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Question

For any two real numbers, an operation * defined by a*b=1+ab is


A

neither commutative nor associative

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B

commutative but not associative

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C

both commutative and associative

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D

associative but not commutative

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Solution

The correct option is B

commutative but not associative


Explanation for the correct option:

Step 1. a*b=1+ab is commutative if;

a*b=b*a

1+ab=1+ba

1+ab=1+ab (True) (a,b, are real numbers)

Step 2. a*b=1+ab is associative if;

(a*b)*c=a*(b*c)

(a*b)*c=(1+ab)*c

=1+(1+ab)c

=1+c+abc

a*(b*c)=a*(1+bc)

=1+a(1+bc)

=1+a+abc

(a*b)*ca*(b*c).

So it is not associative. (a,b, are real numbers but c is not)

Thus, the binary operation * is commutative but not associative.

Hence, Option ‘B’ is Correct.


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