RD Sharma solutions for Mathematics [English] Class 12 chapter 3 - Binary Operations [Latest edition]

Determine whether the following operation define a binary operation on the given set or not : '*' on N defined by a * b = a^b for all a, b ∈ N.

Determine whether the following operation define a binary operation on the given set or not : 'O' on Z defined by a O b = a^b for all a, b ∈ Z.

Determine whether the following operation define a binary operation on the given set or not : '*' on N defined by a * b = a + b - 2 for all a, b ∈ N

Determine whether the following operation define a binary operation on the given set or not : '×₆' on S = {1, 2, 3, 4, 5} defined by

a ×₆ b = Remainder when ab is divided by 6.

Determine whether the following operation define a binary operation on the given set or not :

\[' +_6 ' \text{on S} = \left\{ 0, 1, 2, 3, 4, 5 \right\} \text{defined by}\] 
\[a +_6 b = \begin{cases}a + b & ,\text{ if a} + b < 6 \\ a + b - 6 & , \text{if a} + b \geq 6\end{cases}\]

Determine whether the following operation define a binary operation on the given set or not : '⊙' on N defined by a ⊙ b= a^b + b^a for all a, b ∈ N

Determine whether the following operation define a binary operation on the given set or not :

\[' * ' \text{on Q defined by } a * b = \frac{a - 1}{b + 1} \text{for all a, b} \in Q .\]

Determine whether or not the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this.
On Z⁺, defined * by a * b = a − b

Determine whether or not the definition of *given below gives a binary operation. In the event that * is not a binary operation give justification of this.

On Z⁺, defined * by a * b = ab

Here, Z⁺ denotes the set of all non-negative integers.

Determine whether or not the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this.

On R, define by a*b = ab²

Here, Z⁺ denotes the set of all non-negative integers.

Determine whether or not the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this. 

On Z⁺ define * by a * b = |a − b|

Here, Z⁺ denotes the set of all non-negative integers.

Determine whether or not the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this.

 On Z⁺, define * by a * b = a

Here, Z⁺ denotes the set of all non-negative integers.

Determine whether or not the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this.

On R, define * by a * b = a + 4b²

Here, Z⁺ denotes the set of all non-negative integers.

Let * be a binary operation on the set I of integers, defined by a * b = 2a + b − 3. Find the value of 3 * 4.

Is * defined on the set {1, 2, 3, 4, 5} by a * b = LCM of a and b a binary operation? Justify your answer.

Let S = {a, b, c}. Find the total number of binary operations on S.

Find the total number of binary operations on {a, b}.

Let S be the set of all rational numbers of the form \[\frac{m}{n}\] , where m ∈ Z and n = 1, 2, 3. Prove that * on S defined by a * b = ab is not a binary operation.

Prove that the operation * on the set

\[M = \left\{ \begin{bmatrix}a & 0 \\ 0 & b\end{bmatrix}; a, b \in R - \left\{ 0 \right\} \right\}\] defined by A * B = AB is a binary operation.

The binary operation * : R × R → R is defined as a * b = 2a + b. Find (2 * 3) * 4.

Let * be a binary operation on N given by a * b = LCM (a, b) for all a, b ∈ N. Find 5 * 7.

Let '*' be a binary operation on N defined by
a * b = 1.c.m. (a, b) for all a, b ∈ N
Find 2 * 4, 3 * 5, 1 * 6.

Let '*' be a binary operation on N defined by a * b = 1.c.m. (a, b) for all a, b ∈ N

Check the commutativity and associativity of '*' on N.

Determine which of the following binary operation is associative and which is commutative : * on N defined by a * b = 1 for all a, b ∈ N ?

Determine which of the following binary operations are associative and which are commutative : * on Q defined by \[a * b = \frac{a + b}{2} \text{ for all a, b } \in Q\] ?

Let A be any set containing more than one element. Let '*' be a binary operation on A defined by a * b = b for all a, b ∈ A Is '*' commutative or associative on A ?

Check the commutativity and associativity of the following binary operation '*'. on Z defined by a * b = a + b + ab for all a, b ∈ Z ?

Check the commutativity and associativity of the following binary operations '*'. on N defined by a * b = 2^ab for all a, b ∈ N ?

Check the commutativity and associativity of the following binary operations '*'. on Q defined by a * b = a − b for all a, b ∈ Q ?

Check the commutativity and associativity of the following binary operations '⊙' on Q defined by a ⊙ b = a² + b² for all a, b ∈ Q ?

