Sets ICSE Class 8: Union, Intersection, Disjoint Sets and Complement

What Sets is about

Weightage: CISCE publishes no question paper, marking scheme or chapter weightage for Class VIII Mathematics: your school sets and marks the exam. Sets sits in the Number System theme of the CISCE Curriculum for Upper Primary Classes (VI-VIII).Source: CISCE Curriculum for Upper Primary Classes (VI-VIII), RDCD 2016

Key concepts

Sets, elements and notation

• Sets are named with capital letters and written in braces: $A=\{2,3,5,7\}$.
• $3\in A$ means "3 is an element of $A$"; $4\notin A$ means "4 is not an element of $A$".
• Roster form lists the elements: $\{2,3,5,7\}$.
• Set-builder form describes them: $\{x:x\text{ is a prime number less than }10\}$.
• The empty set $\varnothing$ or $\{\}$ has no elements.
• $n(A)$ is the number of elements in $A$.
• $A$ is a subset of $B$, written $A\subseteq B$, if every element of $A$ is also in $B$.

The universal set $U$ is the set containing all the elements being talked about in a question.

Worked example

Write $A=\{x:x\text{ is a prime number less than }12\}$ in roster form and find $n(A)$.

Try it first, then show the solutionSolution

1. Primes less than 12: 2, 3, 5, 7, 11 (1 is not prime).
2. $A=\{2,3,5,7,11\}$.
3. $n(A)=5$.

Answer: $A=\{2,3,5,7,11\}$; $n(A)=5$

Union of sets

The union $A\cup B$ is the set of elements that are in $A$, or in $B$, or in both.

To find it, list all the elements of $A$, then add the elements of $B$ that are not already listed. Never repeat a common element.

In a Venn diagram, $A\cup B$ is the whole region covered by the two circles.

Facts: $A\cup B=B\cup A$, $A\cup \varnothing =A$, and $A\cup A=A$.

Worked example

If $A=\{1,2,3,4\}$ and $B=\{3,4,5,6\}$, find $A\cup B$.

Try it first, then show the solutionSolution

1. List $A$: 1, 2, 3, 4.
2. Add the new elements of $B$: 5, 6 (3 and 4 are already there).
3. $A\cup B=\{1,2,3,4,5,6\}$.

Answer: $\{1,2,3,4,5,6\}$

Intersection of sets

The intersection $A\cap B$ is the set of elements that are in both $A$ and $B$.

In a Venn diagram it is the overlapping part of the two circles.

Facts: $A\cap B=B\cap A$, $A\cap \varnothing =\varnothing$, and $A\cap A=A$. If $A\subseteq B$, then $A\cap B=A$ and $A\cup B=B$.

A handy link with factors: the intersection of the sets of factors of two numbers is the set of their common factors, and its largest element is the HCF.

Worked example

If $A=\{1,2,3,4\}$ and $B=\{3,4,5,6\}$, find $A\cap B$.

Try it first, then show the solutionSolution

1. Go through $A$ and keep the elements that are also in $B$.
2. 3 and 4 are in both.

Answer: $\{3,4\}$

Disjoint sets

Two sets are disjoint if they have no element in common, that is, $A\cap B=\varnothing$.

In a Venn diagram, disjoint sets are drawn as two circles that do not overlap.

For disjoint sets, $n(A\cup B)=n(A)+n(B)$, because nothing is counted twice.

Examples: odd and even numbers; vowels and consonants; a set and its complement.

Worked example

Are $A=\{1,3,5,7,9\}$ and $B=\{2,4,6,8\}$ disjoint? Find $n(A\cup B)$.

Try it first, then show the solutionSolution

1. $A$ has only odd numbers and $B$ only even numbers, so $A\cap B=\varnothing$.
2. So they are disjoint, and $n(A\cup B)=5+4=9$.

Answer: Yes, disjoint; $n(A\cup B)=9$.

Universal set and complement

The complement of $A$, written $A^{\prime }$, is the set of all elements of the universal set $U$ that are not in $A$.

In a Venn diagram, $U$ is the rectangle and $A^{\prime }$ is the shaded region inside the rectangle but outside circle $A$.

Properties (always true):

• $A\cup A^{\prime }=U$
• $A\cap A^{\prime }=\varnothing$ (so $A$ and $A^{\prime }$ are disjoint)
• $(A^{\prime }{)}^{\prime }=A$
• $U^{\prime }=\varnothing$ and ${\varnothing }^{\prime }=U$

The complement depends on $U$: change $U$ and $A^{\prime }$ changes too.

Worked example

If $U=\{1,2,3,\dots ,10\}$ and $A=\{2,4,6,8,10\}$, find $A^{\prime }$ and check that $A\cup A^{\prime }=U$.

Try it first, then show the solutionSolution

1. Elements of $U$ not in $A$: 1, 3, 5, 7, 9.
2. $A^{\prime }=\{1,3,5,7,9\}$.
3. $A\cup A^{\prime }=\{1,2,\dots ,10\}=U$ ✓.

Answer: $A^{\prime }=\{1,3,5,7,9\}$

Venn diagrams and counting

Two overlapping circles inside a rectangle split $U$ into four regions:
1. only $A$,
2. both ($A\cap B$),
3. only $B$,
4. neither (outside both circles).

Fill the overlap first, then subtract to fill "only $A$" and "only $B$". This gives
$n(A\cup B)=n(A)+n(B)-n(A\cap B)$,
because the overlap is counted twice when you add $n(A)$ and $n(B)$.

Draw-and-label checklist: rectangle labelled $U$; two overlapping circles labelled $A$ and $B$; a number (or the elements) in each of the four regions; the required region shaded.

Worked example

In a class of 40, 25 students like cricket, 20 like football and 10 like both. How many like at least one game, and how many like neither?

Try it first, then show the solutionSolution

1. Both: 10. Only cricket: $25-10=15$. Only football: $20-10=10$.
2. At least one $=15+10+10=35$ (also $25+20-10=35$).
3. Neither $=40-35=5$.

Answer: 35 like at least one; 5 like neither.

Maths in real life: Sets

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.

Topics in this chapter

• Complement of a Set and Survey Problems
• Union and Intersection of Sets

Sets formula sheet

Union

$A\cup B=\{x:x\in A\text{ or }x\in B\}$

In A, in B, or in both.

Intersection

$A\cap B=\{x:x\in A\text{ and }x\in B\}$

In both.

Complement

$A^{\prime }=\{x:x\in U\text{ and }x\notin A\}$

Depends on U.

Disjoint sets

$A\cap B=\varnothing$

Then n(A ∪ B) = n(A) + n(B).

Counting two sets

$n(A\cup B)=n(A)+n(B)-n(A\cap B)$

Subtract the overlap once.

Complement facts

$A\cup A^{\prime }=U;A\cap A^{\prime }=\varnothing$

Also (A′)′ = A.

How the school exam asks Sets

The question patterns to expect, and the method that works for each.

Tricks to remember Sets

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

• ∪ is a cup, ∩ is a cap

  The union sign is an open cup that collects everything; the intersection sign is a cap that keeps only what is shared.

  Use it when: Not mixing up the symbols.

• OR for union, AND for intersection

  Union: in A OR B. Intersection: in A AND B.

  Use it when: Reading set-builder definitions.

• Overlap first

  In Venn counting, always fill the middle (both) before the 'only' parts.

  Use it when: Survey problems.

• Complement = rest of U

  A′ is 'everything in U except A'. Together, A and A′ make U.

  Use it when: Complement questions and checks.

Sets: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section B style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 10-mark question split into 3 + 3 + 4 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Sets

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

• ✗Mistake: Repeating common elements in a union

  Each element is listed once in a set.

  Wrong working

  $\{1,2,3,4\}\cup \{3,4,5\}=\{1,2,3,4,3,4,5\}$

  Right working

  $\{1,2,3,4,5\}$

  ✓Do this: Cross off each element of B as you check it against A.

  Usually costs: 1 mark.

• ✗Mistake: Writing the complement without looking at U

  A′ is everything in U outside A, not 'the opposite' guessed from words.

  ✓Do this: Write U out in full first, then strike out A.

  Usually costs: 1 mark per complement.

• ✗Mistake: Adding n(A) and n(B) when the sets overlap

  The overlap gets counted twice.

  Wrong working

  At least one $=35+30=65$

  Right working

  At least one $=35+30-12=53$

  ✓Do this: Subtract n(A ∩ B) once, or fill the Venn regions.

  Usually costs: Every later part of a survey question.

• ✗Mistake: Writing 'A only' as n(A)

  n(A) includes the overlap.

  ✓Do this: Only A = n(A) − n(A ∩ B).

  Usually costs: 1 to 2 marks.

• ✗Mistake: Using round brackets or forgetting braces

  Sets are written in braces { }.

  ✓Do this: Write every set as { ... }; write the empty set as ∅ or { }, never {0}.

  Usually costs: Presentation marks.

Practice questions on Sets

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

1. Q1 · 1 mark · MCQExam level

   Which pair of sets is disjoint?
   (a) $\{1,2\}$ and $\{2,3\}$
   (b) $\{1,2\}$ and $\{3,4\}$
   (c) $\{a,b\}$ and $\{b,a\}$
   (d) $\{0\}$ and $\{0,1\}$

   Hint

   Look for no common element.

   Answer

   (b)

2. Q2 · 2 marks · Short answerExam level

   $A$ = the set of letters of the word SCHOOL and $B$ = the set of letters of the word CLASS. Find $A\cup B$ and $A\cap B$.

   Hint

   Write each letter only once.

   Answer

   $A=\{S,C,H,O,L\}$, $B=\{C,L,A,S\}$. $A\cup B=\{S,C,H,O,L,A\}$; $A\cap B=\{S,C,L\}$.

3. Q3 · 1 mark · Very short answerExam level

   If $U=\{1,2,\dots ,10\}$ and $A=\{1,3,5,7,9\}$, find $A^{\prime }$.

   Hint

   Strike out A from U.

   Answer

   $A^{\prime }=\{2,4,6,8,10\}$.

4. Q4 · 2 marks · Short answerHarder exam level

   If $U=\{a,b,c,d,e\}$ and $A=\{a,c\}$, verify that $A\cup A^{\prime }=U$ and $A\cap A^{\prime }=\varnothing$.

   Hint

   Find A′ first.

   Answer

   $A^{\prime }=\{b,d,e\}$. $A\cup A^{\prime }=\{a,b,c,d,e\}=U$ and $A\cap A^{\prime }=\varnothing$ ✓.

5. Q5 · 1 mark · Assertion-reasonHarder exam level

   Assertion (A): If $A$ and $B$ are disjoint sets, then $A\cap B=\varnothing$.
   Reason (R): Disjoint sets have no element in common.
   (a) (A) is true and (R) is false.
   (b) (A) is false and (R) is true.
   (c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
   (d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).

   Hint

   What is in the intersection of disjoint sets?

   Answer

   (c)

6. Q6 · 2 marks · Short answerHarder exam level

   If $n(A)=18$, $n(B)=25$ and $n(A\cup B)=35$, find $n(A\cap B)$.

   Hint

   Rearrange the counting formula.

   Answer

   $n(A\cap B)=18+25-35=8$.

7. Q7 · 3 marks · Short answerHarder exam level

   $A=\{x:x\text{ is a factor of }12\}$ and $B=\{x:x\text{ is a factor of }18\}$. Find $A\cap B$. What is its largest element called?

   Hint

   List the factors.

   Answer

   $A=\{1,2,3,4,6,12\}$, $B=\{1,2,3,6,9,18\}$. $A\cap B=\{1,2,3,6\}$, the common factors; the largest, 6, is the HCF.

8. Q8 · 3 marks · Short answerHarder exam level

   If $U=\{1,2,3,4,5,6\}$, $A=\{1,2,3\}$ and $B=\{2,3,4,5\}$, verify that $(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }$.

   Hint

   Work out each side separately.

   Answer

   $A\cap B=\{2,3\}$, so $(A\cap B{)}^{\prime }=\{1,4,5,6\}$. $A^{\prime }=\{4,5,6\}$, $B^{\prime }=\{1,6\}$, $A^{\prime }\cup B^{\prime }=\{1,4,5,6\}$ ✓.

9. Q9 · 3 marks · Short answerHarder exam level

   In a class of 45 students, 25 take music, 18 take art and 8 take both. How many take (i) music only (ii) neither?

   Hint

   Fill the overlap first.

   Answer

   (i) $25-8=17$. (ii) At least one $=25+18-8=35$; neither $=45-35=10$.

10. Q10 · 2 marks · Short answerHarder exam level

    If $A=\{2,4\}$ and $B=\{1,2,3,4\}$, find $A\cup B$ and $A\cap B$. What do you notice, and why?

    Hint

    Is every element of A in B?

    Answer

    $A\cup B=\{1,2,3,4\}=B$ and $A\cap B=\{2,4\}=A$. This happens because $A\subseteq B$.

11. Q11 · 4 marks · Long answerHOTS / competency

    In a group of 50 children, 30 like apples and 25 like bananas. Every child likes at least one of the two fruits. How many like both, apples only and bananas only?

    Hint

    Here n(A ∪ B) = 50.

    Answer

    Both $=30+25-50=5$. Apples only $=25$; bananas only $=20$. Check: $25+5+20=50$ ✓.

12. Q12 · 3 marks · Short answerHOTS / competency

    Let $U=\{1,2,\dots ,30\}$, $A$ = multiples of 4 in $U$ and $B$ = multiples of 6 in $U$. Find $A\cap B$ and describe it using the LCM of 4 and 6.

    Hint

    Common multiples.

    Answer

    $A=\{4,8,12,16,20,24,28\}$, $B=\{6,12,18,24,30\}$. $A\cap B=\{12,24\}$: the multiples of 12, the LCM of 4 and 6, in $U$.

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. 1.What does A ∪ B contain?Show answer

   Every element that is in A, in B, or in both.

2. 2.What does A ∩ B contain?Show answer

   Only the elements common to A and B.

3. 3.When are two sets disjoint?Show answer

   When A ∩ B = ∅.

4. 4.What is A ∪ A′?Show answer

   The universal set U.

5. 5.What is A ∩ A′?Show answer

   The empty set ∅.

6. 6.If A ⊆ B, what is A ∩ B?Show answer

   A.

7. 7.Is {0} the empty set?Show answer

   No: it has one element, 0.

8. 8.n(A) = 10, n(B) = 8, n(A ∩ B) = 3. Find n(A ∪ B).Show answer

   15.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Can you convert between set-builder and roster form?
• ☐Do you list each element only once in a union?
• ☐Can you shade A ∪ B, A ∩ B and A′ in a Venn diagram?
• ☐Do you write U out before finding a complement?
• ☐Do you fill the overlap first in survey problems?
• ☐Can you check that A ∪ A′ = U and A ∩ A′ = ∅?

Sets: frequently asked questions