Question 1: Let be a binary operation on
defined by
Answer:
Given that is an operation that is valid on all natural numbers
and is defined by
According to the problem, binary operation given is assumed to be true.
(i) Let us find the values of
The values of
We know that commutative property is is a binary operation.
(ii) Let’s check the commutativity of given binary operation:
Therefore Commutative property holds for given binary operation
We know that associative property is
Let’s check the associativity of given binary operation:
From(1) and (2) we can say that associative property holds for binary function
Question 2: Determine which of the following binary operations are associative and which are
commutative:
Answer:
Given that is a binary operation on
defined by
We know that commutative property is where
is a binary operation.
Let’s check the commutativity of given binary operation:
Therefore The commutative property holds for given binary operation
We know that associative property is
Let’s check the associativity of given binary operation:
From (1) and (2) we can clearly say that, Associative property holds for given binary operation
We know that commutative property is where
is a binary operation.
Let’s check the commutative property of the given binary operation:
Therefore the commutative property holds for given binary operation
We know that associative property is
Let’s check the associativity of given binary operation:
From (i) and (ii) we can clearly see that associative property does not hold for the binary operation
Question 3: Let be any set containing more than one element. Let
be a binary operation on
defined by
for all
Is commutative or associative on
?
Answer:
Given that is a binary operation on set
defined by
We know that commutative property is where
is a binary operation.
Let’s check the commutativity of given binary operation:
Therefore the commutative property does not hold for given binary operation
We know that associative property is
Let’s check the associativity of given binary operation:
From (i) and (ii) we can clearly say that associativity holds for the binary operation
Question 4: Check the commutativity and associativity of each of the following binary operations:
Answer:
Given that is a binary operation on
defined by
We know that commutative property is where
is a binary operation.
Let’s check the commutativity of given binary operation:
Therefore Commutative property holds for a given binary operation
We know that associative property is
Let’s check the associativity of given binary operation:
From (i) and (ii) we can clearly say that associativity holds for the binary operation
Given that is a binary operation on
defined by
We know that the commutative property is where
is a binary operation.
Let’s check the commutativity of given binary operation:
Therefore the commutative property holds for given binary operation
We know that associative property is
Let’s check the associativity of given binary operation:
From (i) and (ii) we can clearly say that associativity doesn’t hold for the binary operation
Given that is a binary operation on
defined by
We know that the commutative property is where
is a binary operation.
Let’s check the commutativity of given binary operation:
Therefore the commutative property doesn’t hold for a given binary operation
We know that associative property is
Let’s check the associativity of given binary operation:
From (i) and (ii) we can clearly say that associativity doesn’t hold for the binary operation
Given that is a binary operation on
defined by
We know that commutative property is where
is a binary operation.
Let’s check the commutativity of given binary operation:
Therefore Commutative property holds for given binary operation
We know that associative property is
Let’s check the associativity of given binary operation:
From (i) and (ii) we can clearly say that associativity doesn’t hold for the binary operation
We know that commutative property is where
is a binary operation.
Let’s check the comutativity of given binary operation:
Therefore the commutative property holds for given binary operation
We know that associative property is
Let’s check the associativity of given binary operation:
From (i) and (ii) we can clearly say that associativity hold for the binary operation
Given that is a binary operation on
defined by
We know that commutative property is where
is a binary operation.
Let’s check the commutativity of given binary operation:
Therefore Commutative property doesn’t holds for a given binary operation
We know that associative property is
Let’s check the associativity of given binary operation:
From (i) and (ii) we can clearly say that associativity doesn’t hold for the binary operation
Given that is a binary operation on
defined by
We know that the commutative property is where
is a binary operation.
Let’s check the commutativity of given binary operation:
Therefore Commutative property doesn’t holds for a given binary operation
We know that associative property is
Let’s check the associativity of given binary operation:
From (i) and (ii) we can clearly say that associativity doesn’t hold for the binary operation
Given that is a binary operation on
defined by
We know that the commutative property is where
is a binary operation.
Let’s check the commutativity of given binary operation:
Therefore Commutative property holds for a given binary operation
We know that associative property is
Let’s check the associativity of given binary operation:
From (i) and (ii) we can clearly say that associativity holds for the binary operation
Given that is a binary operation on
defined by
We know that commutative property is where
is a binary operation.
