Sets Class 11: Notes, Venn Diagrams and Solved Questions

What Sets is about

Weightage: This chapter is part of Unit I, Sets and Functions, which the syllabus gives 23 of the 80 theory marks in the school annual exam (CBSE adds that there is no fixed chapter-wise weightage).Source: CBSE Mathematics (041) syllabus 2026-27, Class XI course structure, Unit I Sets and Functions

Key concepts

Sets and their representations

NCERT section 1.2

A set is a well-defined collection of distinct objects, called its elements or members. We write $a\in A$ for 'a belongs to A' and $b\notin A$ for 'b does not belong to A'. Sets are named with capital letters.

Standard sets: N (natural numbers), Z (integers), Q (rationals), R (real numbers), $Z^{+}$ (positive integers).

Two ways to write a set:

• Roster (tabular) form: list the elements in braces, separated by commas: $\{2,3,5,7\}$. Order does not matter and repeated elements are written once (the letters of 'SCHOOL' form $\{S,C,H,O,L\}$).
• Set-builder form: describe the common property: $\{x:x\text{ is a prime number less than }10\}$, read 'the set of all x such that…'.

Infinite sets can be written in roster form only if a clear pattern can be shown, e.g. $\{1,3,5,\dots \}$.

Worked example

Write $A=\{x:x\text{ is a prime number less than }20\}$ in roster form, and $B=\{1,4,9,16,25\}$ in set-builder form. [2 marks]

Try it first, then show the solutionSolution

1. A = {2, 3, 5, 7, 11, 13, 17, 19}
2. B is the squares of 1 to 5: $B=\{x:x=n^2,n\in N,1\le n\le 5\}$

Answer: A = {2, 3, 5, 7, 11, 13, 17, 19}; $B=\{x:x=n^2,n\in N,n\le 5\}$

Empty, finite, infinite and equal sets

NCERT section 1.3 to 1.5

• Empty (null, void) set φ or { }: a set with no elements, e.g. $\{x:x\in R,x^2=-1\}$ or the set of even primes greater than 2. Note $\{0\}$ and $\{\varphi \}$ are not empty; each has one element.
• Finite set: empty, or its elements can be counted and the counting ends. $n(A)$ denotes the number of elements.
• Infinite set: not finite, e.g. N, the points on a line, the set of circles through a point.
• Equal sets: A = B if they have exactly the same elements: every element of A is in B and every element of B is in A. Order and repetition do not matter: $\{1,2,2,3\}=\{3,2,1\}$.

Worked example

Which of these are empty, finite or infinite? (a) $\{x:x\in N,2<x<3\}$ (b) the set of months with 31 days (c) $\{x:x\text{ is a multiple of }5\}$ [1 mark]

Try it first, then show the solutionSolution

1. (a) No natural number lies strictly between 2 and 3: empty
2. (b) Seven months: finite; (c) multiples of 5 never end: infinite

Answer: (a) empty (b) finite, n = 7 (c) infinite

Subsets

NCERT section 1.6

A is a subset of B, written $A\subset B$, if every element of A is also an element of B: $A\subset B$ if $a\in A\Rightarrow a\in B$.

• Every set is a subset of itself: $A\subset A$.
• The empty set is a subset of every set: $\varphi \subset A$.
• If $A\subset B$ and $A\ne B$, A is a proper subset of B, and B is a superset of A.
• $A=B$ exactly when $A\subset B$ and $B\subset A$ (a standard way to prove two sets equal).
• A set with one element is a singleton.

Important chain: $N\subset Z\subset Q\subset R$, and the irrationals T satisfy $T\subset R$ with $Q\cap T=\varphi$.

Do not confuse ∈ and ⊂: $2\in \{1,2,3\}$ but $\{2\}\subset \{1,2,3\}$.

Worked example

State true or false: (a) $\{a\}\in \{a,b,c\}$ (b) $\varphi \subset \{1,2\}$ (c) $\{1,2\}\subset \{1,2\}$ (d) $\{x:x\text{ is an even prime}\}\subset \{1,2,3\}$. [2 marks]

Try it first, then show the solutionSolution

1. (a) {a} is a set, not an element of {a, b, c}: false (it should be ⊂)
2. (b) True (φ is a subset of every set); (c) true (every set is a subset of itself); (d) {2} ⊂ {1, 2, 3}: true

Answer: (a) F (b) T (c) T (d) T

Intervals as subsets of R

NCERT section 1.6.1 and 1.6.2

For real numbers $a<b$:

• Open interval $(a,b)=\{x:a<x<b\}$: end points not included.
• Closed interval $[a,b]=\{x:a\le x\le b\}$: both end points included.
• Half-open $[a,b)=\{x:a\le x<b\}$ and $(a,b]=\{x:a<x\le b\}$.
• Unbounded: $(a,\infty )=\{x:x>a\}$, $(-\infty ,b]=\{x:x\le b\}$, $(-\infty ,\infty )=R$. Infinity is never included, so it always takes a round bracket.

