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**novice** A binary operation $\displaystyle *$ is defined on the set $\displaystyle S=\{a,b,c\}$ by $\displaystyle x*y=x$ for all $\displaystyle x,y\inS$. Determine which of the properties (Associative, Identity, Inverse, Commutative) are satisfied by $\displaystyle (S,*).$

Solution:

Let $\displaystyle x,y$, and $\displaystyle z$ be any three elements of $\displaystyle S$ (distinct of not). Then $\displaystyle x*(y*z) = x*y=x$, while $\displaystyle (x*y)*z=x*z=x$. Thus (S,*) is associative. Now (S,*) has no identity since for every element $\displaystyle e \in S$, it follows that $\displaystyle e*a=e*b=e$ and so it is impossible for $\displaystyle e*a=a$ and $\displaystyle e*b=b$. Since (S,*) has no identity, the question of inverses does not apply here. Certainly, $\displaystyle (S,*) $ is not commutative since $\displaystyle a*b=a$ while $\displaystyle b*a=b$.

Question:

I suppose $\displaystyle e$ is an identity element. Since (S,*) has no identity why it contradict itself by saying "(S,*) has no identity since for every element "$\displaystyle e\in S$"?