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# Let $S$ be a non-empty set and o be a binary operation on $S$ defined by $xoy = x ; x, y \in S$. Determine whether o is commutative and associative.

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Toolbox:
• A binary operation $*$ on a set $S$ is associative if $(a*b)*c=a*(b*c)$ for $a,b,c\in S$.It is commutative if $(a*b)=(b*a)$ for $a,b\in S$
Step 1:
$o$ is a binary operation on $S$
$xoy=x$ for $x,y\in S$
Associativity :
Let $x,y,z\in S$
$(xoy)oz=xoz=x$
And $xo(yoz)=xoy=x$
$\therefore (xoy)oz=xo(yoz)$
The binary operation is associative.
Step 2:
Commutativity :
Let $x,y\in S$
$xoy=x$ and $yox=y$
$xoy\neq yox$
The binary operation is not commutative.
answered Sep 13, 2013

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