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?
It is given that the binary operation ∗ on N is defined as a ∗ b = a 3 + b 3 .
Apply the given binary operation on b ∗ a .
b ∗ a = b 3 + a 3 = a 3 + b 3
It shows that the value of a ∗ b is equal to that of b ∗ a . So, the operation is commutative.
Consider different values of the variable as a = 1 , b = 2 and c = 3 .
Apply the given binary operation on ( a ∗ b ) ∗ c .
( a ∗ b ) ∗ c = ( 1 ∗ 2 ) ∗ 3 = ( 1 3 + 2 3 ) ∗ 3 = 9 3 + 3 3 = 756
Apply the given binary operation on a ∗ ( b ∗ c ) .
( a ∗ b ) ∗ c = 1 ∗ ( 2 ∗ 3 ) = 1 ∗ ( 2 3 + 3 3 ) = 1 3 + 35 3 = 42876
As the value obtained by ( a ∗ b ) ∗ c and a ∗ ( b ∗ c ) is different, so the operation is not associative.
This shows that the given operation is commutative, but not associative.
Thus, the correct option is option (B).
It is given that the binary operation
Apply the given binary operation on
It shows that the value of
Consider different values of the variable as
Apply the given binary operation on
Apply the given binary operation on
As the value obtained by
This shows that the given operation is commutative, but not associative.
Thus, the correct option is option (B).