The length of each of these intervals (a to b) is $b-a$.

On the number line a filled dot shows an included end point and an open circle an excluded one. Intervals are used constantly for domains and ranges of functions and for solutions of inequalities.

Worked example

Write in interval form: (a) $\{x:-3<x\le 5\}$ (b) $\{x:x\ge 2\}$. Write $[-1,4)$ in set-builder form. [2 marks]

Try it first, then show the solutionSolution

1. (a) $(-3,5]$; (b) $[2,\infty )$
2. $[-1,4)=\{x:x\in R,-1\le x<4\}$

Answer: (a) (−3, 5] (b) [2, ∞); {x : −1 ≤ x < 4}

Universal set and Venn diagrams

NCERT section 1.7 and 1.8

In a given discussion, the universal set U is the set that contains all the sets under consideration as subsets. It depends on the context: for studying triangles, U can be the set of all triangles in a plane; for number sets, U might be R.

Venn diagrams (after John Venn) picture sets as closed regions: U is a rectangle and its subsets are circles inside it. Elements are written as points inside the regions. Overlapping circles show common elements; a circle inside another shows a subset; separate circles show disjoint sets.

Venn diagrams are a quick way to check set identities (shade both sides and compare) and to see which region a description refers to.

Worked example

Let U = {1, 2, …, 10}, A = {2, 4, 6, 8, 10} and B = {4, 8}. Describe how a Venn diagram shows A and B. [1 mark]

Try it first, then show the solutionSolution

1. Every element of B is in A, so B ⊂ A: draw circle B inside circle A, inside rectangle U.
2. Place 4, 8 in B; 2, 6, 10 in A outside B; 1, 3, 5, 7, 9 in U outside A.

Answer: Circle B inside circle A inside rectangle U, with the elements placed as above.

Union, intersection and difference of sets

NCERT section 1.9

• Union $A\cup B=\{x:x\in A\text{ or }x\in B\}$: elements in A, in B, or in both.
• Intersection $A\cap B=\{x:x\in A\text{ and }x\in B\}$: common elements. If $A\cap B=\varphi$, A and B are disjoint.
• Difference $A-B=\{x:x\in A\text{ and }x\notin B\}$: in A but not in B. In general $A-B\ne B-A$. The sets $A-B$, $A\cap B$ and $B-A$ are pairwise disjoint and their union is $A\cup B$.

Laws:

• Commutative: $A\cup B=B\cup A$, $A\cap B=B\cap A$.
• Associative: $(A\cup B)\cup C=A\cup (B\cup C)$, and the same for ∩.
• Identity: $A\cup \varphi =A$, $A\cap U=A$. Idempotent: $A\cup A=A$, $A\cap A=A$. $U\cup A=U$, $\varphi \cap A=\varphi$.
• Distributive: $A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$ and $A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$.

Worked example

If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find $A\cup B$, $A\cap B$, $A-B$ and $B-A$. [2 marks]

Try it first, then show the solutionSolution

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

Answer: {1, 2, 3, 4, 5, 6}; {3, 4}; {1, 2}; {5, 6}

Complement of a set and its properties

NCERT section 1.10

If U is the universal set and $A\subset U$, the complement of A is
$A^{\prime }=\{x:x\in U\text{ and }x\notin A\}=U-A$

Properties:
1. Complement laws: $A\cup A^{\prime }=U$, $A\cap A^{\prime }=\varphi$.
2. De Morgan's laws: $(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }$ and $(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }$. In words: the complement of a union is the intersection of the complements, and the complement of an intersection is the union of the complements.
3. Double complement: $(A^{\prime }{)}^{\prime }=A$.
4. ${\varphi }^{\prime }=U$ and $U^{\prime }=\varphi$.

Also useful: $A-B=A\cap B^{\prime }$.

