Sets – Class 11 Maths Chapter 1 Notes & NCERT PDF

Class 11MathematicsChapter 1NCERT book: Mathematics - Class 11

Notes and a simple summary of Chapter 1 of the NCERT Class 11 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 1: Sets

Introduction

Almost every branch of modern mathematics — relations, functions, geometry, sequences, probability — is written in the language of sets. The idea was developed by the German mathematician Georg Cantor (1845–1918) while he was studying trigonometric series. This chapter builds that language: how to describe a set, how to compare sets, and how to combine them.

Key Concepts

1.2 Sets and their Representations

A set is a well-defined collection of objects. "Well-defined" means we can decide without doubt whether an object belongs to it. "The five most talented writers of India" is not a set; "the vowels of the English alphabet" is.

Objects of a set are its elements or members; we write a ∈ A (a belongs to A) and b ∉ A (b does not belong to A).

Standard sets: N (natural numbers), Z (integers), Q (rational numbers), R (real numbers), Z⁺, Q⁺, R⁺ (positive ones).

Two ways to write a set:

- Roster (tabular) form — list the elements in braces, separated by commas: {a, e, i, o, u}. Order does not matter and elements are not repeated.

- Set-builder form — state the common property: {x : x is a vowel in the English alphabet}. The colon is read "such that".

1.3 The Empty Set

A set which does not contain any element is called the empty set (null set or void set), denoted φ or { }.

Example: {x : 1 < x < 2, x is a natural number} = φ, because no natural number lies strictly between 1 and 2.

1.4 Finite and Infinite Sets

A set that is empty or has a definite number of elements is finite; n(A) denotes the number of elements.

Otherwise it is infinite — for example, the set of natural numbers or the set of points on a line. Infinite sets cannot always be written in roster form (R cannot).

1.5 Equal Sets

Two sets A and B are equal if they have exactly the same elements; we write A = B.

Repetition does not change a set: {1, 2, 2, 3} = {1, 2, 3}.

1.6 Subsets

A set A is a subset of B (A ⊂ B) if every element of A is also an element of B.

Every set is a subset of itself, and φ is a subset of every set.

A = B if and only if A ⊂ B and B ⊂ A.

If A ⊂ B and A ≠ B, A is a proper subset of B.

A set with only one element is a singleton.

1.6.1 Subsets of R: N ⊂ Z ⊂ Q ⊂ R, and the irrational numbers T = R – Q.

1.6.2 Intervals as subsets of R (a < b):

- open interval (a, b) = {y : a < y < b}

- closed interval [a, b] = {x : a ≤ x ≤ b}

- [a, b) = {x : a ≤ x < b}; (a, b] = {x : a < x ≤ b}

- The length of each of these intervals is b – a. Every interval contains infinitely many points.

1.7 Universal Set

The universal set U is the basic set in a given context — for example, R when we study subsets of real numbers.

1.8 Venn Diagrams

Named after the English logician John Venn, these show U as a rectangle and its subsets as circles, making relationships between sets visible.

1.9 Operations on Sets

Union A ∪ B = {x : x ∈ A or x ∈ B}. Properties: commutative, associative, A ∪ φ = A (identity), A ∪ A = A (idempotent), U ∪ A = U.

Intersection A ∩ B = {x : x ∈ A and x ∈ B}. Properties: commutative, associative, φ ∩ A = φ, U ∩ A = A, A ∩ A = A, and the distributive law A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

If A ∩ B = φ, A and B are disjoint.

Difference A – B = {x : x ∈ A and x ∉ B}. In general A – B ≠ B – A.

1.10 Complement of a Set

A′ = {x : x ∈ U and x ∉ A} = U – A.

Properties:

Complement laws: A ∪ A′ = U, A ∩ A′ = φ

De Morgan's laws: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′

Double complementation: (A′)′ = A

φ′ = U and U′ = φ

Worked Examples

Example 1 (union, intersection, difference). A = {1, 2, 3, 4}, B = {3, 4, 5, 6}.

A ∪ B = {1, 2, 3, 4, 5, 6} (list every element once)

A ∩ B = {3, 4}

A – B = {1, 2} and B – A = {5, 6} — the two differences are different.

