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Sets

NCERT Class 11 · Mathematics Based on NCERT Class 11 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter introduces the fundamental concept of sets, which is the building block of modern mathematics. It covers how to define and represent sets, classify them as empty, finite, or infinite, and perform operations like union, intersection, difference, and complement. The one big idea a student must walk away with is that a set is a well-defined collection of distinct objects, and all operations on sets follow specific laws that mirror arithmetic but with unique properties.
Key Concepts (17)
Set
A well-defined collection of objects.
e.g. The collection of all rivers of India is a set because we can definitely decide whether a given river belongs to it or not.
Element
An object belonging to a set.
e.g. In the set V of vowels in the English alphabet, 'a' is an element.
Roster form
A method of representing a set by listing all its elements, separated by commas, enclosed within braces { }.
e.g. The set of all even positive integers less than 7 is {2, 4, 6}.
Set-builder form
A method of representing a set by stating the common property possessed by all its elements.
e.g. V = {x : x is a vowel in the English alphabet}.
Empty set (Null set / Void set)
A set which does not contain any element, denoted by φ or { }.
e.g. The set of natural numbers between 1 and 2 is empty because there is no natural number between 1 and 2.
Finite set
A set which is empty or consists of a definite number of elements.
e.g. The set of days of the week is finite.
Infinite set
A set which is not finite; it has an indefinite number of elements.
e.g. The set of natural numbers {1, 2, 3, ...} is infinite.
Equal sets
Two sets A and B are said to be equal if they have exactly the same elements, written as A = B.
e.g. A = {1, 2, 3, 4} and B = {3, 1, 4, 2} are equal sets.
Subset
A set A is said to be a subset of a set B if every element of A is also an element of B, written as A ⊂ B.
e.g. The set of rational numbers Q is a subset of the set of real numbers R.
Proper subset
If A ⊂ B and A ≠ B, then A is called a proper subset of B.
e.g. A = {1, 2, 3} is a proper subset of B = {1, 2, 3, 4}.
Singleton set
A set containing exactly one element.
e.g. {a} is a singleton set.
Universal set
The basic set that contains all elements under consideration in a particular context, usually denoted by U.
e.g. For the set of all integers, the universal set can be the set of real numbers R.
Union of sets
The set consisting of all elements of A and all elements of B, with common elements taken only once, denoted by A ∪ B.
e.g. If A = {2, 4, 6, 8} and B = {6, 8, 10, 12}, then A ∪ B = {2, 4, 6, 8, 10, 12}.
Intersection of sets
The set of all elements which are common to both A and B, denoted by A ∩ B.
e.g. If A = {2, 4, 6, 8} and B = {6, 8, 10, 12}, then A ∩ B = {6, 8}.
Disjoint sets
Two sets A and B are called disjoint if they have no common elements, i.e., A ∩ B = φ.
e.g. A = {2, 4, 6, 8} and B = {1, 3, 5, 7} are disjoint sets.
Difference of sets
The set of elements which belong to A but not to B, denoted by A – B.
e.g. If A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8}, then A – B = {1, 3, 5}.
Complement of a set
The set of all elements of the universal set U that are not in A, denoted by A′ or A^c.
e.g. If U = {1, 2, 3, 4, 5, 6} and A = {1, 3, 5}, then A′ = {2, 4, 6}.
Formulas & Equations (6)
A ∪ B = { x : x ∈ A or x ∈ B }
Reads as: A union B is the set of all x such that x belongs to A or x belongs to B
Variables: A and B are sets; x is an element
Use when: Finding the set of all elements in either set A or set B (or both)
⚠ Students often forget to list common elements only once, writing them twice.
A ∩ B = { x : x ∈ A and x ∈ B }
Reads as: A intersection B is the set of all x such that x belongs to A and x belongs to B
Variables: A and B are sets; x is an element
Use when: Finding the set of elements common to both A and B
