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

Question

Find which of the binary operations are commutative and which are associative.
Show that none of the operations given above has identity.

✓

Answer

Let the identity be I.

$\text{a}*\text{I}=\text{a - I}\neq\text{a}$
$\text{a}*\text{I}=\text{a}^2-\text{I}^2\neq\text{a}$
$\text{a}*\text{I}=\text{a + aI}\neq\text{a}$
$\text{a}*\text{I}=(\text{a - I})^2\neq\text{a}$
$\text{a}*\text{I}=\frac{\text{aI}}{4}\neq\text{a}$
$\text{a}*\text{I}=\text{aI}^2\neq\text{a}$

Therefore, none of the operations given above has identity.

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