Class 12 Maths Chapter 1 Relations and Functions – Exercise 1.3 NCERT Solutions

Class 12 Maths Chapter 1 Relations and Functions – Exercise 1.3 NCERT Solutions

Introduction

Exercise 1.3 focuses on binary operations. Students learn how to define operations on sets, check properties like commutativity and associativity, and identify identity and inverse elements. This exercise is essential for algebraic structures such as groups and rings.

Formulas Used

1. Binary Operation: A rule $\ast :A\times A\to A$.

2. Commutative Property:

$$a\ast b=b\ast a$$

3. Associative Property:

$$(a\ast b)\ast c=a\ast (b\ast c)$$

4. Identity Element: $e$ such that $a\ast e=e\ast a=a$.

5. Inverse Element: For each $a$, there exists $a^{-1}$ such that $a\ast a^{-1}=e$.

Students Frequently Make Mistakes

• Forgetting to check closure properly.

• Confusing identity element with inverse element.

• Assuming commutativity without proof.

• Errors in verifying associativity.

• Missing domain restrictions in binary operations.

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

Q1. Define binary operation $\ast$ on $\mathbb{N}$ by $a\ast b=a+b$. Check commutativity and associativity.

• Commutative: $a+b=b+a$. ✔

• Associative: $(a+b)+c=a+(b+c)$. ✔

Q2. Define binary operation $\ast$ on $\mathbb{R}$ by $a\ast b=a-b$. Check commutativity. $a-b\mathrm{\ne }b-a$. ✘ Not commutative.

Q3. Check associativity of above operation. $(a-b)-c=a-(b-c)$. Not equal. ✘ Not associative.

Q4. Define binary operation $\ast$ on $\mathbb{R}$ by $a\ast b=ab$. Check commutativity and associativity.

• Commutative: $ab=ba$. ✔

• Associative: $(ab)c=a(bc)$. ✔

Q5. Find identity element for multiplication on $\mathbb{R}$. Identity = 1, since $a\cdot 1=1\cdot a=a$.

Q6. Find identity element for addition on $\mathbb{R}$. Identity = 0, since $a+0=0+a=a$.

Q7. Find inverse of element $a$ under addition. Inverse = $-a$, since $a+(-a)=0$.

Q8. Find inverse of element $a$ under multiplication. Inverse = $\frac{1}{a},a\mathrm{\ne }0$.

Q9. Define binary operation $\ast$ on $\mathbb{Z}$ by $a\ast b=a+b+1$. Find identity. Let $e$ be identity.

$$a\ast e=a+e+1=a⟹e=-1$$

Identity = $-1$.

Q10. Find inverse of element $a$ under above operation. Inverse $b$ satisfies $a\ast b=-1$.

$$a+b+1=-1⟹b=-a-2$$

Inverse = $-a-2$.

FAQs (10)

FAQ1. What is binary operation? Rule combining two elements of set to give another element of same set.

FAQ2. What is closure property? Result of operation must belong to same set.

FAQ3. What is commutative property? Order of operation does not matter.

FAQ4. What is associative property? Grouping of elements does not matter.

FAQ5. What is identity element? Element that leaves others unchanged under operation.

FAQ6. What is inverse element? Element that combines with given element to give identity.

FAQ7. What is identity for addition?

FAQ8. What is identity for multiplication?

FAQ9. Why check closure? To ensure operation is valid on set.

FAQ10. Why is Exercise 1.3 important? It builds foundation for algebraic structures like groups and rings.

Conclusion

Exercise 1.3 has 10 solved questions and 10 FAQs that strengthen your understanding of binary operations, identity, inverse, commutativity, and associativity. This builds the foundation for algebraic structures in Class 12 Maths.

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