Question 1: 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 2: 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 3: 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

Question 4: Let be the set of all rational numbers except and be defined on by  for all,

Prove that : (i) $latex \ast $ is a binary operation on

(ii) is commutative as well as associative     [CBSE 2014]

Answer:

(i) Sum, difference and product of rational numbers is a unique rational number.

Therefore for each there exists a unique image

is a function

is a binary operation on

(ii)

Therefore is commutative

From (i) and (ii) we get

Therefore is associative.

Question 5: If the binary operation on the set is defined by then element with respect to   [CBSE 2012]

Answer:

Given that binary operation is valid for the set defined by  

Let us assume and the identity element that we need to compute is  

We know that he Identity property is defined as follows:  

Therefore the required Identity element w.r.t    is 5.

Question 6: On the set of integers, if the binary operation is defined by then find the identity element.     [CBSE 2012]

Answer:

Given that binary operation is valid for the set of integers defined by   

Let us assume and the identity element that we need to compute be

We know that he Identity property is defined as follows:  

 

Therefore the required Identity element w.r.t 

Question 7: Let be a binary operation on defined by Show that is commutative and associative. Find the binary element for if any.     [CBSE 2017]

Answer:

Let’s check. Note: Here represent the ordered pairs respectively.

  

Binary elements:  

Answer:

For any we have

Answer: 

Let be the identify element in for the binary operation on Then,

Answer: 

Question 11:  On the set a binary operation is defined by for all Prove that \ast commutative as well as associative on Find the identity element and prove that every element of is invertible.            [CBSE 2015, 2016]

Answer:

We observe the following properties of on

Commutativity: For any we have

[ By commutativity of addition and multiplication on   ]

Hence, is commutative on

Associativity: For any we have

From (i) and (ii), we have

Hence, is associative on

Existence of identity: Let be the identity element. Then,

So, is the identity element for