⚠ Students confuse intersection with union and include all elements instead of only common ones.
A – B = { x : x ∈ A and x ∉ B }
Reads as: A minus B is the set of all x such that x belongs to A and x does not belong to B
Variables: A and B are sets; x is an element
Use when: Finding elements that are in A but not in B
⚠ Students incorrectly think A – B = B – A, which is generally false.
A′ = { x : x ∈ U and x ∉ A }
Reads as: A complement is the set of all x such that x belongs to U and x does not belong to A
Variables: U is the universal set; A is a subset of U; x is an element
Use when: Finding all elements of the universal set not in A
⚠ Students forget to specify the universal set, making the complement ambiguous.
n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
Reads as: Number of elements in A union B equals number in A plus number in B minus number in A intersection B
Variables: n(A) is the number of elements in set A; n(B) is the number in set B; n(A ∩ B) is the number in their intersection
Use when: Solving word problems involving counting elements in two sets
⚠ Students forget to subtract the intersection, leading to double-counting.
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(C ∩ A) + n(A ∩ B ∩ C)
Reads as: Number of elements in A union B union C equals sum of individual counts minus pairwise intersections plus triple intersection
Variables: A, B, C are sets; n denotes number of elements
Use when: Solving word problems involving three sets
⚠ Students forget to add back the triple intersection after subtracting pairwise intersections.
How It Works
Representing a set in Roster Form
  1. List all elements of the set, separated by commas.
  2. Enclose the list within braces { }.
  3. Do not repeat any element; order is immaterial.
💡 For infinite sets with a clear pattern, use three dots (e.g., {1, 3, 5, ...}) but not all infinite sets can be described this way.
Representing a set in Set-Builder Form
  1. Choose a variable (e.g., x) to represent elements.
  2. Write a colon ':' after the variable (read as 'such that').
  3. State the common property that all elements of the set satisfy.
  4. Enclose the entire description within braces { }.
💡 The property must be precise and unambiguous so that any object can be tested for membership.
Determining if a set is Empty, Finite, or Infinite
  1. Check if the set has any elements. If none, it is the empty set.
  2. If the set has a definite number of elements (including zero), it is finite.
  3. If the set has an indefinite number of elements (cannot be counted), it is infinite.
💡 The empty set is always finite. Some infinite sets like the set of real numbers cannot be written in roster form.
Checking if Two Sets are Equal
  1. Check if every element of set A is also an element of set B.
  2. Check if every element of set B is also an element of set A.
  3. If both conditions are true, the sets are equal (A = B).
💡 Repetition of elements does not change a set. Order does not matter.
Finding the Union of Two Sets
  1. List all elements from set A.
  2. Add all elements from set B that are not already listed.
  3. The resulting set is A ∪ B.
💡 Common elements are taken only once. If B ⊂ A, then A ∪ B = A.
Finding the Intersection of Two Sets
  1. Identify elements that belong to both set A and set B.
  2. Collect these common elements into a new set.
  3. The resulting set is A ∩ B.
💡 If A ∩ B = φ, the sets are disjoint. If B ⊂ A, then A ∩ B = B.
Finding the Difference of Two Sets
  1. Start with all elements of set A.
  2. Remove any element that also belongs to set B.
  3. The remaining set is A – B.
💡 A – B is generally not equal to B – A. The sets A – B, A ∩ B, and B – A are mutually disjoint.
Exam Traps — Don't Make These Mistakes
⚠ Confusing 'element of' with 'subset of'
✗ Students write {a} ∈ {a, b, c} or 1 ⊂ {1, 2, 3}.
→ The symbol ∈ is used for an element belonging to a set, while ⊂ is used for a set being a subset of another set. {a} is a set, not an element, so it should be {a} ⊂ {a, b, c}. Similarly, 1 is an element, so it should be 1 ∈ {1, 2, 3}.
