Question

Consider a binary operation ∗ on N defined as a ∗ b =$a^3+b^3$ . Which of the following statement(s) is/are correct?

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

Answer 1

The correct option is B ∗ is commutative but not associative

a * b = $a^3+b^3$
b * a = $b^3+a^3$
a * b = b * a
Therefore, * is not commutative.
$(a*b)*c=\left(a^3+b^3\right)*c={\left(a^3+b^3\right)}^3+c^{3}a*(b*c)=a*\left(b^3+c^3\right)=a^3+{\left(b^3+c^3\right)}^3\therefore (a*b)*c\ne a*(b*c)$
Therefore, * is not associative.