activities gives four groups: only $A$ , only $B$ , both, and neither. Set up unknowns, fill the both region first, then the "only" regions by subtraction, then check the total:
$ $n (U) = (only A) + (only B) + (both) + (neither) .$ $
When the problem is stated with an algebraic relation between the regions, let the smallest region be $x$ and write the other regions in terms of $x$ , then add them to $n (U)$ .

Reading the words. "At least one" is the union. "Exactly one" is union minus both. "Neither" is the complement of the union. "Only A" is $A$ without $B$ .

Complement of a Set and Survey Problems: formulas to remember

Complement size: $n (A^{'}) = n (U) - n (A)$; Inside the universal set.
Neither: $n ((A \cup B)^{'}) = n (U) - n (A \cup B)$; Outside both circles.
Total of four regions: $n (U) = only A + only B + both + neither$; Use to check Venn counts.

How to solve Complement of a Set and Survey Problems questions

Name the universal set, list or count each set, then use the Venn regions.

1State the universal set $U$ clearly.
2Write the sets in roster form (or give counts).
3For complements, list the elements of $U$ not in the set.
4For surveys, find "both" first from the formula, then the "only" regions.
5Add all four regions and compare with $n (U)$ .

Complement of a Set and Survey 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.

Neither tea nor coffee

Suppose in an office of 60 people, 35 take tea, 28 take coffee and 12 take both.

How many take neither?

Try it first, then show the solutionSolution

At least one: $35 + 28 - 12 = 51$ .
Neither: $60 - 51 = 9$ .

Answer: 9 people.

Memory hook: Neither = everyone minus at-least-one.

Tricks to remember Complement of a Set and Survey Problems

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

Everyone is somewhere
The four regions of the Venn diagram must add up to the total.
Use it when: Checking survey answers.
Complement = left outside
The complement is the part of the rectangle outside the circle.
Use it when: Reading Venn diagrams.

Complement of a Set and Survey Problems: solved examples

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

Example 1: Complement from a list · 1 mark · MCQBoard level

If $U = {1, 2, \dots, 10}$ and $A = {2, 4, 6, 8, 10}$ , then $A^{'}$ is:
(a) ${2, 4, 6, 8, 10}$ (b) ${1, 3, 5, 7, 9}$ (c) ${1, 2, \dots, 10}$ (d) $\emptyset$

Try it first, then show the solutionSolution

$A^{'}$ has the elements of $U$ not in $A$ .

Answer: (b)

Example 2: Two complements · 3 marks · Short answerBoard level

$U = {x : x \in N, x \leq 12}$ , $A$ = multiples of 3 and $B$ = factors of 12. Find $A^{'}$ and $B^{'}$ .

Try it first, then show the solutionSolution

$A = {3, 6, 9, 12}$ ; $B = {1, 2, 3, 4, 6, 12}$ .
$A^{'} = {1, 2, 4, 5, 7, 8, 10, 11}$ .
$B^{'} = {5, 7, 8, 9, 10, 11}$ .

Answer: $A^{'} = {1, 2, 4, 5, 7, 8, 10, 11}$ ; $B^{'} = {5, 7, 8, 9, 10, 11}$ .

Example 3: Complement of a union · 3 marks · Short answerHarder board level

With the sets above, verify that $(A \cup B)^{'} = A^{'} \cap B^{'}$ .

Try it first, then show the solutionSolution

$A \cup B = {1, 2, 3, 4, 6, 9, 12}$ , so $(A \cup B)^{'} = {5, 7, 8, 10, 11}$ .
$A^{'} \cap B^{'} = {5, 7, 8, 10, 11}$ (common to the two lists).
They are equal.

Answer: Both sides $= {5, 7, 8, 10, 11}$ .

Example 4: Standard survey · 4 marks · Short answerHarder board level

In a group of 60 people, 35 take tea, 28 take coffee and 12 take both. Find the number who take (i) only tea, (ii) only coffee, (iii) at least one, (iv) neither.

Try it first, then show the solutionSolution

(i) Only tea: $35 - 12 = 23$ .
(ii) Only coffee: $28 - 12 = 16$ .
(iii) At least one: $23 + 16 + 12 = 51$ (also $35 + 28 - 12$ ).
(iv) Neither: $60 - 51 = 9$ .

Answer: (i) 23; (ii) 16; (iii) 51; (iv) 9

Example 5: Survey with neither given · 4 marks · Case-basedHOTS / competency

Of 100 people, 62 like cricket, 48 like football and 15 like neither. Find (i) the number who like both, (ii) the number who like only cricket, (iii) the number who like only football.

