Question

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

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

Answer 1

The correct option is B

commutative but not associative

Explanation for the correct option:

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

$a*b=b*a$

$\Rightarrow$$1+ab=1+ba$

$\Rightarrow$$1+ab=1+ab$ (True) $(\because a,b,$ are real numbers$)$

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

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

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

$=1+(1+ab)c$

$=1+c+abc$

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

$=1+a(1+bc)$

$=1+a+abc$

$\therefore (a*b)*c\ne a*(b*c)$.

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

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

Hence, Option ‘B’ is Correct.