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# Let * be binary operation defined on R by $a*b=1+ab, a,b \in R.$ Then the operation * is

(i) commutative but not associative (ii) associative but not commutative (iii) neither commutative nor associative (iv) both commutative and associative

Toolbox:
• 1. A binary operation * defined on R is commutative if $a*b=b*a$
• 2. A binary operation * defined on R is associative if $(a*b)*c=a*(b*c) \qquad a,b \in R$
Step 1: commutative

* operation defined on R by

$a*b=1+ab \qquad a,b \in R$

$b*a=1+ba$

$=1+ab$

Multiplication is commutative in R

$a*b =b*a$

* operation is commutative

Step 2: Associative

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

$=1+(1+ab)c$

$=1+c+abc$

$a \times (b \times c)=a * (1+bc)$

$=1+a(1+bc)$

$=1+a+abc$

$(a*b)*c \neq a *(b*c)$

* operation is not associative

Solution: * operation is commutative but not associative

(i) option is correct

edited Mar 27, 2013 by meena.p