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Question

Let S be a non-empty set and o be a binary operation on S defined by xoy=x:x,yS. Determine whether o is commutative ans associative.

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Solution

o is a binary operation on S
xoy=xforx,yS

For associative
let x,y,zS(xoy)oz=(xoz)=xxo(yoz)=(xoy)=x(xoy)oz=xo(yoz)
So this binary operation is associative
For commutative
Letx,ySxoy=xyox=yxoyyox
So this binary operation in not commutative

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