Question

Consider a binary operation $\ast$ on $N$ defined as $a\ast b=a^3+b^3$. Choose the correct answer

• A. Is $\ast$ both associative and commutative?
• B. Is $\ast$ commutative but not associative?
• C. s $\ast$ associative but not commutative?
• D. Is $\ast$ neither commutative nor associative?

Answer 1

The correct option is B Is $\ast$ commutative but not associative?

Check commutative

$\ast$ is commuatative if

$a\ast b=b\ast a$

$a\ast b=a^3+b^3;b\ast a=b^3+a^3=a^3+b^3$

Since ,$a\ast b=b\ast a,\forall a,bϵN$

$\ast$ is commutative

Check associative

$\ast$ is associative if $(a\ast b)\ast c=a\ast (b\ast c)$

$(a\ast b)\ast c={(a^3+b^3)}^{\ast }c={(a^3+b^3)}^3+c^{3}a\ast (b\ast c)=a^3\ast (b^3+c^3)=a^3+{(b^3+c^3)}^3$

Since, $(a\ast b)\ast c\ne a\ast (b\ast c),\forall a,b,cϵN$

$\ast$ is not associative.

$\therefore$Is $\ast$ commutative but not associative.