MathsICSE Class 8 · Sets

Union, Intersection and Disjoint Sets: ICSE Class 8 Maths Explained

Updated 17 October 2026 · Fact-checked

The union $A \cup B$ has the elements in $A$ or $B$ or both, each written once. The intersection $A \cap B$ has the elements common to both. If $A \cap B = \varnothing$ the sets are disjoint. Always $n(A \cup B) = n(A) + n(B) - n(A \cap B)$.

The common element 3 sits in the overlap, so it is counted once in the union and subtracted once in 3 + 2 − 1 = 4.

Illustration drawn for QuestionBuddy.

Understand Union and Intersection of Sets

The CISCE curriculum covers the union and intersection of sets, and disjoint sets. In what follows, $A$ and $B$ are sets, and $n (A)$ is the number of elements of $A$ .

Union. $A \cup B$ lists every element that lies in $A$ , in $B$ , or in both. Write each element once, even if it is common: ${1, 2, 3} \cup {3, 4} = {1, 2, 3, 4}$ .

Intersection. $A \cap B$ keeps only the elements that are in both $A$ and $B$ : ${1, 2, 3} \cap {3, 4} = {3}$ .

Disjoint sets. If $A \cap B = \emptyset$ , the sets have no common element, such as the odd numbers and the even numbers.

Counting formula. Adding $n (A) + n (B)$ counts the common elements twice, so subtract once:
$ $n (A \cup B) = n (A) + n (B) - n (A \cap B) .$ $
For disjoint sets, $n (A \cup B) = n (A) + n (B)$ . The regions of a two-set Venn diagram are: only $A$ , only $B$ , both, and (outside) neither.

Properties. $A \cup B = B \cup A$ , $A \cap B = B \cap A$ , $A \cup \emptyset = A$ , $A \cap \emptyset = \emptyset$ , and $A \cap B \subseteq A \subseteq A \cup B$ .

Extension for practice. For three sets, $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ . This distributive law is beyond the stated Class VIII wording and is used only to strengthen skill.

Set-builder to roster. Convert descriptions such as "multiples of 4 less than 30" into lists first; then combine.

Union and Intersection of Sets: formulas to remember

Counting union: $n (A \cup B) = n (A) + n (B) - n (A \cap B)$; Subtract the common elements once.
Disjoint sets: $A \cap B = \emptyset \Rightarrow n (A \cup B) = n (A) + n (B)$; No overlap, no subtraction.
Only in A: $n (A) - n (A \cap B)$; Region of the Venn diagram.

How to solve Union and Intersection of Sets questions

List the sets first, then combine them with the right operation.

1Write each set in roster form.
2For union, list all elements once; for intersection, list only the common ones.
3Check for disjointness by testing whether the intersection is empty.
4For counts, draw a two-circle Venn diagram and fill the overlap first.
5Verify using the counting formula.

Union and Intersection of Sets 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.

Two clubs, one list

Suppose 14 students are in the music club and 9 are in the drama club, and 5 of them are in both.

How many different students are in at least one club?

Try it first, then show the solutionSolution

$n (M \cup D) = 14 + 9 - 5 = 18$ .

Answer: 18 students.

Memory hook: Those in both were counted twice, so subtract them once.

Tricks to remember Union and Intersection of Sets

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

Union is "or", intersection is "and"
U looks like a cup that collects everything; the overlap is the "and" region.
Use it when: Reading symbols.
Overlap first
In a Venn diagram, fill the middle region before the outer parts.
Use it when: Counting problems.

Union and Intersection of Sets: solved examples

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

Example 1: Intersection from lists · 1 mark · MCQBoard level

If $A = {1, 2, 3, 4}$ and $B = {3, 4, 5, 6}$ , then $A \cap B$ is:
(a) ${3, 4}$ (b) ${1, 2, 5, 6}$ (c) ${1, 2, 3, 4, 5, 6}$ (d) $\emptyset$

Try it first, then show the solutionSolution

The common elements are $3$ and $4$ .

Answer: (a)

Example 2: Roster from descriptions · 3 marks · Short answerBoard level

Let $A$ be the set of factors of 12 and $B$ the set of prime numbers less than 10. Find $A \cup B$ and $A \cap B$ .

Try it first, then show the solutionSolution

$A = {1, 2, 3, 4, 6, 12}$ , $B = {2, 3, 5, 7}$ .
$A \cup B = {1, 2, 3, 4, 5, 6, 7, 12}$ .
$A \cap B = {2, 3}$ .

Answer: $A \cup B = {1, 2, 3, 4, 5, 6, 7, 12}$ ; $A \cap B = {2, 3}$ .

Example 3: Multiples and disjointness · 3 marks · Short answerHarder board level

$P$ is the set of multiples of 4 less than 30 and $Q$ the set of multiples of 6 less than 30. Are $P$ and $Q$ disjoint? Find $n (P \cup Q)$ .

Try it first, then show the solutionSolution

$P = {4, 8, 12, 16, 20, 24, 28}$ , $n (P) = 7$ . $Q = {6, 12, 18, 24}$ , $n (Q) = 4$ .
$P \cap Q = {12, 24}$ , not empty, so not disjoint.
$n (P \cup Q) = 7 + 4 - 2 = 9$ .

Answer: Not disjoint; $n (P \cup Q) = 9$ .