Let’s check the commutativity of given binary operation:
Therefore Commutative property holds for given binary operation
We know that associative property is Let’s check the associativity of given binary operation:
From (i) and (ii) we can clearly say that associativity doesn’t hold for the binary operation
Given that is a binary operation on
defined by
We know that the commutative property is where
is a binary operation.
Let’s check the commutativity of given binary operation:
Therefore Commutative property holds for a given binary operation
We know that associative property is
Let’s check the associativity of given binary operation:
From (i) and (ii) we can clearly say that associativity doesn’t hold for the binary operation
Given that is a binary operation on
defined by
We know that the commutative property is where
is a binary operation.
Let’s check the commutativity of given binary operation:
Therefore Commutative property doesn’t hold for a given binary operation
We know that associative property is
Let’s check the associativity of given binary operation:
From (i) and (ii) we can clearly say that associativity doesn’t hold for the binary operation
Given that is a binary operation on
defined by
We know that commutative property is where
is a binary operation.
Let’s check the commutativity of given binary operation:
Therefore Commutative property doesn’t holds for given binary operation
We know that associative property is
Let’s check the associativity of given binary operation:
From (i) and (ii) we can clearly say that associativity doesn’t hold for the binary operation
We know that commutative property is where
is a binary operation.
Let’s check the commutativity of given binary operation:
Commutative property holds for given binary operation
We know that associative property is
Let’s check the associativity of given binary operation:
From (i) and (ii) we can clearly say that associativity hold for the binary operation
Given that is a binary operation on
defined by
We know that the commutative property is where
is a binary operation.
Let’s check the commutativity of given binary operation:
Therefore Commutative property holds for a given binary operation
We know that associative property is
Let’s check the associativity of given binary operation:
From (i) and (ii) we can clearly say that associativity holds for the binary operation
Given that is a binary operation on
defined by
We know that the commutative property is where
is a binary operation.
Let’s check the commutativity of given binary operation:
Therefore Commutative property holds for given binary operation
We know that associative property is
Let’s check the associativity of given binary operation:
From (i) and (ii) we can clearly say that associativity holds for the binary operation
Question 5: If the binary operation is defined by
on the set
of all rational numbers other than 1, show that
is commutative on
Answer:
Question 6: Show that the binary operation on
defined by
is not commutative.
Answer:
Question 7: On the set of integers a binary operation
is defined by
for all
Prove that
is not associative on
Answer:
Question 8: Let be the set of all real numbers except
and let
be an operation defined by
for all
Determine whether
is a binary operation on
If yes, check its commutativity and associativity. Also, solve the equation
Answer:
Let’s check the commutativity of given binary operation:
Let’s check the associativity of given binary operation:
Question 9: On the set of all rational numbers,
is defined by
Answer:
We know that associative property is
Let’s check the associativity of given binary operation:
From (1) and (ii) we can clearly say that associativity doesn’t hold for the binary operation
Question 10: On the set of all integers, a binary operation
is defined by
Prove that
is neither commutative nor associative on
Answer:
Given that is a binary operation on
defined by
We know that commutative property is where
is a binary operation.
Let’s check the commutativity of given binary operation:
Therefore The commutative property doesn’t hold for a given binary operation on
We know that associative property is
Let’s check the associativity of given binary operation:
Hence from (i) and (ii) we can clearly say that associativity doesn’t hold for the binary operation on
Question 11: On the set of all rational numbers if a binary operation
Answer:
We know that associative property is
Let’s check the associativity of given binary operation:
From (i) and (ii) we can clearly say that associativity hold for the binary operation
Question 12: The binary operation is defined by
of all rational numbers. Show that
is associative
Answer:
We know that associative property is
Let’s check the associativity of given binary operation:
From (i) and (ii) we can clearly say that associativity hold for the binary operation
Question 13: On the set of all rational numbers a binary operation
Answer:
We know that associative property is
Let’s check the associativity of given binary operation:
From (i) and (ii) we can clearly say that associativity does’nt hold for the binary operation
Question 14: Let be the set of all rational numbers except
and
be defined on
by
for all,
Prove that : (i) 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.