Sets — Mathematics Class 11 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Sets (Class 11) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 4 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Sets (CBSE & HBSE)
Foundation of all higher mathematics — every later chapter (functions, probability, calculus) speaks the language of sets. CBSE asks 5–6 marks: 1 MCQ + 1 short answer (set operations) + 1 application (Venn-diagram counting). HBSE prefers De Morgan verification and union/intersection cardinality formulas.
Sets — Definition, Notation, Types
What is a Set?
A set is a well-defined collection of distinct objects. "Well-defined" means: given any object, we can decide unambiguously whether it belongs to the set or not.
The collection of "tall students in your class" is not a set (tall is subjective); the collection of "students above 170 cm" is a set.
Notation
- Sets are denoted by capital letters: A, B, C, …
- Elements by lowercase: a, b, x, y, …
- x ∈ A means "x belongs to A"; x ∉ A means "x does not belong to A".
Ways to Represent a Set
| Method | Example |
|---|---|
| Roster (Tabular) | A = {1, 2, 3, 4, 5} |
| Set-builder (Rule) | A = {x : x ∈ ℕ, x ≤ 5} |
| Descriptive | A = first five natural numbers |
In roster form: elements are separated by commas, order does not matter, and repetition is ignored (so {1, 2, 2, 3} = {1, 2, 3}).
Standard Number Sets (must memorise the symbols)
| Symbol | Set |
|---|---|
| ℕ | Natural numbers {1, 2, 3, …} |
| W | Whole numbers {0, 1, 2, …} |
| ℤ | Integers {…, −2, −1, 0, 1, 2, …} |
| ℚ | Rational numbers (p/q form) |
| ℝ | Real numbers |
| ℂ | Complex numbers |
| ℤ⁺, ℝ⁺ | Positive integers / reals |
Types of Sets
Empty / Null Set (∅ or { }) — contains no elements. Caution: {0} is not empty (it contains 0), and {∅} contains one element (the empty set itself).
Singleton set — exactly one element: {7}.
Finite set — countable, finite number of elements. Cardinality n(A) = number of elements.
Infinite set — cannot be exhausted by counting (ℕ, ℝ).
Equal sets — A = B iff every element of A is in B and vice-versa (same elements, in any order).
Equivalent sets — A ~ B iff n(A) = n(B). Equal ⇒ equivalent, but equivalent ⇏ equal.
CBSE Tip: Always specify the universal set when using set-builder notation — "x ∈ ℕ" or "x ∈ ℝ" can change the resulting set entirely.
Subsets, Power Set, Universal Set, Intervals
Subset
A ⊆ B iff every element of A is also in B. Read "A is a subset of B".
- Every set is a subset of itself: A ⊆ A.
- The empty set is a subset of every set: ∅ ⊆ A.
- A = B ⇔ A ⊆ B and B ⊆ A (anti-symmetry).
Proper Subset
A ⊂ B iff A ⊆ B and A ≠ B. (At least one element of B is missing from A.)
Some textbooks use ⊂ for any subset and ⊊ for proper; NCERT uses ⊂ as proper.
Power Set
P(A) = set of all subsets of A, including ∅ and A itself.
If n(A) = n, then n(P(A)) = 2ⁿ.
Example: A = {a, b}. P(A) = {∅, {a}, {b}, {a, b}}. n(P(A)) = 2² = 4.
Universal Set (U)
The "largest" set for the discussion — every set considered is a subset of U. Choice of U depends on context:
- In number theory: U = ℕ or ℤ.
- In geometry: U = the plane / ℝ².
- In probability: U = sample space.
Intervals — Sets of Real Numbers
| Notation | Set | Type |
|---|---|---|
| (a, b) | {x : a < x < b} | Open interval |
| [a, b] | {x : a ≤ x ≤ b} | Closed interval |
| (a, b] | {x : a < x ≤ b} | Half-open |
| [a, b) | {x : a ≤ x < b} | Half-open |
| (−∞, b] | {x : x ≤ b} | Unbounded |
| [a, ∞) | {x : x ≥ a} | Unbounded |
| (−∞, ∞) | ℝ | All reals |
Note: ∞ and −∞ are not numbers; they only indicate unboundedness — always use a parenthesis, never a bracket, next to them.