Example 4: Working backwards with the formula · 4 marks · Short answerHarder board level

In a hostel, 25 students play cricket, 18 play badminton and 36 play at least one of the two. How many play both, only cricket, and only badminton?

Try it first, then show the solutionSolution

$36 = 25 + 18 - n (A \cap B)$ , so $n (A \cap B) = 7$ .
Only $A$ : $25 - 7 = 18$ . Only $B$ : $18 - 7 = 11$ .
Check: $18 + 11 + 7 = 36$ .

Answer: 7; 18; 11

Example 5: A distributive check · 4 marks · Long answerHOTS / competency

For $A = {1, 2, 3, 4, 5, 6}$ , $B = {2, 4, 6, 8}$ , $C = {3, 6, 9}$ , verify that $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ .

Try it first, then show the solutionSolution

$B \cup C = {2, 3, 4, 6, 8, 9}$ , so $A \cap (B \cup C) = {2, 3, 4, 6}$ .
$A \cap B = {2, 4, 6}$ and $A \cap C = {3, 6}$ .
$(A \cap B) \cup (A \cap C) = {2, 3, 4, 6}$ . Both sides are equal.

Answer: Both sides $= {2, 3, 4, 6}$ .

Where students lose marks in Union and Intersection of Sets

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

Mistake: Writing common elements twice in a union
A set lists each element once.
Wrong working
${1, 2, 3} \cup {3, 4} = {1, 2, 3, 3, 4}$ .
Right working
${1, 2, 3, 4}$ .
Do this: Cross off repeated elements.
Usually costs: The answer mark.
Mistake: Adding n(A) and n(B) only
This counts the overlap twice.
Wrong working
$n (A \cup B) = 14 + 9 = 23$ .
Right working
$14 + 9 - 5 = 18$ .
Do this: Fill the intersection first in the Venn diagram.
Usually costs: Method and answer marks.

Exam tips

Write the roster of each set before combining.
Name the regions of the Venn diagram in your answer for counting problems.
Use curly braces and commas; examiners deduct for brackets of the wrong type.

Practice questions on Union and Intersection of Sets

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
Which pair of sets is disjoint?
(a) ${1, 2, 3}, {3, 4, 5}$ (b) ${2, 4, 6}, {1, 3, 5}$ (c) ${a, b}, {b, c}$ (d) ${1, 2}, {2, 3, 4}$
Hint
Disjoint means no common element.
Answer
(b)
Q2 · 2 marks · Short answerBoard level
Let $A$ be the set of letters of MATHS and $B$ the set of letters of SMART. Find $A \cap B$ and $A \cup B$ .
Hint
Write each set once with no repeats.
Answer
$A = {M, A, T, H, S}$ , $B = {S, M, A, R, T}$ . $A \cap B = {M, A, T, S}$ ; $A \cup B = {M, A, T, H, S, R}$ .
Q3 · 3 marks · Short answerHarder board level
$A$ is the set of natural numbers less than 8, and $B$ is the set of even natural numbers up to 12. Find $A \cap B$ and $A \cup B$ .
Hint
List A and B first.
Answer
$A = {1, \dots, 7}$ , $B = {2, 4, 6, 8, 10, 12}$ . $A \cap B = {2, 4, 6}$ ; $A \cup B = {1, 2, 3, 4, 5, 6, 7, 8, 10, 12}$ .
Q4 · 4 marks · Short answerHarder board level
$n (A) = 40$ , $n (B) = 27$ and $n (A \cap B) = 12$ . Find $n (A \cup B)$ and the number of elements in exactly one of the two sets.
Hint
Exactly one = union minus both.
Answer
$n (A \cup B) = 40 + 27 - 12 = 55$ . Exactly one: $55 - 12 = 43$ (also $28 + 15 = 43$ ).
Q5 · 4 marks · Long answerHOTS / competency
Given $A \cup B = {1, 2, 3, 4, 5, 6, 7, 8}$ , $A \cap B = {3, 6}$ , $n (A) = n (B) = 5$ , and $1, 2, 4 \in A$ . Find $A$ and $B$ .
Hint
Place the elements of the union into only-A, only-B or both.
Answer
Both: $3, 6$ . $A$ has 3 more elements: $1, 2, 4$ , so $A = {1, 2, 3, 4, 6}$ . The rest $5, 7, 8$ go to $B$ only, so $B = {3, 5, 6, 7, 8}$ (5 elements).

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.Formula for n(A ∪ B)?Show answer
n(A) + n(B) - n(A ∩ B).
2.What are disjoint sets?Show answer
Sets with no common element, A ∩ B = ∅.
3.Is A ∩ B a subset of A?Show answer
Yes.
4.What is A ∪ ∅?Show answer
A.

Union, Intersection and Disjoint Sets: ICSE Class 8 Maths Explained

Understand Union and Intersection of Sets

Union and Intersection of Sets: formulas to remember

How to solve Union and Intersection of Sets questions

Union and Intersection of Sets in real life

Tricks to remember Union and Intersection of Sets

Union and Intersection of Sets: solved examples

Where students lose marks in Union and Intersection of Sets

Exam tips

Practice questions on Union and Intersection of Sets

Test your recall

Union and Intersection of Sets: frequently asked questions

What if both sets are equal?

Can disjoint sets have n(A ∪ B) different from n(A) + n(B)?