Check the commutativity and associativity of the following binary operation 'o' on Q defined by \[\text{a o b }= \frac{ab}{2}\] for all a, b ∈ Q ?

Check the commutativity and associativity of the following binary operation '*' on Q defined by a * b = ab² for all a, b ∈ Q ?

Check the commutativity and associativity of the following binary operation '*' on Q defined by a * b = a + ab for all a, b ∈ Q ?

Check the commutativity and associativity of the following binary operation  '*' on R defined by a * b = a + b − 7 for all a, b ∈ R ?

Check the commutativity and associativity of the following binary operation'*' on Q defined by a * b = (a − b)² for all a, b ∈ Q ?

 Check the commutativity and associativity of the following binary operation'*' on Q defined by a * b = ab + 1 for all a, b ∈ Q ?

Check the commutativity and associativity of the following binary operation '*' on N, defined by a * b = a^b for all a, b ∈ N ?

Check the commutativity and associativity of the following binary operation '*' on Z defined by a * b = a − b for all a, b ∈ Z ?

Check the commutativity and associativity of the following binary operation '*' on Q defined by \[a * b = \frac{ab}{4}\] for all a, b ∈ Q ?

Check the commutativity and associativity of the following binary operation  '*' on Z defined by a * b = a + b − ab for all a, b ∈ Z ?

Check the commutativity and associativity of the following binary operation '*' on N defined by a * b = gcd(a, b) for all a, b ∈ N ?

If the binary operation o is defined by aob = a + b − ab on the set Q − {−1} of all rational numbers other than 1, shown that o is commutative on Q − [1].

Show that the binary operation * on Z defined by a * b = 3a + 7b is not commutative ?

On the set Z of integers a binary operation * is defined by a * b = ab + 1 for all a , b ∈ Z. Prove that * is not associative on Z.

Let S be the set of all real numbers except −1 and let '*' be an operation defined by a * b = a + b + ab for all a, b ∈ S. Determine whether '*' is a binary operation on S. If yes, check its commutativity and associativity. Also, solve the equation (2 * x) * 3 = 7.

On Q, the set of all rational numbers, * is defined by \[a * b = \frac{a - b}{2}\] , shown that * is no associative ?

On Z, the set of all integers, a binary operation * is defined by a * b = a + 3b − 4. Prove that * is neither commutative nor associative on Z.

On the set Q of all ration numbers if a binary operation * is defined by \[a * b = \frac{ab}{5}\] , prove that * is associative on Q.

The binary operation * is defined by \[a * b = \frac{ab}{7}\] on the set Q of all rational numbers. Show that * is associative.

On Q, the set of all rational numbers a binary operation * is defined by \[a * b = \frac{a + b}{2}\] Show that * is not associative on Q.

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 \[\in\] S:

Prove that * is a binary operation on S ?

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 \[\in\] S:

Prove that * is commutative as well as associative ?

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

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

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

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

Let * be a binary operation on Z defined by
a * b = a + b − 4 for all a, b ∈ Z Show that '*' is both commutative and associative ?

Let * be a binary operation on Z defined by
a * b = a + b − 4 for all a, b ∈ Z Find the identity element in Z ?

Let * be a binary operation on Z defined by
a * b = a + b − 4 for all a, b ∈ Z Find the invertible elements in Z ?

Let * be a binary operation on Q₀ (set of non-zero rational numbers) defined by \[a * b = \frac{ab}{5} \text{for all a, b} \in Q_0\]

 Show that * is commutative as well as associative. Also, find its identity element if it exists.

Let * be a binary operation on Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Show that '*' is both commutative and associative on Q − {−1}.

Let * be a binary operation on Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Find the identity element in Q − {−1} ?

Let * be a binary operation on Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Show that every element of Q − {−1} is invertible. Also, find the inverse of an arbitrary element ?

Let A = R₀ × R, where R₀ denote the set of all non-zero real numbers. A binary operation '⊙' is defined on A as follows (a, b) ⊙ (c, d) = (ac, bc + d) for all (a, b), (c, d) ∈ R₀ × R :

Show that '⊙' is commutative and associative on A ?

Let A = R₀ × R, where R₀ denote the set of all non-zero real numbers. A binary operation '⊙' is defined on A as follows (a, b) ⊙ (c, d) = (ac, bc + d) for all (a, b), (c, d) ∈ R₀ × R :

Find the identity element in A ?

