# RD SHARMA Solutions for Class 12-science Maths Chapter 3 - Binary Operations

## Chapter 3 - Binary Operations Exercise Ex. 3.1

Determine whether the following operation defines a binary operation on the

We have,

A ⊙ b = a^{b} + b^{a} for all a, b, ϵ N

Let a ϵ N and b ϵ N

⇒ a^{b} ϵ N and b^{a} ϵ N

⇒ a^{b} + b^{a} ϵ N

⇒ a ⊙ b ϵ N

Thus, the operation ‘⊙’ defines a binary relation on N

Let * be a binary opeartor o the set I of integers, defined by a*b = 2a + b - 3.

find the value of 3*4

It is given that, a*b = 2a + b - 3

now,

3*4 = 2 × 3 + 4 - 3

= 10 - 3

= 7

The binary operation & : R × R → R is defined as a * b = 2a + b.

Find (2*3)*4.

## Chapter 3 - Binary Operations Exercise Ex. 3.2

Check the commutativity and associativity of each of the following binary operations:

'⊙' on Q defined by a⊙b
= a^{2} + b^{2} for all a, b ϵ Q.

'⊙'
on Q defined by a⊙b = a^{2}
+ b^{2} for all a, b ϵ Q

Commutativity:

For a, b ϵ Q

a⊙b
= a^{2} + b^{2} = b^{2} + a^{2} = b⊙a

So, '⊙' is commutative on Q.

Associativity:

For a, b, c ϵ Q

(a⊙b) ⊙c
= (a^{2} + b^{2}) ⊙c = (a^{2}
+ b^{2})^{2} + c^{2 }

a⊙(b ⊙c)
=a ⊙(
b^{2} +c^{2})= a^{2} +(b^{2} + c^{2 })^{2}

(a⊙b) ⊙c ≠ a⊙(b ⊙c)

So, '⊙' is not associative on Q.

Check the commutativity and associativity of the following binary operation :

'*' on R defined by a*b = a + b - 7 for all a, b ε Q

Check the commutativity and associativity of the following binary operation :

'*' on Q defined by a*b = (a - b)^{2} for all a, b ε Q.

Check the commutativity and associativity of the following binary operation :

'*' on Q defined by a*b = ab + 1 for all a, b ε Q

Check the commutativity and associativity of the following binary operation :

'*' on N, defined by a*b = ab for all a, b, ε N.

Check the commutativity and associativity of the following binary operation :

'*' on Z, a*b = a - b for all a, b, ε Z.

Check the commutativity and associativity of the following binary operation :

Check the commutativity and associativity of each of the following binary operations:

'*' on Z defined by a * b = a + b - ab for all a, b ϵ Z.

'*' on Q defined by a*b = a + b - ab for all a, b ϵ Z

Commutativity:

For a, b ϵ Z

a*b = a + b - ab = b + a - ba = b*a

So, '*' is commutative on Z.

Associativity:

For a, b, c ϵ Z

(a*b) *c = (a + b - ab) *c

= a + b - ab + c + ac + bc - abc

a*(b*c )= a*( b + c - bc)

= a + b +c - bc + ab + ac + - abc

(a*b) *c ≠ a*(b*c )

So, '*' is not associative on Z.

Check the commutativity and associativity of each of the following binary operations:

'*' on Q defined by a * b = gcd (a, b) for all a, b ϵ N.

'*' on Q defined by a*b = gcd (a, b) for all a, b ϵ N

Commutativity:

For a, b ϵ Q

a*b = gcd (a, b) = gcd (b, a) = b*a

So, '*' is commutative on N.

Associativity:

For a, b, c ϵ N

(a*b) *c = (gcd (a, b)) *c

= gcd (a, b, c)

=a*( gcd (b, c))

=a*(b*c)

(a*b) *c = a*(b*c )

So, '*' is associative on N.

Now consider (a * b) * c.

Thus, we have, (a * b) * c = (a + b + ab) * c

= a + b + ab + c +(a + b + ab)c

= a + b + ab + c + ac + bc + abc

= a + b + c + ab + ac + bc + abc ---(i)

Let *S* be the set of all rational numbers
except 1 and ∗ be
defined on *S* by a ∗ b = a + b - ab, for all
a, b ∊ s.

Prove that:

(i) ∗ is a binary operation on S

(ii) ∗ is commutative as well as associative.

