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# RS Aggarwal Class 12 Chapter 3 Solutions (Binary Operations)

RS Aggarwal Solutions for Class 12 Chapter 3 are prepared to help you practice extensively the Binary Operations of functions from the Cartesian Products of a set with the same set. In this chapter, we will learn about the commutative, associative and distributive properties of binary operations. You will also be introduced to the identity element and the invertible element for a binary operation besides learning to describe a binary operation on a finite set with the help of a composition table.

There are a sum total of 14 questions with comprehensive coverage of each topic in the syllabus in 2 exercises 3A and 3B. The questions range from very short answer types to the questions where you will be required to show if a given binary operation exhibits a property or not. You will also get to explore more difficult problems where you will have to define operations for given functions.

Instasolv has identified the most recurring doubts and problems of the students in an orderly fashion, therefore you will find these RS Aggarwal Solutions for Class 12 in resonance with your ability. This will also help you to practice time management in the stressful exam environment, and will, therefore, help you to entrance examinations. With relations and functions in your syllabus, these exercise questions are highly recommended before your school exams.

## Topics Covered in RS Aggarwal Solutions for Class 12 Chapter 3 Binary Operations

Introduction to Binary Operations:

A Binary Operation on a set S is a function from the cartesian product of the set with itself to the set itself. It is often denoted by ‘*’.

Therefore, *: S x S →S where * is the binary operation.

The binary operation will associate two elements a, b ∈ S to a separate element a * b ∈ S.

Commutative Property of Binary Operations:

If a * b = b * a for every a, b ∈ S, then the binary operation, denoted as *, is termed as a commutative binary operation on the set S.

For instance, the binary operations of addition (denoted as +) and multiplication (denoted as x) exhibit commutative property on Z whereas the binary operation of subtraction does not show commutativity on the set of integers denoted by Z.

Associative Property of Binary Operations:

If (a * b) * c = a * (b * c) for every a, b, c ∈ S where * is a binary operation the set S, then such a binary operation is known to exhibit the property of associativity.

The binary operation of addition and the binary operation of multiplication displays associativity on the set of integers Z while the operation of subtraction is not an associative binary operation on the set Z.

In case of a non-empty set, union and intersection are both commutative and associative binary operations on the power set of the same.

Distributive Property of Binary Operations:

If we have two binary operations denoted by ‘ * ‘ and ‘⊙’ and S as an empty set,

then * would be distributive over ⊙ if

a * (b ⊙ c) = (a * b) ⊙ (a * c) and

(b ⊙ c) * a = (b * a) ⊙ (c * a)for all a, b, c ∈ S

The binary of multiplication is distributive over the operation of addition for the set Z while the binary operation of addition does not display distributivity over the binary operation of multiplication for the set Z.

In case of a non-empty set, union exhibits distributivity over the intersection on its power set while the intersection is distributive over union also holds true for its power set.

Identity Element:

If for an element e

a * e = a = e * a holds true where * is a binary operation on the set S and a, e ∈ S,

then e is known as an Identity Element for *on S.

For example, 1 is an identity element for the binary operation of multiplication on the set of Integers.

Inverse Element:

If for an identity element e in a given set Sand a binary operation denoted as *,

a * b = e = b * a, thus a ∈ S is an invertible function for b ∈ S and vice versa.

## Highlights for Chapter 3 Binary Operations

• A binary operation is a function from the cartesian product of a set with itself with the given set itself.
• A binary operation will be (i) show commutative property if a * b = b * a for every

a, b ∈ S; (ii) show associative property if (a * b)* c = a*(b*c) for every

a, b, c ∈ S; (iii) show distributive property over another binary operation if

a*(b ⊙ c)=(a*b) ⊙ (a*c) and (b ⊙ c)*a=(b*a) ⊙ (c*a)for all a, b, c ∈ S.

### Discussion of Exercises of RS Aggarwal Solutions for Class 12 Chapter 3

1. The first exercise solutions will help you to learn various binary operations and the properties that they show with a diverse collection of problems to practice from.
2. You will be required to identify commutativity, associativity and distributivity of the binary operations besides defining the operations in some solutions.
3. The last exercise requires you to find the identity element and invertible element for a given operation in the variety of sets given in the exercise questions.
4. These questions are in line with your syllabus and will, therefore, give you a perfect idea about the patterns of questions asked in the exam.
5. Practising these questions will not only boost your time management skills in an exam but will also amplify your analytical skills.

## Benefits of RS Aggarwal Solutions for Class 12 Maths Chapter 1 by Instasolv

1. Instasolv has prepared  RS Aggarwal Class 12 Maths Solutions in layman’s language in elaborate and apt steps to aid you clear your understanding of binary operations.
2. The solutions at Instasolv are free of cost and will help you use your time effectively.
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