CBSE Class 11 Mathematics Chapter 1 Notes: Sets

Chapter Overview

Sets is the opening chapter of CBSE Class 11 Mathematics and belongs to the Sets and Functions unit (Unit I). It introduces the language of modern mathematics — sets, operations on sets, and Venn diagrams — which you will use in every subsequent chapter from Relations to Probability. Mastering Sets is non-negotiable because the notation and logic carry forward to Class 12 and competitive exams like JEE.

This guide covers every NCERT topic in the chapter with definitions, worked concepts, tables, formulae, common mistakes, and exam-focused tips.

Key Definitions

Types of Sets — Comparison

Operations on Sets

Union, Intersection, Difference, Complement

Properties of Set Operations

De Morgan's Laws

De Morgan's laws relate the complement of unions and intersections:

1. (A ∪ B)' = A' ∩ B' — The complement of a union is the intersection of complements. 2. (A ∩ B)' = A' ∪ B' — The complement of an intersection is the union of complements.

These extend to three or more sets:

• (A ∪ B ∪ C)' = A' ∩ B' ∩ C'
• (A ∩ B ∩ C)' = A' ∪ B' ∪ C'

Verification of De Morgan's Laws (Example)

Let U = {1, 2, 3, 4, 5, 6}, A = {1, 2, 3}, B = {2, 4, 5}.

• A ∪ B = {1, 2, 3, 4, 5}, so (A ∪ B)' = {6}
• A' = {4, 5, 6}, B' = {1, 3, 6}, so A' ∩ B' = {6}
• Hence (A ∪ B)' = A' ∩ B'. Verified.

Venn Diagrams

Venn diagrams represent sets as overlapping closed curves inside a rectangle (the universal set U).

• Two sets: The rectangle is U. Two overlapping circles represent A and B. The overlapping region is A ∩ B.
• Three sets: Three overlapping circles divide the rectangle into 8 regions — useful for solving cardinality problems.

Reading Venn Diagrams for Cardinality

For two sets A and B within U:

• Only A (not B): n(A) − n(A ∩ B)
• Only B (not A): n(B) − n(A ∩ B)
• Both A and B: n(A ∩ B)
• Neither A nor B: n(U) − n(A ∪ B)

Important Formulae — Cardinality

Important Numericals

1. Cardinality Problem (Two Sets)

In a class of 50 students, 30 like Maths and 25 like Science. If 10 like both, how many like at least one subject?

2. Cardinality Problem (Three Sets)

In a group of 100 people: 40 read Hindi newspapers, 30 read English, and 20 read Tamil. 10 read Hindi and English, 8 read English and Tamil, 5 read Hindi and Tamil, and 3 read all three. How many read at least one newspaper?

3. De Morgan's Law Verification

If U = {1, 2, 3, 4, 5}, A = {1, 3, 5}, B = {2, 3, 4}, verify (A ∩ B)' = A' ∪ B'.

A ∩ B = {3}, so (A ∩ B)' = {1, 2, 4, 5} A' = {2, 4}, B' = {1, 5}, so A' ∪ B' = {1, 2, 4, 5} Verified: (A ∩ B)' = A' ∪ B'.

4. Power Set

Find the power set of A = {a, b}.

n(A) = 2, so P(A) has 2² = 4 subsets: P(A) = {∅, {a}, {b}, {a, b}}

5. Venn Diagram Problem

In a survey of 200 people: 120 like tea, 80 like coffee, 40 like both. Find how many like neither.

n(T ∪ C) = 120 + 80 − 40 = 160 Neither = n(U) − n(T ∪ C) = 200 − 160 = 40 people

Intervals as Subsets of Real Numbers

Intervals are special subsets of R that appear frequently in later chapters (especially calculus and inequalities).

Note: ∞ is not a real number, so we always use parentheses (not brackets) with ∞.

Practice Problems

Problem 1

If A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}, find A ∪ B, A ∩ B, A − B, and B − A.

• A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8}
• A ∩ B = {4, 5}
• A − B = {1, 2, 3}
• B − A = {6, 7, 8}

Problem 2

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6, 7}, verify (A ∪ B)' = A' ∩ B'.

• A ∪ B = {1, 2, 3, 4, 5, 6, 7}, so (A ∪ B)' = {8, 9}
• A' = {6, 7, 8, 9}, B' = {1, 2, 8, 9}, so A' ∩ B' = {8, 9}
• Verified.