## Chapter 3 - Binary Operations Exercise Ex. 3.3

## Chapter 3 - Binary Operations Exercise Ex. 3.4

## Chapter 3 - Binary Operations Exercise Ex. 3.5

## Chapter 3 - Binary Operations Exercise MCQ

If a*b = a^{2} + b^{2}, then the value of (4*5) * 3 is

a. (4^{2} + 5^{2}) + 3^{2}

b. (4 + 5)^{2} + 3^{2}

c. 41^{2} + 3^{2}

d. (4 + 5 + 3)^{2}

If a*b denote the bigger among a and b and if a. b = (a * b) + 3, then 4.7 =

a. 14

b. 31

c. 10

d. 8

On the power set p of a non-empty set A, we define an operation Δ by

Then which are the following statements is true about Δ

a. Commutative and associative without an identity

b. Commutative but not associative with an identity

c. Associative but not commutative without an identity

d. Associative and commutative with an identity

If the binary operation* on Z is defined by a*b = a^{2} - b^{2} + ab + 4, then value of (2*3) * 4 is

a. 233

b. 33

c. 55

d. - 55

For the binary operation * on Z is defined by a*b = a + b + 1 the identity element is

a. 0

b. -1

c. 1

d. 2

If a binary operation * is defined on the set Z of integers as a * b = 3a - b, then the value of (2*3) * 4 is

a. 2

b. 3

c. 4

d. 5

Q^{+} denote the set of all positive rational numbers. If the binary operation ⊙ on Q^{+} is defined as then the inverse of 3 is

a. 4/3

b. 2

c. 1/3

d. 2/3

If G is the set of all matrices of the form , where x ∊ R - {0}, then the identity element with respect to the multiplication of matrices as binary operation, is

Q+ is the set of all positive rational numbers with the binary operation * defined by for all a, b ∊ Q^{+}. Then inverse of an element a ∊ Q^{+} is

a. A

b.

c.

d.

If the binary operation is defined on the set Q+ of all positive rational numbers by

Let * be a binary operation defined on set Q - {1} by the rule

a*b = a + b - ab. Then, the identity element for * is

Which of the following is true?

a. * defined by a * b = is a binary operation on Z

b. * defined by a * b = is a binary operation on Q

c. All binary commutative operations are associative

d. Subtraction is a binary operation on N.

The binary operations * defined on n by a * b = a+ b + ab for all a, b ∊ N is.

a. Commutative only

b. Associative only

c. Commutative and associative both

d. None of these

If a binary operation * is defined by a * b = a^{2} + b^{2} + ab + 1, then (2 * 3) * 2 is equal to

a. 20

b. 40

c. 400

d. 445

Let * be a binary operation on R defined by a * b = ab + 1. Then, * is

a. Commutative but not associative

b. Associative but not commutative

c. Neither commutative nor associative

d. Both commutative and associative

Subtraction of integers is

a. Commutative but not associative

b. Commutative and associative

c. Associative but not commutative

d. Neither commutative nor associative

The law a + b = b + a is called

a. Closure law

b. Associative law

c. Commutative law

d. Neither commutative nor associative

An operation * is defined on the set Z of non-zero integers by for all a, b ∊ Z. Then the property satisfied is

a. Closure

b. Commutative

c. Associative

d. None of these

On Z an operation * is defined by a * b = a^{2} + b^{2} for all a, b ∊ Z. Then operation * on Z is

a. Commutative and associative

b. Associative but not commutative

c. Not associative

d. Not a binary operation

A binary operation * on Z defined by a * b = 3a + b for all a, b ∊ Z, is

a. Commutative

b. Associative

c. Not commutative

d. Commutative and associative

Let * be a binary operation on Q^{+} defined by a * b = for all a, b ∊ Q^{+}. The inverse of 0.1 is

a. 10^{5}

b. 10^{4}

c. 10^{6}

d. None of these

Let * be a binary operation on N defined by a * b = a + b + 10 for all a, b ∊ N. The identify element for * in N is

a. -10

b. 0

c. 10

d. Non-existent

Consider the binary operation * defined on Q - {1} by the rule a * b = a + b - ab for all a, b ∊ Q - {1}. The identity element in Q - {1} is

a. 0

b. 1

c.

d. -1

For the binary operation * defined on R - {- 1} by the rule a * b = a + b + ab for all a, b ∊ R - {1}, the inverse of a is

a. -a

b.

c.

d. a^{2}

For the multiplication of matrices as a binary operation on the set of all matrices of the form , a, b ∊ R the inverse is

On the set Q^{+} of all positive rational numbers a binary operation * is defined by a * b = for all a, b ∊ Q^{+}. The inverse of 8 is

a.

b.

c. 2

d. 4

Let * be a binary option defined on Q+ by the rule a * b = for all a, b, ∊ Q+. The inverse of 4 * 6 is

a.

b.

c.

d. None of these

The number of binary operations that can be defined on a set of 2 element is

a. 8

b. 4

c. 16

d. 64

The number of commutative binary operations that can be defined on a set of 2 elements is

a. 8

b. 6

c. 4

d. 2

## Chapter 3 - Binary Operations Exercise Ex.3VSAQ

Write the inverse of 5 under multiplication modulo 11 on the set {1, 2, ..., 10}.

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