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
defined on
Existence of inverse: Let and let
be the inverse of
Then,
Question 12: Consider the infimum binary operation on set
defined by
minimum of
and
Write the composition table of the operation
Answer:
We have,
So, we have the following composition table for
Show that is the identity for this operation and each element
of the set is invertible with
being the inverse of
[CBSE 2011]
Answer:
The operation is defined as
Thus, is the identity element for the given operation