Class 12 Math Chapter 1 Ex 1.1 – Types of Relations – Solutions

Introduction – Types of Relations

Types of relations in the most important topics in the Class 12th Math . Relations can be classified into different types based on specific properties:

1. Empty Relation: A relation with no elements, i.e., R=∅.
2. Universal Relation: A relation containing all possible pairs from A×A.
3. Reflexive Relation: Every element is related to itself, i.e., (a,a)∈R for all a∈A.
4. Symmetric Relation: If (a,b)∈R, then (b,a)∈R.
5. Transitive Relation: If (a,b)∈R and (b,c)∈R then (a,c)∈R.
6. Equivalence Relation: A relation that is reflexive, symmetric, and transitive.

Exercise 1.1 Solutions – Types of Relations

The exercise focuses on identifying and verifying the properties of relations, specifically reflexive, symmetric, transitive, or equivalence.

Example 1:

Given A={1,2,3} and a relation R={(1,1),(2,2),(3,3)}. Check whether R is reflexive, symmetric, and transitive.

Solution:

1. Reflexive: All elements of A are related to themselves, i.e., (1,1),(2,2),(3,3)∈R. Hence, R is reflexive.
2. Symmetric: Since RRR only contains self-pairs and no (a,b)≠(b,a). R is symmetric.
3. Transitive: For (1,1),(2,2),(3,3), the condition of transitivity holds trivially. R is transitive.

Thus, R is an equivalence relation.

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Example 2:

Let A={1,2,3}, and R={(1,2),(2,3)}. Check if R is transitive.

Solution:

• (1,2)∈R and (2,3)∈R, but (1,3)∉R. Thus, R is not transitive.

FAQs on Types of Relations

1. What is a relation in mathematics?
   A relation is a subset of the Cartesian product AXB times B that defines a connection between elements of sets A and B.
2. What are the types of relations?
   The main types are:
   • Empty Relation
   • Universal Relation
   • Reflexive Relation
   • Symmetric Relation
   • Transitive Relation
   • Equivalence Relation
3. What is an equivalence relation?
   An equivalence relation is a relation that is reflexive, symmetric, and transitive.
4. What is the difference between reflexive and symmetric relations?
   • Reflexive: Each element is related to itself, i.e., (a,a)∈R in R for all a∈A.
   • Symmetric: If (a,b)∈R, then (b,a)∈R.
5. Can a relation be both symmetric and transitive but not reflexive?
   Yes, a relation can be symmetric and transitive without being reflexive.
6. What is an empty relation?
   An empty relation has no elements, i.e., R=∅.
7. What is a universal relation?
   A universal relation contains all possible pairs from A×A.
8. How do you identify a transitive relation?
   A relation RR is transitive if (a,b)∈R and (b,c)∈R implies (a,c)∈R.
9. What are real-life examples of relations?
   Examples include:
   • “Is a sibling of” (symmetric relation).
   • “Is greater than” (transitive relation).
   • “Is equal to” (equivalence relation).
10. Can a relation belong to more than one type?
    Yes, for example, an equivalence relation is reflexive, symmetric, and transitive.