Question

For any two real number, an operation defined by $a\ast b=1+ab$ is.

• A. Commutative but not associative
• B. Associative but not commutative
• C. Neither Commutative nor associative
• D. Both commutative and associative

Answer 1

The correct option is A Commutative but not associative

A binary operation $\ast$ on $A$ is associative if $\forall a,b,c\in A$,

$(a\ast b)\ast c=a\ast (b\ast c)$ and

A binary operation $\ast$ on $A$ is commutative if $\forall a,b\in A$,

$a\ast b=b\ast a$

$a\ast b=1+ab$ is commutative as:

$a\ast b=b\ast a\Rightarrow 1+ab=1+ba\Rightarrow 1+ab=1+ab$

which is true.

$a\ast b=1+ab$ is not associative as:

$(a\ast b)\ast c=a\ast (b\ast c)\Rightarrow (1+ab)\ast c=a\ast (1+bc)\Rightarrow 1+(1+ab)c=1+a(1+bc)\Rightarrow 1+c+abc=a+a+abc$

which is not true.

Hence, the binary operation $\ast$ is commutative but not associative.