Question

Consider a binary operation $\ast$ on N defind as $a\ast b=a^3+b^3$. Choose the correct answer.
$\left(a\right)\ast$ both associative and commutative
$\left(b\right)\ast$ commutative but not associative
$\left(c\right)\ast$ is associative but not commutative
$\left(d\right)\ast$ is neither commutative nor associative

Answer 1

On N, the operation $\ast$ is defined as $a\ast b=a^3+b^3.$
For, $a,b\in N$, we have
$a\ast b=a^3+b^3=b^3+a^3=b\ast a$ [Addition is commutative in N]
Therefore, the operation $\ast$ is commutative.
It can be observed that
$(1\ast 2)\ast 3=(1^3+2^3)\ast 3=9\ast 3=9^3+3^3=729+27=7561\ast (2\ast 3)=1\ast (2^3+3^3)=1\ast (8+27)=1\ast 35=1^3+{35}^3=1+(35{)}^3=1+42875=42876$
Therefore, $(1\ast 2)\ast 3\ne 1\ast (2\ast 3)$ where $1,2,3\in N$
Therefore, the operation $\ast$ is not associative.
Hence, the operation $\ast$ is commutative but not associative.
Thus, the correct answer is (b).