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Formal Definition

A binary operation $*$ on a set $S$ is commutative if $a*b=b*a$ for all $a,b \in S$ .

Informal Definition

A binary operation is commutative if the elements can be placed in a different order and still get the same outcome.


For $a, b \in \mathbb{R}$ under the operation of addition, defined by $+$
$a + b = b + a$
so $+$ is commutative in $\mathbb{R}$.


For $a,b \in \mathbb{R}$ under the operation of subtraction, defined by $-$
$a - b \neq b - a$ as $a - b = -1(b-a)$
therefore, $-$ is not commutative in $\mathbb{R}$.

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