Number of Subsets
For a set of n elements:
- Total subsets: 2ⁿ
- Proper subsets: 2ⁿ − 1 (exclude A itself)
- Non-empty subsets: 2ⁿ − 1 (exclude ∅)
HBSE Tip: When asked to list all subsets, organise by cardinality (0-element, 1-element, 2-element, …, n-element) to avoid omissions.
Operations on Sets
Union (A ∪ B)
A ∪ B = {x : x ∈ A or x ∈ B} — elements in at least one of A, B.
Properties:
- Commutative: A ∪ B = B ∪ A
- Associative: (A ∪ B) ∪ C = A ∪ (B ∪ C)
- Identity: A ∪ ∅ = A
- Idempotent: A ∪ A = A
- A ⊆ B ⇒ A ∪ B = B
Intersection (A ∩ B)
A ∩ B = {x : x ∈ A and x ∈ B} — elements in both A and B.
Properties:
- Commutative, Associative, Idempotent (same as union).
- Identity: A ∩ U = A; A ∩ ∅ = ∅.
- A ⊆ B ⇒ A ∩ B = A.
Disjoint sets: A ∩ B = ∅ (no common element).
Difference (A − B)
A − B = {x : x ∈ A and x ∉ B} — elements of A that are not in B.
Note: A − B ≠ B − A in general.
Complement (A′ or Aᶜ)
With respect to universal set U: A′ = U − A = {x : x ∈ U and x ∉ A}.
Properties:
- (A′)′ = A (double complement)
- A ∪ A′ = U, A ∩ A′ = ∅
- U′ = ∅, ∅′ = U
Distributive Laws
- A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
- A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
De Morgan's Laws
- (A ∪ B)′ = A′ ∩ B′
- (A ∩ B)′ = A′ ∪ B′
Mnemonic: the complement "flips" union ↔ intersection and distributes over each.
Useful Identity (Symmetric Difference)
A △ B = (A − B) ∪ (B − A) = (A ∪ B) − (A ∩ B). Symmetric: A △ B = B △ A.
CBSE Tip: When proving identities, list elements of each side and show set-equality both ways — "⊆" then "⊇".
Venn Diagrams & Counting (Inclusion–Exclusion)
Venn Diagrams
Closed regions inside a rectangle (the universal set) depict each set. Overlaps show intersections.
Convention: rectangles = U; circles = A, B, C; shading shows the result of the operation.
Two-Set Inclusion–Exclusion
n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
Derivation: each element of A∩B is counted twice when we add n(A) + n(B); subtracting n(A∩B) corrects the over-count.
If A and B are disjoint (A ∩ B = ∅), then n(A ∪ B) = n(A) + n(B).
Three-Set Inclusion–Exclusion
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C).
Useful Corollaries
| Quantity | Formula |
|---|---|
| Only A | n(A) − n(A ∩ B) − n(A ∩ C) + n(A ∩ B ∩ C) (3-set) |
| Exactly one of A, B | n(A) + n(B) − 2 n(A ∩ B) |
| Neither A nor B | n(U) − n(A ∪ B) |
| Complement count | n(A′) = n(U) − n(A) |
Problem-Solving Strategy
- Identify each given count (n(A), n(B), n(A ∩ B), etc.).
- Sketch the Venn diagram with regions labelled.
- Fill regions inside-out: start with the innermost intersection, then pair-wise intersections (subtracting), then singletons.
- Verify the totals.
Sample Application
In a class of 60, 40 study Maths, 35 Physics, 20 both. How many study neither?
- n(A ∪ B) = 40 + 35 − 20 = 55.
- Neither = 60 − 55 = 5.
HBSE Tip: Survey-based problems (sports, languages, subjects) are standard 4-mark questions — always draw the Venn diagram before computing.
Frequently asked questions
Are these Sets notes free?
Yes — the Sets notes for Mathematics (Class 11) on Siksha Sarovar are completely free to read, with no account required.
Do these notes follow CBSE and HBSE?
Yes. The Sets notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Sets chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Sets.