# For any two-real numbers, an operation defined by \[a * b{\text{ }} = \;1 + ab\] is

$\left( A \right)$. Commutative but not associative

$\left( B \right)$. Associative but not commutative

$\left( C \right)$. Neither commutative nor associative

$\left( D \right)$. Both commutative and associative

Last updated date: 26th Mar 2023

•

Total views: 309k

•

Views today: 6.88k

Answer

Verified

309k+ views

Hint: Use commutative and associative property for the given operation.

We have been given the operator \[ * \] such that:

\[{\text{ }}a * b = 1 + ab{\text{ (1) ; }}a,b{\text{ }} \in {\text{R}}\]

Since \[(1 + ab){\text{ }}\]also belongs to \[R{\text{ }}\](Real Numbers Set),

Operator \[ * \] satisfies closure property

\[a * b\] is a binary operation.

For binary operation to be commutative, we would have the following condition:

\[a * b = b * a {\text{(2)}}\]

We need to check condition (2) for operation (1)

\[

a * b = 1 + ab \\

b * a = 1 + ba \\

\]

Since multiplication operator is commutative, we have

\[

ab = ba \\

\Rightarrow a * b = 1 + ab = 1 + ba = b * a \\

\]

Hence condition (2) is satisfied.

Therefore, operation (1) is commutative.

For binary operation to be associative, we would have the following condition:

\[a * \left( {b * c} \right) = \left( {a * b} \right) * c{\text{ (3)}}\]

We need to check for condition (3) for operator (1)

\[

a * \left( {b * c} \right) = a * \left( {1 + bc} \right) = 1 + a(1 + bc) = 1 + a + abc \\

\left( {a * b} \right) * c = \left( {1 + ab} \right) * c = 1 + \left( {1 + ab} \right)c = 1 + c + abc \\

\]

Since \[1 + a + abc \ne 1 + c + abc\], condition (3) is not satisfied.

Therefore, operation (1) is not associative.

Hence the correct option is $\left( A \right)$. Commutative but not associative.

Note: Always try to remember the basic conditions for associativity and commutativity. Commutativity does not imply associativity.

We have been given the operator \[ * \] such that:

\[{\text{ }}a * b = 1 + ab{\text{ (1) ; }}a,b{\text{ }} \in {\text{R}}\]

Since \[(1 + ab){\text{ }}\]also belongs to \[R{\text{ }}\](Real Numbers Set),

Operator \[ * \] satisfies closure property

\[a * b\] is a binary operation.

For binary operation to be commutative, we would have the following condition:

\[a * b = b * a {\text{(2)}}\]

We need to check condition (2) for operation (1)

\[

a * b = 1 + ab \\

b * a = 1 + ba \\

\]

Since multiplication operator is commutative, we have

\[

ab = ba \\

\Rightarrow a * b = 1 + ab = 1 + ba = b * a \\

\]

Hence condition (2) is satisfied.

Therefore, operation (1) is commutative.

For binary operation to be associative, we would have the following condition:

\[a * \left( {b * c} \right) = \left( {a * b} \right) * c{\text{ (3)}}\]

We need to check for condition (3) for operator (1)

\[

a * \left( {b * c} \right) = a * \left( {1 + bc} \right) = 1 + a(1 + bc) = 1 + a + abc \\

\left( {a * b} \right) * c = \left( {1 + ab} \right) * c = 1 + \left( {1 + ab} \right)c = 1 + c + abc \\

\]

Since \[1 + a + abc \ne 1 + c + abc\], condition (3) is not satisfied.

Therefore, operation (1) is not associative.

Hence the correct option is $\left( A \right)$. Commutative but not associative.

Note: Always try to remember the basic conditions for associativity and commutativity. Commutativity does not imply associativity.

Recently Updated Pages

If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

KCN reacts readily to give a cyanide with A Ethyl alcohol class 12 chemistry JEE_Main

Trending doubts

What was the capital of Kanishka A Mathura B Purushapura class 7 social studies CBSE

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Tropic of Cancer passes through how many states? Name them.

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

Name the Largest and the Smallest Cell in the Human Body ?