Class 12 RD Sharma Solutions- Chapter 3 Binary Operations - Exercise 3.3

Binary operations are fundamental concepts in mathematics and computer science that involve operations on pairs of elements. In Chapter 3 of RD Sharma’s Class 12 textbook, we explore the binary operations in detail focusing on their properties and applications. Exercise 3.3 covers practical problems and examples to help students understand these concepts better.

Binary Operations

A binary operation is a function that combines two elements from the set to produce another element within the same set. The Common binary operations include addition, subtraction, multiplication, and division. These operations are defined by the specific rules that govern how pairs of elements interact to produce results.

Question 1. Find the identity element in the set I+ of all positive integers defined by a * b = a + b for all a, b ∈ I⁺.

Solution:

Let e be the identity element in I⁺ with respect to * such that

a * e = a = e * a, ∀ a ∈ I⁺

a * e = a and e * a = a, ∀ a ∈ I⁺

a + e = a and e + a = a, ∀ a ∈ I⁺

e = 0, ∀ a ∈ I⁺

Hence, 0 is the identity element in I⁺ with respect to *.

Question 2. Find the identity element in the set of all rational numbers except – 1 with respect to * defined by a * b = a + b + ab

Solution:

Let e be the identity element in I+ with respect to * such that

a * e = a = e * a, ∀ a ∈ Q – {-1}

a * e = a and e * a = a, ∀ a ∈ Q – {-1}

a + e + ae = a and e + a + ea = a, ∀ a ∈ Q – {-1}

e + ae = 0 and e + ea = 0, ∀ a ∈ Q – {-1}

e (1 + a) = 0 and e (1 + a) = 0, ∀ a ∈ Q – {-1}

e = 0, ∀ a ∈ Q – {-1} [because a not equal to -1]

Hence, 0 is the identity element in Q – {-1} with respect to *.

Question 3. If the binary operation * on the set Z is defined by a*b = a + b - 5, then find the identity element with respect to *.

Solution: 

We are given the binary operator * defined on Z as

a*b = a + b - 5 for all a, b ∈ Q

Let e be the identity elements with respect to *

Then, a*e = e*a = a  [By identity property]

⇒ a + e - 5 = a

⇒ e = 5

Therefore, the required identity element with respect to * is 5.

Question 4. On the set Z integers, if the binary operation * is defined by a*b = a + b + 2, then find the identity elements.

Solution: 

The binary operator * is defined on Z, and is given by

a*b = a + b +2 for all a, b ∈ Z.

Let a ∈ Z and e ∈ Z be the identity element with respect to *, then

a*e = e*a = a    [By identity property]

⇒ a + e + 2 = a

⇒ e = -2 ∈ Z

Therefore, the identity element with respect to * is -2.

Read More:

• Arithmetic Operations of Binary Numbers
• Binary Operation

Summary

Exercise 3.3 in Chapter 3 of RD Sharma's Class 12 mathematics textbook likely focuses on more advanced topics related to binary operations and algebraic structures. This exercise may cover concepts such as groups, abelian groups, subgroups, and various properties of these structures. It might include problems on proving group axioms, identifying subgroups, and exploring the relationships between different algebraic structures. The exercise may also delve into cyclic groups, order of elements, and Lagrange's theorem.