Let A = R₀ × R, where R₀ denote the set of all non-zero real numbers. A binary operation '⊙' is defined on A as follows (a, b) ⊙ (c, d) = (ac, bc + d) for all (a, b), (c, d) ∈ R₀ × R :

Find the invertible elements in A ?

Let 'o' be a binary operation on the set Q₀ of all non-zero rational numbers defined by   \[a o b = \frac{ab}{2}, \text{for all a, b} \in Q_0\].

Show that 'o' is both commutative and associate ?

Let 'o' be a binary operation on the set Q₀ of all non-zero rational numbers defined by \[a o b = \frac{ab}{2}, \text{ for all a, b } \in Q_0\] :

 Find the identity element in Q₀.

Let 'o' be a binary operation on the set Q₀ of all non-zero rational numbers defined by  \[a o b = \frac{ab}{2}, \text{ for all a, b } \in Q_0\]:

Find the invertible elements of Q_{0 ?}

On R − {1}, a binary operation * is defined by a * b = a + b − ab. Prove that * is commutative and associative. Find the identity element for * on R − {1}. Also, prove that every element of R − {1} is invertible.

Let R₀ denote the set of all non-zero real numbers and let A = R₀ × R₀. If '*' is a binary operation on A defined by

(a, b) * (c, d) = (ac, bd) for all (a, b), (c, d) ∈ A

Show that '*' is both commutative and associative on A ?

Let R₀ denote the set of all non-zero real numbers and let A = R₀ × R₀. If '*' is a binary operation on A defined by

(a, b) * (c, d) = (ac, bd) for all (a, b), (c, d) ∈ A

Find the identity element in A ?

Let R₀ denote the set of all non-zero real numbers and let A = R₀ × R₀. If '*' is a binary operation on A defined by

(a, b) * (c, d) = (ac, bd) for all (a, b), (c, d) ∈ A

Find the invertible element in A ?

Let * be the binary operation on N defined by a * b = HCF of a and b.
Does there exist identity for this binary operation one N ?

Let A  \[=\] R  \[\times\] R and \[*\]  be a binary operation on A defined by \[(a, b) * (c, d) = (a + c, b + d) .\] . Show that \[*\] is commutative and associative. Find the binary element for \[*\] on A, if any.

Construct the composition table for ×₄ on set S = {0, 1, 2, 3}.

Construct the composition table for +₅ on set S = {0, 1, 2, 3, 4}.

Construct the composition table for ×₅ on Z₅ = {0, 1, 2, 3, 4}.

For the binary operation ×₁₀ on set S = {1, 3, 7, 9}, find the inverse of 3.

For the binary operation ×₇ on the set S = {1, 2, 3, 4, 5, 6}, compute 3⁻¹ ×₇ 4.

Find the inverse of 5 under multiplication modulo 11 on Z₁₁.

Write the multiplication table for the set of integers modulo 5.

Consider the binary operation * defined by the following tables on set S = {a, b, c, d}.

Show that the binary operation is commutative and associative. Write down the identities and list the inverse of elements.

Consider the binary operation 'o' defined by the following tables on set S = {a, b, c, d}.

Show that the binary operation is commutative and associative. Write down the identities and list the inverse of elements.

Define a binary operation * on the set {0, 1, 2, 3, 4, 5} as \[a * b = \begin{cases}a + b & ,\text{ if a  + b} < 6 \\ a + b - 6 & , \text{if a + b} \geq 6\end{cases}\]

Show that 0 is the identity for this operation and each element a ≠ 0 of the set is invertible with 6 − a being the inverse of a.

Write the identity element for the binary operation * on the set R₀ of all non-zero real numbers by the rule \[a * b = \frac{ab}{2}\] for all a, b ∈ R₀.

On the set Z of all integers a binary operation * is defined by a * b = a + b + 2 for all a, b ∈ Z. Write the inverse of 4.

Define a binary operation on a set.

Define a commutative binary operation on a set.

Define an associative binary operation on a set.

Write the total number of binary operations on a set consisting of two elements.

Write the identity element for the binary operation * defined on the set R of all real numbers by the rule

\[a * b = \frac{3ab}{7} \text{ for all a, b} \in R .\] ?

Let * be a binary operation, on the set of all non-zero real numbers, given by \[a * b = \frac{ab}{5} \text { for all a, b } \in R - \left\{ 0 \right\}\]

Write the value of x given by 2 * (x * 5) = 10.