✓ Write {a} ⊂ {a, b, c} and 1 ∈ {1, 2, 3}.
⚠ Repeating elements in roster form
✗ Students write {a, a, b, c} for a set.
→ A set contains distinct elements. Repetition does not change the set, but it is incorrect form. The correct roster form should list each element only once.
✓ Write {a, b, c}.
⚠ Assuming A – B = B – A
✗ Students write that A – B = B – A in general.
→ The difference of sets is not commutative. For example, if A = {1, 2} and B = {2, 3}, then A – B = {1} and B – A = {3}, which are not equal.
✓ State that A – B ≠ B – A in general, and give a counterexample.
⚠ Forgetting the universal set when writing complement
✗ Students write A′ = {x : x ∉ A} without specifying U.
→ The complement of a set is defined relative to a universal set. Without U, the complement is ambiguous because it is not clear what elements are being considered.
✓ Write A′ = {x : x ∈ U and x ∉ A}.
⚠ Misidentifying empty set as 'not a set'
✗ Students say φ has no elements, so it is not a set.
→ The empty set is a valid set; it is the set with no elements. It is a subset of every set.
✓ State that φ is a set (the empty set) and φ ⊂ A for any set A.
⚠ Confusing 'belongs to' with 'subset' for nested sets
✗ Students write {3, 4} ⊂ A when A = {1, 2, {3, 4}, 5}.
→ In this example, {3, 4} is an element of A, not a subset. The elements of A are 1, 2, {3, 4}, and 5. So {3, 4} ∈ A, but {3, 4} ⊄ A because 3 and 4 are not elements of A individually.
✓ Write {3, 4} ∈ A, and {{3, 4}} ⊂ A.
Compare & Contrast
Method of representing a set
Roster form
Set-builder form
Roster form lists all elements explicitly within braces, while set-builder form describes a common property that all elements satisfy.
Operation on sets
Union of sets (A ∪ B)
Intersection of sets (A ∩ B)
Union includes all elements from both sets (common elements taken once), while intersection includes only elements common to both sets.
Number of elements
Finite set
Infinite set
A finite set has a definite number of elements (including zero), while an infinite set has an indefinite, non-terminating number of elements.
Number of elements
Empty set (φ)
Singleton set ({a})
The empty set has zero elements, while a singleton set has exactly one element.
Exam Focus
Marks pattern: 1-2 MCQs (1 mark each), 1-2 Very Short Answer (1 mark), 1 Short Answer (3 marks), 1 Long Answer (5 marks) — total approximately 8-10 marks
Most asked: Representation of sets (roster and set-builder forms) · Subsets and proper subsets · Union, intersection, and difference of sets · Venn diagrams and word problems using n(A ∪ B) formula · Complement of a set
Don't skip: Word problems involving union and intersection of two or three sets (using formulas for n(A ∪ B) and n(A ∪ B ∪ C)); Proving set identities using Venn diagrams or properties
One-Line Revision (read aloud before the exam)
  1. A set is a well-defined collection of distinct objects.
  2. Sets are represented in roster form by listing elements within braces, e.g., {1, 2, 3}.
  3. Sets are represented in set-builder form by stating a common property, e.g., {x : x is a natural number}.
  4. The empty set φ has no elements and is a subset of every set.
  5. A finite set has a definite number of elements; an infinite set does not.
  6. Two sets are equal if they have exactly the same elements, regardless of order or repetition.
  7. A ⊂ B means every element of A is also in B.
  8. The union A ∪ B contains all elements from A and B, with common elements taken once.
  9. The intersection A ∩ B contains only elements common to both A and B.
  10. If A ∩ B = φ, the sets are disjoint.
  11. The difference A – B contains elements in A but not in B.
  12. The complement A′ contains all elements of the universal set not in A.
  13. n(A ∪ B) = n(A) + n(B) – n(A ∩ B).
  14. For three sets, n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A∩B) – n(B∩C) – n(C∩A) + n(A∩B∩C).
  15. Venn diagrams use rectangles for universal set and circles for subsets.
  16. The sets A – B, A ∩ B, and B – A are mutually disjoint.

