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Relations and Functions
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Class 12MathsChapter 1NCERTCBSE
Relations and Functions – Class 12 Maths Chapter 1
📌 Quick Overview: Chapter 1 of Class 12 Maths extends the concept of relations and functions from Class 11. This chapter introduces types of relations, types of functions, composition of functions, and invertible functions — all key topics for CBSE board exams and JEE.
📋 Topics Covered in this Chapter:
- Types of Relations
- Reflexive Relation
- Symmetric Relation
- Transitive Relation
- Equivalence Relation
- Types of Functions
- One-One (Injective) Function
- Onto (Surjective) Function
- Bijective Function
- Composition of Functions
- Invertible Functions
- Binary Operations
1. What is a Relation?
A relation is a way of connecting elements of one set to elements of another set (or the same set). Formally, if A and B are two sets, then any subset of A × B (the Cartesian product) is called a relation from A to B.
Definition: A relation R from set A to set A (i.e., on set A) is a subset of A × A.
If (a, b) ∈ R, we write aRb and say "a is related to b."
If (a, b) ∈ R, we write aRb and say "a is related to b."
Example: Let A = {1, 2, 3}. Define R = {(1,1), (2,2), (3,3), (1,2), (2,3)}. This is a valid relation on A.
2. Types of Relations
2.1 Empty Relation
Definition: A relation R on set A is called an empty relation if no element of A is related to any element of A.
R = ∅ (empty set), i.e., R ⊆ A × A but R = ∅.
R = ∅ (empty set), i.e., R ⊆ A × A but R = ∅.
2.2 Universal Relation
Definition: A relation R on set A is called a universal relation if every element of A is related to every element of A.
R = A × A.
R = A × A.
⚠️ Note: Both the empty relation and the universal relation are sometimes called trivial relations.
2.3 Reflexive Relation
Definition: A relation R on set A is called reflexive if every element of A is related to itself.
Condition: (a, a) ∈ R, for every a ∈ A.
Condition: (a, a) ∈ R, for every a ∈ A.
Example: A = {1, 2, 3}, R = {(1,1), (2,2), (3,3), (1,2)} → Reflexive ✓
R = {(1,2), (2,3)} → Not reflexive ✗ (because (1,1) ∉ R)
R = {(1,2), (2,3)} → Not reflexive ✗ (because (1,1) ∉ R)
2.4 Symmetric Relation
Definition: A relation R on set A is called symmetric if:
Condition: Whenever (a, b) ∈ R ⇒ (b, a) ∈ R, for all a, b ∈ A.
Condition: Whenever (a, b) ∈ R ⇒ (b, a) ∈ R, for all a, b ∈ A.
Example: R = {(1,2), (2,1), (3,3)} → Symmetric ✓
R = {(1,2), (2,3)} → Not symmetric ✗ (because (2,1) ∉ R)
R = {(1,2), (2,3)} → Not symmetric ✗ (because (2,1) ∉ R)
2.5 Transitive Relation
Definition: A relation R on set A is called transitive if:
Condition: Whenever (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R, for all a, b, c ∈ A.
Condition: Whenever (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R, for all a, b, c ∈ A.
Example: R = {(1,2), (2,3), (1,3)} → Transitive ✓
R = {(1,2), (2,3)} → Not transitive ✗ (because (1,3) ∉ R)
R = {(1,2), (2,3)} → Not transitive ✗ (because (1,3) ∉ R)
2.6 Equivalence Relation ⭐ (Most Important)
Definition: A relation R on set A is called an equivalence relation if it is simultaneously:
✅ Reflexive ✅ Symmetric ✅ Transitive
✅ Reflexive ✅ Symmetric ✅ Transitive
Example: Let A = set of all triangles. Define R: "Triangle T₁ is similar to Triangle T₂."
• Reflexive: Every triangle is similar to itself ✓
• Symmetric: If T₁ ~ T₂, then T₂ ~ T₁ ✓
• Transitive: If T₁ ~ T₂ and T₂ ~ T₃, then T₁ ~ T₃ ✓
∴ R is an equivalence relation.
• Reflexive: Every triangle is similar to itself ✓
• Symmetric: If T₁ ~ T₂, then T₂ ~ T₁ ✓
• Transitive: If T₁ ~ T₂ and T₂ ~ T₃, then T₁ ~ T₃ ✓
∴ R is an equivalence relation.
