NCERT Class 12 Mathematics Mathematics Part-I: Chapter 1 — RELATIONS AND FUNCTIONS
This chapter, NCERT Class 12 Mathematics Part-I, Chapter 1, delves into the fundamental concepts of Relations and Functions. It revisits the definitions of relations, domain, co-domain, and range introduced in Class XI, emphasizing that a relation from set A to set B is an arbitrary subset of A × B. The chapter introduces the mathematical definition of a relation and notation like 'a R b'. It then explores different types of relations within a set A, including the empty relation (R = φ) and the universal relation (R = A × A), also termed trivial relations. The text highlights the importance of equivalence relations and introduces the foundational properties required for them: reflexive, symmetric, and transitive relations. Understanding these types of relations is crucial for further study in mathematics and for CBSE learning.
Quick info
| Board | CBSE / NCERT |
|---|---|
| Class | Class 12 |
| Subject | Mathematics |
| Book | Mathematics Part-I |
| Chapter | Chapter 1 — RELATIONS AND FUNCTIONS |
| Language | English |
| PDF type | NCERT Textbook |
| Session | CBSE 2026 |
| Reading time | 5 minutes |
| Word count | 899 |
Learning outcomes
- Understand the definition of a relation from a set A to a set B.
- Identify and differentiate between empty and universal relations.
- Define and recognize reflexive, symmetric, and transitive relations.
- Grasp the foundational concepts for understanding equivalence relations.
Vocabulary
| Word | Meaning |
|---|---|
| Relation | A subset of the Cartesian product of two sets. |
| Domain | The set of all first elements of the ordered pairs in a relation. |
| Co-domain | The set from which the second elements of the ordered pairs are taken. |
| Range | The set of all second elements of the ordered pairs in a relation. |
| Empty Relation | A relation where no element of a set is related to any element of the same set (R = φ). |
| Universal Relation | A relation where every element of a set is related to every element of the same set (R = A × A). |
| Reflexive Relation | A relation R in set A is reflexive if (a, a) ∈ R for every a ∈ A. |
| Symmetric Relation | A relation R in set A is symmetric if (a1, a2) ∈ R implies (a2, a1) ∈ R for all a1, a2 ∈ A. |
| Transitive Relation | A relation R in set A is transitive if (a1, a2) ∈ R and (a2, a3) ∈ R implies (a1, a3) ∈ R for all a1, a2, a3 ∈ A. |
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Practice questions
- Let A = {1, 2, 3}. Which of the following relations is an empty relation? Answer: R = {(a, b): a - b = 5}
- Let A = {1, 2, 3}. Which of the following relations is a universal relation? Answer: R = {(a, b): |a - b| ≥ 0}
- Define a symmetric relation. Answer: A relation R in a set A is symmetric if (a1, a2) ∈ R implies that (a2, a1) ∈ R, for all a1, a2 ∈ A.
- Define a transitive relation. Answer: A relation R in a set A is transitive if (a1, a2) ∈ R and (a2, a3) ∈ R implies that (a1, a3) ∈ R, for all a1, a2, a3 ∈ A.
Practice MCQs
Q1. What is a relation R in a set A called if R = φ?
Explanation: An empty relation is defined as a relation R in a set A where no element of A is related to any element of A, meaning R is the empty set (φ).
Q2. What is a relation R in a set A called if R = A × A?
Explanation: A universal relation is a relation R in a set A where every element of A is related to every element of A, meaning R is the Cartesian product A × A.
Q3. For a relation R in a set A to be reflexive, which condition must be satisfied?
Explanation: A relation R in a set A is reflexive if every element 'a' in set A is related to itself, i.e., (a, a) is an element of R.
Q4. If (a1, a2) ∈ R and (a2, a1) ∈ R for all a1, a2 ∈ A, the relation R is:
Explanation: The definition of a symmetric relation states that if an ordered pair (a1, a2) is in the relation R, then the reversed pair (a2, a1) must also be in R.
Q5. If (a1, a2) ∈ R and (a2, a3) ∈ R implies (a1, a3) ∈ R for all a1, a2, a3 ∈ A, the relation R is:
Explanation: This is the definition of a transitive relation. If a relation holds between a1 and a2, and between a2 and a3, it must also hold between a1 and a3.
Frequently asked questions
What is the difference between a relation and a function?
A function is a special type of relation where each element in the domain maps to exactly one element in the co-domain. A relation can have elements in the domain mapping to multiple elements in the co-domain.
What are the two types of trivial relations?
The two types of trivial relations are the empty relation (R = φ) and the universal relation (R = A × A).
What is the condition for a relation R in set A to be reflexive?
A relation R in set A is reflexive if for every element 'a' in A, the pair (a, a) is in R.
When is a relation R in set A called symmetric?
A relation R in set A is symmetric if whenever (a1, a2) is in R, then (a2, a1) is also in R for all a1, a2 in A.
What does it mean for a relation R in set A to be transitive?
A relation R in set A is transitive if whenever (a1, a2) is in R and (a2, a3) is in R, then (a1, a3) must also be in R for all a1, a2, a3 in A.
What is the significance of studying reflexive, symmetric, and transitive relations?
These properties are fundamental for understanding equivalence relations, which are crucial in various areas of mathematics.
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NCERT Class 12 Mathematics — Mathematics Part-I — Chapter 1 — RELATIONS AND FUNCTIONS. Verified by NCERT Help Editorial Team. Reviewed on 29 Jul 2026. Last updated 10 Aug 2026.