MCQs — 10 Questions with Answers

Q1. Which of the following is a well-defined set?

  • A. The collection of five most famous mathematicians of India
  • B. The collection of all natural numbers less than 100 ✓
  • C. The collection of beautiful flowers in a garden
  • D. The collection of intelligent students in your school

Answer: B — Only option B has a clear, objective criterion (natural numbers < 100) that allows definite membership decision; all others depend on subjective judgment.

Q2. Write the set {1, 4, 9, 16, 25, 36} in set-builder form.

  • A. {x : x is a perfect square and x < 40}
  • B. {x : x = n², where n ∈ N and 1 ≤ n ≤ 6} ✓
  • C. {x : x is divisible by perfect squares}
  • D. {x : x is an odd perfect square}

Answer: B — Option B correctly identifies that all elements are perfect squares of natural numbers from 1 to 6; option A is incomplete as 49 < 40 is false.

Q3. The set of solution of the equation x² – 5x + 6 = 0 in roster form is:

  • A. {2, 3} ✓
  • B. {-2, -3}
  • C. {1, 6}
  • D. {0, 5}

Answer: A — Factoring gives (x – 2)(x – 3) = 0, so x = 2 or x = 3; the solution set is {2, 3}.

Q4. Which representation correctly describes the set of positive integers greater than 5 and less than 10?

  • A. Roster form: {5, 6, 7, 8, 9, 10}
  • B. Roster form: {6, 7, 8, 9} ✓
  • C. Set-builder form: {x : x ∈ Z and x ≥ 5 and x ≤ 10}
  • D. Roster form: {5, 6, 7, 8, 9}

Answer: B — Integers strictly greater than 5 and strictly less than 10 are 6, 7, 8, 9; endpoints are excluded because of strict inequalities.

Q5. The set A = {x : x is a vowel in the English alphabet} in roster form is:

  • A. {a, e, i, o, u, y}
  • B. {a, e, i, o, u} ✓
  • C. {A, E, I, O, U}
  • D. {a, e, i, o, u, and sometimes y}

Answer: B — The five pure vowels in English are a, e, i, o, u; y is sometimes a vowel but not a pure vowel by definition.

Q6. If P = {x : x is a prime factor of 30}, which statement is NOT correct?

  • A. 3 ∈ P
  • B. 5 ∈ P
  • C. 15 ∉ P
  • D. 6 ∈ P ✓

Answer: D — Prime factors of 30 are 2, 3, 5 only; 6 = 2 × 3 is composite, not prime, so 6 ∉ P.

Q7. Consider the statements: (I) In roster form, elements can be repeated. (II) Set {1, 2, 3} and {3, 2, 1} are the same. Which is true?

  • A. Both (I) and (II) are true
  • B. (I) is true and (II) is false
  • C. Only (II) is true ✓
  • D. Both (I) and (II) are false

Answer: C — Statement I is false: repetition is avoided in roster form. Statement II is true: order is immaterial, so {1, 2, 3} = {3, 2, 1}.

Q8. Write {n/(n+1) : n ∈ N and n ≤ 4} in roster form.

  • A. {1/2, 2/3, 3/4, 4/5} ✓
  • B. {1, 2/3, 3/4, 4/5}
  • C. {1/2, 2/3, 3/4, 1}
  • D. {0, 1/2, 2/3, 3/4}

Answer: A — For n = 1, 2, 3, 4: n/(n+1) gives 1/2, 2/3, 3/4, 4/5 respectively.

Q9. The set of letters in the word 'SCHOOL' in roster form is:

  • A. {S, C, H, O, O, L}
  • B. {S, C, H, O, L} ✓
  • C. {L, O, H, C, S}
  • D. {S, C, H, O, L, O}

Answer: B — Repetition is omitted in roster form, so O appears only once; the correct set is {S, C, H, O, L}, order irrelevant.

Q10. A student claims that the set {x : x is the largest natural number} is a valid set. Analyze this claim.

  • A. The claim is correct because natural numbers form a valid set
  • B. The claim is incorrect because there is no largest natural number; the set is not well-defined ✓
  • C. The claim is correct because the set uses set-builder notation
  • D. The claim is incorrect because natural numbers are not included in standard mathematics

Answer: B — Natural numbers are infinite with no upper bound, so no largest natural number exists; this makes the condition not well-defined and membership undecidable.

Flashcards

What is a well-defined collection?

A collection where you can definitely decide whether any given object belongs to it or not.

What is the difference between roster and set-builder form?

Roster form lists all elements separated by commas in braces {1, 2, 3}; set-builder form describes elements by a common property {x : x is odd}.

What does the symbol ∈ mean?

The symbol ∈ means 'belongs to' and is used to show that an element is a member of a set.

Can elements repeat in a set written in roster form?

No, in roster form each element is listed only once; all elements are taken as distinct.

What does the colon ':' represent in set-builder notation?

The colon ':' stands for 'such that' and introduces the common property that all elements possess.

Give examples of three standard sets in mathematics.

N = natural numbers, Z = integers, Q = rational numbers, R = real numbers.

Convert {2, 4, 6, 8} to set-builder form.

{x : x is an even positive integer and x ≤ 8} or {x : x = 2n, where n ∈ N and n ≤ 4}.

Is the collection 'five most talented actors in India' a set? Why or why not?

No, because the criterion for 'most talented' varies from person to person, making it not well-defined.

What property must a collection have to be called a set?

It must be well-defined, meaning there is a clear rule to determine whether any object belongs to it or not.

If A = {1, 4, 9, 16, 25}, what property do all elements share?