⭐ Equivalence Class: For an equivalence relation R on A, the equivalence class of element a ∈ A is the set of all elements related to a:
[a] = {b ∈ A : (a, b) ∈ R}
[a] = {b ∈ A : (a, b) ∈ R}
3. Quick Comparison Table
| Type | Condition | Key Word |
|---|---|---|
| Reflexive | (a, a) ∈ R ∀ a ∈ A | Every element related to itself |
| Symmetric | (a, b) ∈ R ⇒ (b, a) ∈ R | Reverse pair also exists |
| Transitive | (a, b), (b, c) ∈ R ⇒ (a, c) ∈ R | Chain pairs covered |
| Equivalence | All three above | Reflexive + Symmetric + Transitive |
4. Types of Functions
A function f: A → B is a special relation where every element of A is associated with exactly one element of B.
4.1 One-One Function (Injective)
Definition: f: A → B is called one-one (or injective) if different elements of A have different images in B.
Condition: f(a) = f(b) ⇒ a = b, for all a, b ∈ A.
Equivalently: a ≠ b ⇒ f(a) ≠ f(b).
Condition: f(a) = f(b) ⇒ a = b, for all a, b ∈ A.
Equivalently: a ≠ b ⇒ f(a) ≠ f(b).
Example: f: R → R defined by f(x) = 3x + 5 is one-one.
Proof: f(a) = f(b) ⇒ 3a + 5 = 3b + 5 ⇒ 3a = 3b ⇒ a = b ✓
Proof: f(a) = f(b) ⇒ 3a + 5 = 3b + 5 ⇒ 3a = 3b ⇒ a = b ✓
4.2 Onto Function (Surjective)
Definition: f: A → B is called onto (or surjective) if every element of B has at least one pre-image in A.
Condition: For every b ∈ B, there exists a ∈ A such that f(a) = b.
In other words: Range of f = Co-domain of f.
Condition: For every b ∈ B, there exists a ∈ A such that f(a) = b.
In other words: Range of f = Co-domain of f.
Example: f: R → R defined by f(x) = x2 is NOT onto.
(Because negative numbers like -1 have no pre-image in R since x2 ≥ 0 always.)
(Because negative numbers like -1 have no pre-image in R since x2 ≥ 0 always.)
4.3 Bijective Function ⭐
Definition: f: A → B is called bijective (or one-one onto) if it is both:
✅ One-One (Injective) ✅ Onto (Surjective)
✅ One-One (Injective) ✅ Onto (Surjective)
Example: f: R → R defined by f(x) = 2x + 3 is bijective.
• One-one: f(a) = f(b) ⇒ 2a+3 = 2b+3 ⇒ a = b ✓
• Onto: For any y ∈ R, x = (y−3)/2 ∈ R and f(x) = y ✓
• One-one: f(a) = f(b) ⇒ 2a+3 = 2b+3 ⇒ a = b ✓
• Onto: For any y ∈ R, x = (y−3)/2 ∈ R and f(x) = y ✓
5. Composition of Functions
Definition: Let f: A → B and g: B → C. The composition of f and g, written as gof, is defined by:
(gof)(x) = g(f(x)), for all x ∈ A.
(gof)(x) = g(f(x)), for all x ∈ A.
gof: A → C, where (gof)(x) = g(f(x))
fog: B → B, where (fog)(x) = f(g(x)) [if domains match]
fog: B → B, where (fog)(x) = f(g(x)) [if domains match]
⚠️ Important: In general, gof ≠ fog. Composition of functions is NOT commutative, but it IS associative.
i.e., ho(gof) = (hog)of
i.e., ho(gof) = (hog)of
Example: Let f(x) = 2x and g(x) = x + 3.
gof(x) = g(f(x)) = g(2x) = 2x + 3
fog(x) = f(g(x)) = f(x+3) = 2(x+3) = 2x + 6
Clearly gof ≠ fog.
gof(x) = g(f(x)) = g(2x) = 2x + 3
fog(x) = f(g(x)) = f(x+3) = 2(x+3) = 2x + 6
Clearly gof ≠ fog.
6. Invertible Functions
Definition: A function f: A → B is called invertible if there exists a function g: B → A such that:
gof =
gof =
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