Question 1: Determine whether each of the following operations define binary operation on the given set or not:

(i)

(ii)

(iii)

(vi)

Answer:

(i)

Given that is an operation that is valid in the Natural Numbers and it is defined as given:

According to the problem it is given that on applying the operation for two given natural numbers it gives a natural number as a result of the operation,

So, we can state that,

From (1) and (2) we can see that both L.H.S and R.H.S gave only Natural numbers as a result.

Thus we can clearly state that is a Binary Operation on .

(ii)

Given that is an operation that is valid in the Integers and it is defined as given:

According to the problem it is given that on applying the operation for two given integers it gives Integers as a result of the operation,

Let us values of on substituting in the R.H.S side we get,

From (2), we can see that ab doesn’t give only Integers as a result. So, this cannot be stated as a binary function.

Therefore the operation does not define a binary function on

(iii)

Given that is an operation that is valid in the Natural Numbers and it is defined as given:

According to the problem it is given that on applying the operation for two given natural numbers it gives a natural number as a result of the operation,

Let us take the values of substituting in the R.H.S side we get,

From (2), we can see that doesn’t give only Natural numbers as a result.

So, this cannot be stated as a binary function.

Therefore the operation does not define a binary operation on .

Given that is an operation that is valid for the numbers in the Set and it is defined as given:

According to the problem it is given that on applying the operation for two given numbers in the set it gives one of the numbers in the set as a result of the operation,

Let us take the values of

We know that 12 is a multiple of 6. So, on dividing 12 with 6 we get 0 as remainder which is not in the given set

The operation does not define a binary operation on set

(vi)

Given that is an operation that is valid in the Natural Numbers and it is defined as given:

According to the problem it is given that on applying the operation for two given natural numbers it gives a natural number as a result of the operation,

We also know that the sum of two natural numbers is a natural number.

Therefore the operation defines the binary operation on N.

Given that is an operation that is valid in the Rational Numbers and it is defined as given:

Since

According to the problem it is given that on applying the operation for two given rational numbers it gives a rational number as a result of the operation,

Therefore the operation does not define a binary operation on Q.

Question 2: Determine whether or not each of the definition of given below gives a binary operation. In the event that is not a binary operation give justification of this.

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

Answer:

Given that is an operation that is valid in the Positive integers and it is defined as given:

According to the problem it is given that on applying the operation for two given positive integers it gives a positive integer as a result of the operation,

Therefore the operation does not define a binary operation on

Given that is an operation that is valid in the Positive integers and it is defined as given:

According to the problem it is given that on applying the operation for two given positive integers it gives a Positive integer as a result of the operation,

Therefore the operation defines a binary operation on

Given that is an operation that is valid in the Real Numbers ‘R’ and it is defined as given:

According to the problem it is given that on applying the operation for two given real numbers it gives a real number as a result of the operation,

Therefore the operation defines a binary operation on

Given that is an operation that is valid in the Positive integers and it is defined as given:

According to the problem it is given that on applying the operation for two given positive integers it gives a Positive integer as a result of the operation,

Therefore the operation does not define a binary function on

Given that is an operation that is valid in the Positive integers and it is defined as given:

According to the problem it is given that on applying the operation for two given positive integers it gives a Positive integer as a result of the operation,

It is told from the problem

The operation defines a binary operation on

Given that is an operation that is valid in the Real Numbers and it is defined as given:

According to the problem it is given that on applying the operation for two given real numbers it gives a Real number as a result of the operation,

We also know that the sum of two real numbers gives a real number. So,

From (1) and (2),

Therefore the operation defines a binary operation on

Question 3: Let be a binary operation on the set of integers, defined by . Find the value of **[CBSE 2011]**

Answer:

Given that is a binary operation on the set I of integers.

The operation is defined by

We need to find the value of

Since 3 and 4 belongs to the set of integers we can use the binary operation.

Therefore the value of is 7.

Question 4: Is defined on the set of and a binary operation? Justify your answer.

Answer:

Given that is an operation that is valid on the set defined by

According to the problem it is given that on applying the operation for two given numbers in the set it gives a number in the set as a result of the operation.

Let us take the values of

We know that L.C.M of two prime numbers is given by the product of that two prime numbers.

Therefore the operation does not define a binary operation on

Question 5: Let Find the total number of binary operations on

Answer:

Given set we need to find the total number of binary operations possible for the set

We know that the total number of binary operations on a set with elements is given by

Here

Therefore the total number of binary operations possible on set is

Question 6: Find the total number of binary operations only

Answer:

Given set we need to find the total number of binary operations possible for the set

We know that the total number of binary operations on a set with elements is given by

Here

Therefore the total number of binary operations possible on set is

Question 7:Prove that the operation on the set

defined by is a binary operation.

Answer:

Given that * is an operation that is valid on the set.

defined by is a binary operation.

According to the problem it is given that on applying the operation for two given numbers in the set it gives a number in the set as a result of the operation.

Therefore the operation defines a binary operation on

Question 8: Let be the set of all rational numbers of the form where and Prove that on defined by is not a binary operation.

Answer:

Given that is an operation that is valid on the set which consists of all rational numbers of the form

According to the problem it is given that on applying the operation for two given numbers in the set it gives a number in the set as a result of the operation.

Therefore the operation does not define a binary operation on

Question 9: The binary operation is defined as **[CBSE 2012]**

Answer:

Given that is an operation that is valid for the following Domain and Range and is defined by

We need to find the value of

According to the problem the binary operation involving is true for all real values of a and b.

Therefore the value of

Question 10: Let be a binary operation on **[CBSE 2012]**

Answer:

Given that is an operation that is valid for the natural numbers and is defined by

We need to find the value of

According to the Problem, Binary operation is assumed to be true for the values of to be natural.

We know that LCM of two prime numbers is the product of that given two prime numbers.

Therefore the value of