Try it first, then show the solutionSolution

At least one: $100 - 15 = 85$ .
(i) Both $= 62 + 48 - 85 = 25$ .
(ii) Only cricket $= 62 - 25 = 37$ . (iii) Only football $= 48 - 25 = 23$ .
Check: $37 + 23 + 25 + 15 = 100$ .

Answer: (i) 25; (ii) 37; (iii) 23

Where students lose marks in Complement of a Set and Survey Problems

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

Mistake: Taking the complement without a universal set
The complement is only defined relative to the universal set.
Wrong working
"The complement of the even numbers is the odd numbers" when $U$ is ${1, \dots, 20}$ with no mention of limits.
Right working
$A^{'} = {1, 3, \dots, 19}$ , that is, only the odd numbers inside $U$ .
Do this: Write U first.
Usually costs: The answer mark.
Mistake: Counting "at least one" as "exactly one"
Exactly one excludes those in both.
Wrong working
Exactly one $= 51$ (the union).
Right working
Exactly one $= 51 - 12 = 39$ .
Do this: Underline the key phrase in the question.
Usually costs: One part of a multi-part question.

Exam tips

Draw a rectangle for U, with two overlapping circles inside it.
Fill the middle first, then the outer parts.
Check that the four regions add up to the total.

Practice questions on Complement of a Set and Survey 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
For any set $A$ in a universal set $U$ , $A \cup A^{'}$ equals:
(a) $A$ (b) $\emptyset$ (c) $U$ (d) $A^{'}$
Hint
Together, a set and its complement make up everything.
Answer
(c)
Q2 · 2 marks · Short answerBoard level
Let $U$ be the set of distinct letters in the word CHEMISTRY and $A$ the set of vowels in it. Write $A^{'}$ and find $n (A^{'})$ .
Hint
List the letters of U first.
Answer
$U = {C, H, E, M, I, S, T, R, Y}$ , $n (U) = 9$ ; $A = {E, I}$ . $A^{'} = {C, H, M, S, T, R, Y}$ and $n (A^{'}) = 9 - 2 = 7$ .
Q3 · 3 marks · Short answerHarder board level
$U = {1, 2, \dots, 20}$ , $P$ is the set of primes and $Q$ the set of odd numbers. Find $P \cap Q^{'}$ and $(P \cup Q)^{'}$ .
Hint
$Q^{'}$ is the set of even numbers in $U$ .
Answer
$P = {2, 3, 5, 7, 11, 13, 17, 19}$ , $Q^{'} = {2, 4, \dots, 20}$ . $P \cap Q^{'} = {2}$ . $P \cup Q$ = all odd numbers plus $2$ , so $(P \cup Q)^{'} = {4, 6, 8, 10, 12, 14, 16, 18, 20}$ .
Q4 · 4 marks · Short answerHarder board level
Of 80 learners, 50 play chess, 40 play carrom and 20 play neither. Find how many play both, only chess and only carrom.
Hint
First find the number who play at least one.
Answer
At least one: $80 - 20 = 60$ . Both $= 50 + 40 - 60 = 30$ . Only chess $= 20$ ; only carrom $= 10$ . Check: $20 + 10 + 30 + 20 = 80$ .
Q5 · 4 marks · Case-basedHOTS / competency
In a survey of 70 children, the number liking only maths is twice the number liking both maths and science, the number liking only science is 5 more than the number liking both, and 5 like neither. Find the number liking both, and the total liking maths.
Hint
Let the number liking both be x.
Answer
Let both $= x$ : $2 x + x + (x + 5) + 5 = 70$ , so $4 x = 60$ , $x = 15$ . Only maths $30$ , only science $20$ . Liking maths: $30 + 15 = 45$ .

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.What is A ∪ A'?Show answer
The universal set U.
2.What is n(A') ?Show answer
n(U) - n(A).
3.What does "neither" mean in a Venn diagram?Show answer
The part of U outside both circles.
4.What is (A')'?Show answer
A.

Complement of a Set and Survey Problems with Venn Diagrams (ICSE Class 8)

Understand Complement of a Set and Survey Problems

Complement of a Set and Survey Problems: formulas to remember

How to solve Complement of a Set and Survey Problems questions

Complement of a Set and Survey Problems in real life

Tricks to remember Complement of a Set and Survey Problems

Complement of a Set and Survey Problems: solved examples

Where students lose marks in Complement of a Set and Survey Problems

Exam tips

Practice questions on Complement of a Set and Survey Problems

Test your recall

Complement of a Set and Survey Problems: frequently asked questions

Is the complement a subset of U?

What is U' ?