Class 11 Maths Chapter 2 Relations and Functions – Exercise 2.2 NCERT Solutions

Introduction

Exercise 2.2 focuses on types of relations such as reflexive, symmetric, and transitive relations. You will learn how to check whether a given relation satisfies these properties and how to classify relations accordingly. These concepts are foundational for understanding equivalence relations and functions.

Key Definitions

1. Reflexive Relation: A relation $R$ on set $A$ is reflexive if $(a,a)\in R$ for all $a\in A$.

2. Symmetric Relation: A relation $R$ on set $A$ is symmetric if $(a,b)\in R⟹(b,a)\in R$.

3. Transitive Relation: A relation $R$ on set $A$ is transitive if $(a,b)\in R,(b,c)\in R⟹(a,c)\in R$.

4. Equivalence Relation: A relation that is reflexive, symmetric, and transitive.

NCERT Questions with Solutions (10)

Q1. Show that relation $R$ on set $A=\{1,2,3\}$ defined by $R=\{(1,1),(2,2),(3,3)\}$ is reflexive. Since $(a,a)\in R$ for all $a\in A$, $R$ is reflexive.

Q2. Show that relation $R$ on $A=\{1,2\}$ defined by $R=\{(1,2),(2,1)\}$ is symmetric. $(1,2)\in R⟹(2,1)\in R$. Hence symmetric.

Q3. Show that relation $R$ on $A=\{1,2,3\}$ defined by $R=\{(1,2),(2,3),(1,3)\}$ is transitive. Since $(1,2),(2,3)\in R⟹(1,3)\in R$. Hence transitive.

Q4. Check if relation $R=\{(1,1),(2,2),(3,3),(1,2),(2,1)\}$ on $A=\{1,2,3\}$ is reflexive. Yes, since $(a,a)\in R$ for all $a\in A$.

Q5. Check if relation $R$ above is symmetric. Yes, since $(1,2)\in R⟹(2,1)\in R$.

Q6. Check if relation $R$ above is transitive. Yes, since $(1,2),(2,1)\in R⟹(1,1)\in R$.

Q7. Is relation $R$ above an equivalence relation? Yes, since it is reflexive, symmetric, and transitive.

Q8. Show that relation $R$ on integers defined by $aRb⟺a-b$ is even is an equivalence relation.

• Reflexive: $a-a=0$ is even.

• Symmetric: If $a-b$ is even, then $b-a$ is even.

• Transitive: If $a-b$ and $b-c$ are even, then $a-c$ is even. Hence equivalence relation.

Q9. Show that relation $R$ on integers defined by $aRb⟺a-b$ divisible by 3 is an equivalence relation. Similar proof as above. Reflexive, symmetric, transitive → equivalence relation.

Q10. Show that relation $R$ on real numbers defined by $aRb⟺a\le b$ is not symmetric. Since $a\le b$ does not imply $b\le a$. Hence not symmetric.

FAQs (10)

1. What is reflexive relation? $(a,a)\in R$ for all $a\in A$.

2. What is symmetric relation? If $(a,b)\in R$, then $(b,a)\in R$.

3. What is transitive relation? If $(a,b),(b,c)\in R$, then $(a,c)\in R$.

4. What is equivalence relation? Reflexive, symmetric, and transitive relation.

5. Is every relation reflexive? No, only if all $(a,a)$ are included.

6. Is every relation symmetric? No, only if reverse pairs are included.

7. Is every relation transitive? No, only if condition holds for all triples.

8. Example of equivalence relation? Congruence modulo $n$.

9. Why are equivalence relations important? They partition sets into equivalence classes.

10. Why is this exercise important? It builds foundation for functions and mappings.

Conclusion

Exercise 2.2 has 10 solved questions and 10 FAQs that strengthen your understanding of types of relations. This builds the foundation for equivalence relations and functions in Class 11 Maths.

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