Problem 3

In a group of 70 students, 45 play cricket, 30 play football, and 15 play both. How many play neither?

n(C ∪ F) = 45 + 30 − 15 = 60 Neither = 70 − 60 = 10 students

Problem 4

If A ⊂ B and B ⊂ C, prove that A ⊂ C.

Since A ⊂ B, every element of A is in B. Since B ⊂ C, every element of B is in C. Therefore every element of A is in C, so A ⊂ C. This is the transitive property of subsets.

NCERT Exercise Mapping

Common Mistakes

1. Confusing ∅ with {∅}: ∅ has zero elements. {∅} has one element (the empty set itself). 2. Writing repeated elements in roster form: {1, 2, 2, 3} is wrong — sets have distinct elements. Write {1, 2, 3}. 3. Forgetting ∅ is a subset of every set: Students often miss counting the empty set when listing subsets or computing the power set. 4. Wrong sign in cardinality formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). The minus sign is crucial — forgetting it overcounts the intersection. 5. Confusing subset with element: 1 ∈ {1, 2, 3} (element), but {1} ⊆ {1, 2, 3} (subset). An element is not the same as a singleton set. 6. Ignoring order in set-builder form: {x : x ∈ N, x < 5} is correct; writing conditions in the wrong order can make the set meaningless. 7. Misapplying De Morgan's laws: The complement distributes as intersection over union and union over intersection — it does NOT distribute as union over union or intersection over intersection.

Exam Tips

• 1-mark questions: Expect definitions (empty set, power set, universal set) or simple calculations like number of subsets.
• 2-mark questions: Verify De Morgan's laws with a given universal set, or write the roster/set-builder form.
• 4-mark questions: Cardinality problems with two or three sets using the union formula. Always draw a Venn diagram to support your answer.
• 6-mark questions: Word problems requiring Venn diagrams — read carefully, assign regions, and show all steps.
• Always state the formula before substituting values.
• For Venn diagram problems, label all regions and verify that the total equals n(U).
• In set-builder form, ensure the variable and its property are clearly stated.

Special Results and Theorems

Number of Elements in Power Set

If A has n elements, then P(A) has 2ⁿ elements. This follows because each element is either present or absent in a subset — 2 choices for each of n elements.

Difference of Sets — Symmetric Difference

The symmetric difference of A and B is: A Δ B = (A − B) ∪ (B − A)

Properties:

• A Δ B = B Δ A (commutative)
• A Δ ∅ = A
• A Δ A = ∅
• A Δ B = (A ∪ B) − (A ∩ B)

Important Results for Cardinality

• n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
• n(A − B) = n(A) − n(A ∩ B)
• n(A Δ B) = n(A) + n(B) − 2n(A ∩ B)
• n(A' ∩ B') = n(U) − n(A ∪ B)

More Worked Examples

Example 1

If A = {x : x² − 5x + 6 = 0} and B = {x : x² − 3x + 2 = 0}, find A ∪ B and A ∩ B.

A = {2, 3} (roots of x² − 5x + 6). B = {1, 2} (roots of x² − 3x + 2). A ∪ B = {1, 2, 3}. A ∩ B = {2}.

Example 2

If n(U) = 50, n(A) = 30, n(B) = 25, n(A ∩ B) = 10, find n(A' ∩ B').

n(A ∪ B) = 30 + 25 − 10 = 45. n(A' ∩ B') = n(U) − n(A ∪ B) = 50 − 45 = 5.

Study Plan for This Chapter

Quick Revision Notes

• A set is a well-defined collection of distinct objects.
• Represent sets in roster form (list elements) or set-builder form (describe property).
• Empty set ∅ is a subset of every set. Every set is a subset of itself.
• Number of subsets of a set with n elements = 2ⁿ.
• Power set P(A) = set of all subsets of A.
• A ∪ B = all elements in A or B. A ∩ B = common elements.
• A − B = elements in A but not in B. A' = U − A.
• De Morgan's: (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'.
• n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
• For three sets: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C).
• Disjoint sets: n(A ∩ B) = 0.
• Venn diagrams visually represent set operations — always label U and all regions.
• Intervals are subsets of R: open (a,b), closed [a,b], half-open [a,b), unbounded [a,∞).
• A Δ B = (A − B) ∪ (B − A) = (A ∪ B) − (A ∩ B).
• n(A − B) = n(A) − n(A ∩ B). n(A Δ B) = n(A) + n(B) − 2n(A ∩ B).
• For any set A: ∅ ⊆ A, A ⊆ A, and A ⊆ U.
• Two sets are disjoint if and only if A ∩ B = ∅.
• If A ⊆ B, then A − B = ∅ and A ∩ B = A and A ∪ B = B.