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: