This is Exercise 4 from page 28 of Analysis I by Amann and Escher. I searched and found two questions that appear to concern this same exercise. However, the first one is cryptic to me and the second one doesn't answer the second part of the question. Therefore I would like to solicit feedback on the correctness and clarity of my attempt.
Exercise:
My attempt:
Consider elements $x$ and $r$ from $X$, $x \neq r$. We see that $x \circledast x = r$ from (ii), and that $x \circledast r = x$ from (i).
If the operation $\circledast$ were commutative, then we would have $r \circledast x = x \circledast r = x$. Using (ii), we see that
$$(r \circledast x) \circledast (x \circledast r) = x \circledast x = r,$$
which implies that $x = r$. Since this contradicts our assumption that $x$ and $r$ are distinct, we conclude that the operation $\circledast$ is not commutative.
To show that there is no identity element, assume that there were one, and call it $e$. Using (i), we see that $e \circledast r = e $, but because $e$ is the identity, it is also true that $e \circledast r = r$, so we see that $e = r$ and therefore $r$ is the identity element. But we showed above that $r$ doesn't commute with $x$, so $r$ is not the identity, and therefore $\circledast$ has no identity element.
Edit:
I have one more question. I looked at the Wikipedia page for "Anticommutative property", and it says that an anticommutative operation is one for which reversing the order of the two arguments produces the inverse. That definition looks similar to the one copied above; is it exactly the same?