Let * be a binary operation on `R` defined by `a*b=a b+1` . Then, * is commutative but not associative associative but not commutative neither commutative nor associative (d) both commutative and associative

Options

A. (a) commutative but not associative
B. (b) associative but not commutative
C. (c) neither commutative nor associative
D. (d) both commutative and associative

Answer

(a) commutative but not associative

Explanation

Given that, `a∗b=1+ab`,`forall a,binR`

`a∗b=ab+1=b⋅a`


So, `∗` is a commutative binary operation.

`a∗(b∗c)=a∗(1+bc)=1+a(1+bc)`

`a∗(b∗c)=1+a+abc` `…(i)`

`(a∗b)∗c=(1+ab)∗c`

`=1+(1+ab)c=1+c+abc` `...(ii)`


From Eq `(i)` and `(ii)`,

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