Question

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

• A. commutative but not associative
• B. associative but not commutative
• C. neither commutative nor associative
• D. both commutative and associative

Answer 1

Answer: (A)

For Commutative
Given, $a^{*} b=1+a b$
Now, $b^{*} a=1+b a=1+a b$
$\therefore a^{*} b=b^{*} a$
So, it is commutative.
For Associative
$\left(a^{*} b\right)^{*} c =(1+a b)^{*} c$
$=1+(1+a b) c$
$=1+c+a b c$
and $a^{*}\left(b^{*} c\right) =a^{*}(1+b c)$
$=1+a(1+b c)$
$=1+a+a b c$
$\therefore \left(a^{*} b\right)^{*} c \neq a^{*}\left(b^{*} c\right)$
So, it is not associative.