Example 2 (complement). U = {1, 2, …, 10}, A = {1, 3, 5, 7, 9}. Then A′ = {2, 4, 6, 8, 10}.

Example 3 (De Morgan check). With the same U, take X = {2, 3, 4}, Y = {3, 5, 7}.

X ∪ Y = {2, 3, 4, 5, 7}, so (X ∪ Y)′ = {1, 6, 8, 9, 10}.

X′ = {1, 5, 6, 7, 8, 9, 10}, Y′ = {1, 2, 4, 6, 8, 9, 10}; X′ ∩ Y′ = {1, 6, 8, 9, 10}. Both sides agree.

Example 4 (intervals). {x : –3 ≤ x < 5} = [–3, 5); {x : –5 < x ≤ 7} = (–5, 7]. Length of [–3, 5) is 5 – (–3) = 8.

Example 5 (empty set). {x : x is a natural number, x < 5 and x > 7} = φ, since no number is both less than 5 and greater than 7.

Formula Sheet

A ∪ A′ = U, A ∩ A′ = φ

(A ∪ B)′ = A′ ∩ B′; (A ∩ B)′ = A′ ∪ B′

(A′)′ = A; φ′ = U; U′ = φ

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Length of (a, b), [a, b], [a, b), (a, b] = b – a

Exam Focus

Convert between forms. "Write {x : x is an integer, –3 ≤ x < 7} in roster form" → list –3, –2, …, 6 (note 7 is excluded).

Proof-style questions. To show two sets are equal, show each is a subset of the other: take any x in the left side and prove it is in the right side, then the reverse.

Venn diagram shading. Shade A – B as the part of circle A outside B; shade (A ∪ B)′ as everything outside both circles.

Key Terms

Set (समुच्चय), Element (अवयव)

Roster form (सूची रूप), Set-builder form (समुच्चय निर्माण रूप)

Empty set (रिक्त समुच्चय), Finite / Infinite set (परिमित / अपरिमित समुच्चय)

Subset (उपसमुच्चय), Proper subset (उचित उपसमुच्चय), Singleton (एकल समुच्चय)

Universal set (सार्वत्रिक समुच्चय), Interval (अंतराल)

Union (सम्मिलन), Intersection (सर्वनिष्ठ), Difference (अंतर)

Complement (पूरक), Disjoint sets (असंयुक्त समुच्चय), Venn diagram (वेन आरेख)

Common Mistakes

Writing {φ} for the empty set — {φ} is a set with one element, not the empty set.

Confusing ∈ with ⊂: 1 ∈ {1, 2} but {1} ⊂ {1, 2}.

Repeating elements or thinking order matters in roster form.

Mixing up open and closed ends: [a, b) includes a but not b.

Assuming A – B = B – A.

💡 Key Learning Points

✓Recognise a set as a well-defined collection and write it in roster and set-builder form
✓Identify empty, finite, infinite, equal and universal sets, subsets and proper subsets
✓Express subsets of R as open, closed and half-open intervals and find their length
✓Compute union, intersection, difference and complement, using Venn diagrams
✓Apply the laws of set operations, including De Morgan's laws and the distributive law

👨‍🏫 Teaching Tips

→Start with examples and non-examples of 'well-defined' collections so students see why the word matters
→Have students convert the same set between roster and set-builder form in pairs
→Use a number line to draw each type of interval with open and filled circles at the ends
→Draw Venn diagrams on the board and let students shade A ∪ B, A ∩ B, A – B and A′
→Verify De Morgan's laws with a concrete U before discussing the general proof
→Contrast ∈ and ⊂ with quick oral true/false questions
→Mention Cantor and Russell's Paradox from the Historical Note to show why precise definitions matter

📋 Assessment Questions

Can the student write a given set in both roster and set-builder form?
Can the student distinguish ∈ from ⊂ and φ from {φ}?
Can the student write set-builder descriptions as intervals and vice versa?
Can the student compute A ∪ B, A ∩ B, A – B and A′ for given sets?
Can the student verify De Morgan's laws for specific sets?

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