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

‘x₆’ on S={1,2,3,4,5} defined by a x₆ b = Remainder when ab is divided by 6.

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

‘+₆’ on S = {0,1,2,3,4,5} defined by a +₆ b=

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

‘O’ on N defined by a O b = a^b + b^a for all a,b N.

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

‘*’ on Q defined by for all a,b Q.

Determine whether or not each definition * 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

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

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

On R, defined * by a*b = ab²

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

Determine whether or not each definition * 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|

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

Determine whether or not each definition * 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

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

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

On R, defined * 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}

Prove that the operation * on the set

defined by A*B=AB is a binary operation.

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

The binary operation * defined on 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 = L.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 = L.C.M(a,b) for all a,b∈N.

Check the commutativity and associativity of ‘*’ on N.

Determine which of the following binary operations are associative and which are 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 for all a,b 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 each of the following binary operations:

‘*’ on Z defined by a*b = a + b + ab for all a,b∈Z

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

‘*’ on N defined by a*b = 2^ab for all a,b∈N

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

‘*’ on Q defined by a*b = a – b for all a,b∈Q

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

‘Ο’ on Q defined by aΟb = a² + b² for all a,b∈Q

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

‘o’ on Q defined by a o b = ab/2 for all a,b Q.

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

‘*’ on Q defined by a*b = ab² for all a,b∈Q

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

‘*’ on Q defined by a*b = a + ab for all a,b∈Q

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

‘*’ on R defined by a*b = a + b – 7 for all a,b∈Q

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

‘*’ on Q defined by a*b = (a – b)² for all a,b∈Q

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

*’ on Q defined by a*b = ab + 1 for all a,b∈Q

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

‘*’ on N defined by a*b = a^b for all a,b∈N

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

‘*’ on Z defined by a*b = a – b for all a,b∈Z

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

‘*’ on Q defined by a*b = ab/4 for all a,b∈Q

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∈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∈Q

If the binary operation ο is defined by aοb = a + b – ab on the set Q – { – 1} of all rational numbers other than – 1. Show that ο 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 ab∈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 , show that * is not 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.

The binary operation * is defined by on the set Q if all rational numbers. Show that * is associative.

On Q, the set of all rational numbers a binary operation * is defined by . 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∈S.

Prove that:

i. * is a binary operation on S

ii. * 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, then 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.

i. Show that ‘ * ’ is both commutative and associative.

ii. Find the identity element in Z.

iii. Find the invertible elements in Z

Let * be a binary operation on Q₀ (Set of non – zero rational numbers) defined by a * b = 3ab/5 for all a, b ∈ Q₀.

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}. Then,

i. Show that ‘ * ’ is both commutative and associative on Q – {– 1}.

ii. Find the identity element in Q – {– 1}.

iii. Show that every element of Q – {– 1}. Is invertible. Also, find the inverse of an arbitrary element.

Let A = R₀ x R, where R₀ denote the set of all non – zero real numbers. A binary operation ‘O’ is defined on A as follows: (a, b) ^O (c, d) = (ac, bc + d) for all (a, b), (c, d) ∈ R₀ x R.

i. Show that ‘O’ is commutative and associative on A

ii. Find the identity element in A

iii. Find the invertible elements in A.

Let ‘o’ be a binary operation on the set Q₀ if all non - zero rational numbers defined by a o b = ab/2, for all a, b ∈ Q₀.

i. Show that ‘o’ is both commutative and associate.

ii. Find the identity element in Q₀.

iii. Find the invertible elements of Q₀.

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 x R₀. If ‘0’ is a binary operation on A defined by (a, b)0(c, d) = (ac, bd), (c, d)∈A.

i. Show that ‘0’ is both commutative and associative on A

ii. Find the identity element in A

iii. 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 on N?

Let A = RxR 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 x₄ 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 x₆ on set S = {0, 1, 2, 3, 4, 5}.

Construct the composition table for x₅ on Z₅ = {0, 1, 2, 3, 4}

For the binary operation x₁₀ on set S = {1, 3, 7, 9}, find the inverse of e3.

For the binary operation x₇ on the set of S = {1, 2, 3, 4, 5, 6} compute 3 ^{– 1}x₇4.

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 * and 0 defined by the following tables on set S = {a, b, c, d}.

Consider the binary operation * and 0 defined by the following tables on set S = {a, b, c, d}.

Write the identity element for the binary operation * on the set R₀ of all non-zero real numbers by the rule 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 for all a, b ∈ R.

Let * be a binary operation, on the set of all non-zero real numbers, given by for all a, b ∈ R – {0}. 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 (3 ×₅ 4^–1)^–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 for all a, b ∈ 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.

Mark the correct alternative in each of the following:

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

Mark the correct alternative in each of the following:

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

Mark the correct alternative in each of the following:

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

Mark the correct alternative in each of the following:

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 each of the following:

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

Mark the correct alternative in each of the following:

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

Mark the correct alternative in each of the following:

Q⁺ denote the set of all positive rational numbers. If the binary operation on Q⁺ is defined as a , then the inverse of 3 is

Mark the correct alternative in each of the following:

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

Mark the correct alternative in each of the following:

Q⁺ is the set of all positive rational numbers with the binary operation * defined by for all a, b, ∈ Q⁺. The inverse of an element a ∈ Q⁺ is

Mark the correct alternative in each of the following:

If the binary operation is defined on the set Q⁺ of all positive rational numbers by . Then, is equal to

Mark the correct alternative in each of the following:

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

Mark the correct alternative in each of the following:

Which of the following is true?

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

Mark the correct alternative in each of the following:

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

Mark the correct alternative in each of the following:

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

Mark the correct alternative in each of the following:

Subtraction of integers is

Mark the correct alternative in each of the following:

The law a + b = b + a is called

Mark the correct alternative in each of the following:

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

Mark the correct alternative in each of the following:

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

Mark the correct alternative in each of the following:

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

Mark the correct alternative in each of the following:

Let * be a binary operation on Q⁺ defined by for all a, b ∈ Q⁺. The inverse of 0.1 is

Mark the correct alternative in each of the following:

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

Mark the correct alternative in each of the following:

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

Mark the correct alternative in each of the following:

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

Mark the correct alternative in each of the following:

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

Mark the correct alternative in each of the following:

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

Mark the correct alternative in each of the following:

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

Mark the correct alternative in each of the following:

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

Mark the correct alternative in each of the following:

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