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

Define identity element for a binary operation defined on a set.

Write the composition table for the binary operation multiplication modulo 10 (×₁₀) on the set S = {2, 4, 6, 8}.

For the binary operation multiplication modulo 10 (×₁₀) defined on the set S = {1, 3, 7, 9}, write the inverse of 3.

For the binary operation multiplication modulo 5 (×₅) defined on the set S = {1, 2, 3, 4}. Write the value of \[\left( 3 \times_5 4^{- 1} \right)^{- 1}.\] 

Write the composition table for the binary operation ×₅ (multiplication modulo 5) on the set S = {0, 1, 2, 3, 4}.

A binary operation * is defined on the set R of all real numbers by the rule \[a * b = \sqrt{  a^2 + b^2} \text{for all a, b } \in R .\]

Write the identity element for * on R.

Let +₆ (addition modulo 6) be a binary operation on S = {0, 1, 2, 3, 4, 5}. Write the value of \[2 +_6 4^{- 1} +_6 3^{- 1} .\]

Let * be a binary operation defined by a * b = 3a + 4b − 2. Find 4 * 5.

If the binary operation * on the set Z of integers is defined by a * b = a + 3b², find the value of 2 * 4.

Let * be a binary operation on N given by a * b = HCF (a, b), a, b ∈ N. Write the value of 22 * 4.

Let * be a binary operation on set of integers I, defined by a * b = 2a + b − 3. Find the value of 3 * 4.

If a * b = a² + b², then the value of (4 * 5) * 3 is _____________ .

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

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

\[X ∆ Y = \left( \overline{X} \cap Y \right) \cup \left( X \cap \overline{Y} \right)\]

Then which are of the following statements is true about ∆.

If the binary operation * on Z is defined by a * b = a² − b² + ab + 4, then value of (2 * 3) * 4 is ____________ .

Mark the correct alternative in the following question:-

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

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 ___________ .

Q⁺ denote the set of all positive rational numbers. If the binary operation a ⊙ on Q⁺ is defined as \[a \odot = \frac{ab}{2}\] ,then the inverse of 3 is __________ .

If G is the set of all matrices of the form

\[\begin{bmatrix}x & x \\ x & x\end{bmatrix}, \text{where x } \in R - \left\{ 0 \right\}\] 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 \[a * b = \frac{ab}{2}\] for all a, b ∈ Q⁺. The inverse of an element a ∈ Q⁺ is ______________ .

If the binary operation ⊙ is defined on the set Q⁺ of all positive rational numbers by \[a \odot b = \frac{ab}{4} . \text{ Then }, 3 \odot \left( \frac{1}{5} \odot \frac{1}{2} \right)\] is equal to __________ .

Let * be a binary operation defined on set Q − {1} by the rule a * b = a + b − ab. Then, the identify element for * is ____________ .

Which of the following is true ?

The binary operation * defined on N by a * b = a + b + ab for all a, b ∈ N is ________________ .

The binary operation * is defined by a * b = a² + b² + ab + 1, then (2 * 3) * 2 is equal to ______________ .

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

Subtraction of integers is ___________________ .

The law a + b = b + a is called _________________ .

An operation * is defined on the set Z of non-zero integers by \[a * b = \frac{a}{b}\]  for all a, b ∈ Z. Then the property satisfied is _______________ .

On Z an operation * is defined by a * b = a² + b² for all a, b ∈ Z. The operation * on Z is _______________ .

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

Let * be a binary operation on Q⁺ defined by \[a * b = \frac{ab}{100} \text{ for all a, b } \in Q^+\] The inverse of 0.1 is _________________ .

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

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 _______________ .

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 ________________ .

For the multiplication of matrices as a binary operation on the set of all matrices of the form \[\begin{bmatrix}a & b \\ - b & a\end{bmatrix}\] a, b ∈ R the inverse of \[\begin{bmatrix}2 & 3 \\ - 3 & 2\end{bmatrix}\] is ___________________ .

On the set Q⁺ of all positive rational numbers a binary operation * is defined by \[a * b = \frac{ab}{2} \text{ for all, a, b }\in Q^+\]. The inverse of 8 is _________ .

Let * be a binary operation defined on Q⁺ by the rule

\[a * b = \frac{ab}{3} \text{ for all a, b } \in Q^+\] The inverse of 4 * 6 is ___________ .

The number of binary operation that can be defined on a set of 2 elements is _________ .

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