Question

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

(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

a binary operation * on N defined as a * b = a³ + b³ .

we can see that a * b = a³ + b³
b * a = b³ + a³ = a³ + b³
here, a * b = b * a
therefore, * is commutative.

take a , b , c belongs to N
now, (a * b ) * c = (a³ + b³) * c = (a³ + b³)³ + c³
a * (b * c) = a * ( b³ + c³) = a³ + (b³ + c³)³
here, (a * b) * c ≠ a * (b * c)
therefore, * is not associative .

$\textbf{hence, option (B) is correct.}$