Class 12 Maths Chapter 1 Relations and Functions – Exercise 1.4 NCERT Solutions

Introduction

Exercise 1.4 focuses on equivalence relations. Students learn how to prove reflexivity, symmetry, and transitivity of relations, and how to partition sets using equivalence relations. This exercise is crucial for understanding algebraic structures and set theory.

Formulas Used

1. Reflexive Relation:

$$(a,a)\in R\mathrm{\forall }a\in A$$

2. Symmetric Relation:

$$(a,b)\in R⟹(b,a)\in R$$

3. Transitive Relation:

$$(a,b),(b,c)\in R⟹(a,c)\in R$$

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

Students Frequently Make Mistakes

• Forgetting to check all three properties separately.

• Assuming symmetry without proof.

• Confusing equivalence relation with partial order relation.

• Missing counterexamples when relation fails.

• Errors in partitioning sets into equivalence classes.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. Show that relation $R$ on $\mathbb{Z}$ defined by $aRb⟺a-b$ divisible by 5 is equivalence.

• Reflexive: $a-a=0$ divisible by 5. ✔

• Symmetric: If $a-b$ divisible by 5, then $b-a$ divisible by 5. ✔

• Transitive: If $a-b$ and $b-c$ divisible by 5, then $a-c$ divisible by 5. ✔ Hence equivalence.

Q2. Show that relation $R$ on $\mathbb{Z}$ defined by $aRb⟺a-b$ divisible by 2 is equivalence.

• Reflexive: $a-a=0$ divisible by 2. ✔

• Symmetric: If $a-b$ divisible by 2, then $b-a$ divisible by 2. ✔

• Transitive: If $a-b$ and $b-c$ divisible by 2, then $a-c$ divisible by 2. ✔ Hence equivalence.

Q3. Show that relation $R$ on $\mathbb{R}$ defined by $aRb⟺\mathrm{\mid }a-b\mathrm{\mid }\le 2$ is not equivalence. Fails transitive property. Example: $a=1,b=2,c=4$.

Q4. Show that relation $R$ on set $A=\{1,2,3\}$ defined by $R=\{(a,b):a=b\}$ is equivalence.

• Reflexive: $(1,1),(2,2),(3,3)\in R$. ✔

• Symmetric: If $(a,b)\in R$, then $a=b⟹(b,a)\in R$. ✔

• Transitive: If $(a,b),(b,c)\in R$, then $a=b=c⟹(a,c)\in R$. ✔

Q5. Show that relation $R$ on $\mathbb{N}$ defined by $aRb⟺a^2=b^2$ is equivalence.

• Reflexive: $a^2=a^2$. ✔

• Symmetric: If $a^2=b^2$, then $b^2=a^2$. ✔

• Transitive: If $a^2=b^2$ and $b^2=c^2$, then $a^2=c^2$. ✔

Q6. Show that relation $R$ on $\mathbb{R}$ defined by $aRb⟺a-b\in \mathbb{Q}$ is equivalence.

• Reflexive: $a-a=0\in \mathbb{Q}$. ✔

• Symmetric: If $a-b\in \mathbb{Q}$, then $b-a=-(a-b)\in \mathbb{Q}$. ✔

• Transitive: If $a-b\in \mathbb{Q},b-c\in \mathbb{Q}$, then $a-c=(a-b)+(b-c)\in \mathbb{Q}$. ✔

Q7. Show that relation $R$ on $\mathbb{R}$ defined by $aRb⟺a-b\in \mathbb{Z}$ is equivalence. Similar proof as above. ✔

Q8. Show that relation $R$ on set of lines in plane defined by $l_{1}R{l}_2⟺l_1\parallel l_2$ is equivalence.

• Reflexive: Line parallel to itself. ✔

• Symmetric: If $l_1\parallel l_2$, then $l_2\parallel l_1$. ✔

• Transitive: If $l_1\parallel l_2,l_2\parallel l_3$, then $l_1\parallel l_3$. ✔

Q9. Show that relation $R$ on set of triangles defined by “congruent” is equivalence. Congruence is reflexive, symmetric, transitive. ✔

Q10. Show that relation $R$ on set of students defined by “same age” is equivalence.

• Reflexive: Student has same age as self. ✔

• Symmetric: If A same age as B, then B same age as A. ✔

• Transitive: If A same age as B, B same age as C, then A same age as C. ✔

FAQs (10)

FAQ1. What is equivalence relation? Relation that is reflexive, symmetric, transitive.

FAQ2. What is reflexive relation? Each element related to itself.

FAQ3. What is symmetric relation? If $aRb$, then $bRa$.

FAQ4. What is transitive relation? If $aRb$ and $bRc$, then $aRc$.

FAQ5. Is congruence of triangles equivalence? Yes.

FAQ6. Is parallelism of lines equivalence? Yes.

FAQ7. Is “greater than” relation equivalence? No, fails reflexive and symmetric.

FAQ8. Is “equal age” relation equivalence? Yes.

FAQ9. What is partition of set by equivalence relation? Division into disjoint equivalence classes.

FAQ10. Why is Exercise 1.4 important? It builds foundation for equivalence classes and algebraic structures.

Conclusion

Exercise 1.4 has 10 solved questions and 10 FAQs that strengthen your understanding of equivalence relations. This completes the Relations and Functions chapter in Class 12 Maths.

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