All elements are perfect squares of natural numbers; A = {x : x = n², where n ∈ N and 1 ≤ n ≤ 5}.

Important Board Questions

What is a well-defined set? Give two examples of well-defined collections and two examples of collections that are NOT sets. [2 marks]

A well-defined set has an unambiguous criterion for membership. Example sets: odd numbers less than 10, prime factors of 12. Non-sets: famous actors, beautiful paintings (subjective criteria).

Convert the following sets between roster and set-builder forms: (i) {2, 4, 6, 8, 10} (ii) {x : x is a natural number and x² < 50}. Explain your reasoning in each case. [5 marks]

For (i), identify the common property (even numbers up to 10). For (ii), find which natural numbers satisfy n² < 50 by testing: 1, 4, 9, 16, 25, 36, 49. Show both conversions with clear property descriptions.

Write the set A = {1/2, 2/3, 3/4, 4/5, 5/6} in set-builder form and verify your answer by converting back to roster form. Also express the general term for the nth element of this set. [6 marks]

Observe that each numerator is one less than the denominator and numerators range from 1 to 5. The set-builder form is A = {x : x = n/(n+1), where n ∈ N and 1 ≤ n ≤ 5}. The general term is aₙ = n/(n+1). Convert back by substituting n = 1, 2, 3, 4, 5 to verify all elements match.

Assertion-Reason Questions for Sets

Q1. Assertion (A):

A set is a well-defined collection of objects.

Reason (R):

A well-defined collection means we can definitively determine whether any given object belongs to the collection or not.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true, and the reason correctly explains why a set must be well-defined. A set requires an objective criterion for membership that leaves no ambiguity.

Q2. Assertion (A):

The sets {2, 4, 6} and {6, 4, 2} are equal.

Reason (R):

In set notation, the order of elements is significant because sets are ordered collections.

Answer: C — A is true but R is false — The assertion is true (the sets are equal because order doesn't matter in sets), but the reason is false. Sets are unordered collections; order is NOT significant. The reason contradicts the fundamental property of sets.

Q3. Assertion (A):

The set A = {x : x is a letter of the word 'BOOK'} has 4 elements.

Reason (R):

The word 'BOOK' has 4 letters, so the set must have 4 elements.

Answer: D — A is false but R is true — Both the assertion and reason are false. The word 'BOOK' contains the letters B, O, O, K. When forming a set, repetitions are not listed, so the set is {B, O, K}, which has 3 elements, not 4. Students often count repeated letters as distinct elements.

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Case-Based Questions (CBQ) for Sets

Case Study:

A library in India maintains a database of its book collection. The librarian categorizes books as follows: Set S contains all Science books, Set H contains all History books, Set M contains all Mathematics books, Set F contains all Fiction books, and Set R contains all books available in Reference section. A student visits the library and finds that among 150 books surveyed, 45 belong to S, 30 belong to H, 40 belong to M, 25 belong to F, and 10 books belong to R. However, some books fall into multiple categories (like a Mathematics History book). The librarian needs to understand the nature of each category as a well-defined set.

Q1. Which of the following best describes why Set S (Science books) is a well-defined set in this library?

  • A. Because all science books are expensive
  • B. Because we can objectively determine whether any book is a science book or not ✓
  • C. Because there are exactly 45 books in this set
  • D. Because science books are written by famous authors

Answer: B

Q2. If a book is both a Mathematics book and a History book, how should it be represented in set notation?

  • A. The book belongs to M only, not to H
  • B. The book belongs to both M and H; it is an element of both sets ✓
  • C. The book cannot exist because it must belong to only one category
  • D. The book belongs to a special 'overlap set', separate from M and H

Answer: B

Q3. The librarian notices that 150 books total were surveyed, but 45 + 30 + 40 + 25 + 10 = 150. What does this tell us?

  • A. No book belongs to more than one category ✓
  • B. Every book in the library belongs to at least one of these categories
  • C. The sum equals 150 by pure coincidence
  • D. Some books belong to multiple categories, but the arithmetic still works

Answer: A

Q4. The Reference section (Set R) contains 10 books. Which of the following statements about Set R is definitely true?

  • A. All books in R are unique and do not overlap with S, H, M, or F
  • B. Set R is not a true set because reference books are subjectively chosen
  • C. Set R is well-defined because we can determine which books are reference books ✓
  • D. The books in R must be older than books in other sets

Answer: C

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