Question 1: Let be a binary operation on defined by for all

(i) Show that is both commutative and associative.

(ii) Find the identity element in

(iii) Find the invertible elements in

Answer:

(i) Let us prove the commutativity of

Now, let us prove the associativity of

(ii) Let be the identity element in with respect to

Thus, is the identity element in with respect to

(iii)

Question 2: Let be a binary operation on (set of non-zero rational numbers) defined by Show that is commutative as well as associative. Also, find its identity element, if it exists

Answer:

Let us prove the commutativity of

Let us prove the associativity of

Thus is associative on

Let us find the identity element

Let be the identity element in with respect to

Question 3: Let be a binary operation on defined by for all Then,

(i) Show that is both commutative and associative on

(ii) Find the identity element in

(iii) Show that every element of is invertible. Also, find the inverse of an arbitrary element.

Answer:

(i) Let us check the commutativity of

Let us prove that associativity of \ast $

(ii) Let be the identity element in with respect to

(iii)

Question 4: Let denote the set of all non-zero real numbers. A binary operation is defined on as follows:

(i) Show that is commutative and associative on

(ii) Find the identity element in

(iii) Find the invertible elements in

Answer:

(i)

Such that,

Question 5: Let be a binary operation on the set of all non-zero

(i) Show that is both commutative and associate.

(ii) Find the identity element in

(iii) Find the invertible elements of

Answer:

(i) Commutativity:

Associativity:

Question 6: On a binary operation is defined by Prove that is commutative and associative. Find the identity element for Also, prove that every element of is invertible.

Answer:

A general binary operation is nothing but an association of any pair of elements from an arbitrary set to another element of This gives rise to a general definition as follows:

A binary operation on a set is a function We denote

Let us proceed with the solution.

Question 7: Let denote the set of all non-zero real numbers and let If is a binary operation on

(i) Show that is both commutative and associative on

(ii) Find the identity element in

(iii) Find the invertible element in

Answer:

(i) Show that is both commutative and associative on

Commutativity:

Associativity:

(ii) Find the identity element in

(iii) Find the invertible element in

Question 8: Let be the binary operation on defined by HCF of and Does there exist identity for this binary operation on ?

Answer:

Therefore,

Here, if the operation is applied to the above numbers as follows:

Now, Identity Element:

Therefore, this operation does not have any identity.

Question 9: 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: