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.