Find which of the binary operations are commutative and which are… - Vidyadip

MCQ

Find which of the binary operations are commutative and which are associative. Consider a binary operation $*$ on $N$ defined as $a * b = a^3 + b^3.$ Choose the correct answer.

A
Is $*$ both associative and commutative?
✓
Is $*$ commutative but not associative?
C
Is $*$ commutative but not associative?
D
Is $*$ neither commutative nor associative?

✓

Answer

Correct option: B.

Is $*$ commutative but not associative?

$a * b = a^3 + b^3 = b^3 + a^3 = b * a$
$\therefore$ The operation is commutative.
Again, $(a * b) * c = a * (a^3 + b^3) = a^3(a^3 + b^3)^3$
And $(a * b) * c= (a^3 + b^3) * c = (a^3 + b^3)^3 + c^3 \neq\text{a}*(\text{b}*\text{c})$
$\therefore$ The operation $*$ is not associative.
Therefore, option $(B)$ is correct.
Is $*$ commutative but not associative?

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