Worked example

Let U = {1, 2, …, 8}, A = {1, 2, 3} and B = {3, 4, 5}. Verify that $(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }$. [2 marks]

Try it first, then show the solutionSolution

1. $A\cup B=\{1,2,3,4,5\}$, so $(A\cup B{)}^{\prime }=\{6,7,8\}$
2. $A^{\prime }=\{4,5,6,7,8\}$, $B^{\prime }=\{1,2,6,7,8\}$, so $A^{\prime }\cap B^{\prime }=\{6,7,8\}$: equal ✓

Answer: Both sides equal {6, 7, 8}.

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

• Set Operations, Complements and De Morgan's Laws
• Subsets, Intervals and Set-Builder Notation

Sets formula sheet

Union, intersection, difference

$A\cup B=\{x:x\in A\text{ or }x\in B\},A\cap B=\{x:x\in A\text{ and }x\in B\},A-B=A\cap B^{\prime }$

Complement

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

Complement laws

$A\cup A^{\prime }=U,A\cap A^{\prime }=\varphi ,(A^{\prime }{)}^{\prime }=A,{\varphi }^{\prime }=U,U^{\prime }=\varphi$

De Morgan's laws

$(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime },(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }$

Distributive laws

$A\cap (B\cup C)=(A\cap B)\cup (A\cap C),A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$

Intervals

$(a,b)=\{x:a<x<b\},[a,b]=\{x:a\le x\le b\}$

Half-open: [a, b) and (a, b]; length b − a.

Equality of sets

$A=B⟺A\subset B\text{ and }B\subset A$

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.

• ∪ = or, ∩ = and

  Union collects everything from either set (cup holds more); intersection keeps only what is shared (cap is narrower).

  Use it when: Reading set expressions.

• Break the bar, flip the sign

  De Morgan: take the complement inside and change ∪ to ∩ or ∩ to ∪.

  Use it when: Complement of unions and intersections.

• Element in, set sub

  ∈ for an element of a set, ⊂ for a set inside a set.

  Use it when: True/false items.

• Infinity never sits in a square bracket

  $[2,\infty )$, never $[2,\infty ]$.

  Use it when: Interval notation.

• Minus means 'and not'

  $A-B=A\cap B^{\prime }$.

  Use it when: Proofs with differences.

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 E style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 4-mark question split into 1 + 1 + 1 + 1 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: Writing repeated elements in roster form

  Copying letters of a word.

  Wrong working

  Letters of APPLE: {A, P, P, L, E}

  Right working

  {A, P, L, E}

  ✓Do this: List each distinct element once.

  Usually costs: 1 mark.

• ✗Mistake: Confusing ∈ and ⊂

  Thinking braces don't matter.

  ✓Do this: $a\in A$ but $\{a\}\subset A$.

  Usually costs: 1 mark per item.

• ✗Mistake: Treating {0} or {φ} as empty

  Seeing zero or φ inside.

  ✓Do this: Each has one element, so neither is empty.

  Usually costs: 1 mark.

• ✗Mistake: Assuming A − B = B − A

  Treating difference like intersection.

  ✓Do this: Difference depends on order: $A-B$ keeps A's elements not in B.

  Usually costs: 1 mark.

• ✗Mistake: Wrong brackets for intervals

  Mixing up 'less than' and 'less than or equal to'.

  ✓Do this: < or > gives a round bracket; ≤ or ≥ gives a square bracket.

  Usually costs: 1 mark.

• ✗Mistake: Writing power-set answers

  Old notes.

  ✓Do this: The power set was removed for 2026-27.

  Usually costs: Time.

NCERT exercise guide: Sets

What each exercise in the NCERT chapter tests, how to approach it, and a question of the same kind solved step by step. Questions here are written in our own words.

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 of these is a proper subset of {1, 2, 3}?
   (a) {1, 2, 3}
   (b) {1, 4}
   (c) {2, 3}
   (d) {0}

   Hint

   Subset, but not equal.

   Answer

   (c) {2, 3}

2. Q2 · 1 mark · MCQExam level

   The set $\{x:x\in R,3\le x<7\}$ in interval form is:
   (a) (3, 7)
   (b) [3, 7]
   (c) [3, 7)
   (d) (3, 7]

   Hint

   ≤ gives a square bracket.

   Answer

   (c) [3, 7)

3. Q3 · 1 mark · Assertion-reasonExam level

   Assertion (A): The empty set is a subset of every set.
   Reason (R): The empty set has no element that fails to belong to any given set.
   (a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.

   Hint

   Definition of subset.

   Answer

   (a)

4. Q4 · 1 mark · Very short answerExam level

   Write the set of letters of the word 'MATHEMATICS' in roster form and state its number of elements.

   Hint

   Repeat letters once only.

   Answer

   {M, A, T, H, E, I, C, S}; 8 elements.

5. Q5 · 2 marks · Very short answerHarder exam level

   If U = {1, 2, …, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}, find $(A\cup B{)}^{\prime }$ and $A^{\prime }\cap B^{\prime }$.

   Hint

   De Morgan.

   Answer

   $A\cup B=\{2,3,4,5,6,7,8\}$, $(A\cup B{)}^{\prime }=\{1,9\}$; $A^{\prime }=\{1,3,5,7,9\}$, $B^{\prime }=\{1,4,6,8,9\}$, $A^{\prime }\cap B^{\prime }=\{1,9\}$.

6. Q6 · 2 marks · Very short answerHarder exam level

   Find $A\cap B$ and $A\cup B$ for A = (−∞, 2] and B = (0, ∞).

   Hint

   Number line.

   Answer

   $A\cap B=(0,2]$; $A\cup B=(-\infty ,\infty )=R$.

7. Q7 · 2 marks · Very short answerHarder exam level

   Are the sets A = {x : x is a solution of $x^2-5x+6=0$} and B = {2, 3} equal? Justify.

   Hint

   Solve the quadratic.

   Answer

   $x^2-5x+6=(x-2)(x-3)=0$ gives x = 2, 3. So A = {2, 3} = B.

8. Q8 · 2 marks · Very short answerHarder exam level

   Draw Venn diagrams to show (a) $(A\cap B{)}^{\prime }$ (b) $A^{\prime }\cup B^{\prime }$, and state what you notice.

   Hint

   Shade outside the overlap.

   Answer

   Both shade everything in U except the overlap of A and B. They are the same region, illustrating De Morgan's law $(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }$.

9. Q9 · 3 marks · Short answerHarder exam level

   Prove that $A\cap B=A$ if and only if $A\subset B$.

   Hint

   Two directions.

   Answer

   If $A\cap B=A$: any $x\in A$ lies in $A\cap B$, so $x\in B$; thus $A\subset B$. Conversely, if $A\subset B$: $A\cap B\subset A$ always, and any $x\in A$ is in B, so $x\in A\cap B$; thus $A\subset A\cap B$. Hence $A\cap B=A$.

10. Q10 · 3 marks · Short answerHOTS / competency

    Show that $A\cup (A\cap B)=A$ for any sets A and B (absorption law).

    Hint

    A ∩ B ⊂ A.

    Answer

    Since $A\cap B\subset A$, every element of $A\cup (A\cap B)$ is in A, so LHS ⊂ A. Also $A\subset A\cup (A\cap B)$. Hence the two are equal.

11. Q11 · 2 marks · Very short answerHOTS / competency

    Give an example of three sets A, B, C such that $A\cap B$, $B\cap C$ and $A\cap C$ are non-empty but $A\cap B\cap C=\varphi$.

    Hint

    Make each pair share a different element.

    Answer

    A = {1, 2}, B = {2, 3}, C = {1, 3}: $A\cap B=\{2\}$, $B\cap C=\{3\}$, $A\cap C=\{1\}$, but no element is in all three.

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 is a set?Show answer

   A well-defined collection of distinct objects.

2. 2.Is {0} an empty set?Show answer

   No, it has one element, 0.

3. 3.Write {x : 1 < x ≤ 4, x ∈ R} as an interval.Show answer

   (1, 4].

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

   When their intersection is empty.

5. 5.State De Morgan's laws.Show answer

   $(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }$ and $(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }$.

6. 6.Write A − B using complement.Show answer

   $A-B=A\cap B^{\prime }$.

7. 7.Is φ a subset of every set?Show answer

   Yes.

8. 8.What is $A\cup A^{\prime }$?Show answer

   U, the universal set.

Quick revision checklist

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

• ☐Can you decide whether a given collection is a set?
• ☐Can you convert between roster and set-builder forms?
• ☐Can you identify empty, finite, infinite and equal sets?
• ☐Can you use ∈ and ⊂ correctly?
• ☐Can you write sets as intervals and draw them?
• ☐Can you find unions and intersections of intervals?
• ☐Can you find A ∪ B, A ∩ B, A − B and A′ for given sets?
• ☐Can you shade Venn diagrams for expressions such as (A ∩ B)′?
• ☐Can you verify De Morgan's laws for given sets?
• ☐Can you prove a simple set identity by the element method?

Sets: frequently asked questions