A binary operation $\circledast$ on a set $X$ is called anticommutative if
- $\exists r\in X: x\circledast r = x,\;\; x\in X$ and
- $x\circledast y=r\Leftrightarrow (x\circledast y)\circledast(y\circledast x)=r\Leftrightarrow x=y$
I have to prove that an anticommutative bin. operation on $X$ is not commutative and that there is no identity element $e\in X$, if $X$ has more than two elements.
I proved is as follows: Let $\circledast$ be an anticommutative binary operation on a set $X$, which contains two distinct elements $x$ and $y$ with $x\circledast y=y\circledast x$. Because of 2., it follows that $(x\circledast y)\circledast(y\circledast x)=r$, which, since $\circledast$ is anticommutative, is equivalent to $x=y$, contradicting our assumption that $x$ and $y$ are distinct elements.
END OF PROOF
I think that this proves that any anticommutative binary operation on a set that contains at least two elements is not commutative and does not have an identity element $e$. However, it bothers me that the exercise says I should prove the statement for anticom. bin. operations on sets of at least three elements. Is my proof incomplete?
Your proof looks fine. In this question and this book they say exactly the same things except the condition is that $X$ has more than one element. It looks like perhaps you've got an earlier edition or an incorrect copy of the exercise?