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

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