Exercises I.4-4

[muzimuzhi]

To simplify input, denote $\circ$ insdead of $\circledast$

$\circ$ is not commutative

• Prove by contradiction. Assume $\circ​$ is commutative.
• There exists $x \in X​$, $x \neq r​$.
• From (ii), $x \circ x = r$.
• By commutativity, $x \circ r = r \circ x = x$.
• Hence $r = \color{red}{x} \circ \color{blue}{x} = \color{red}{(x \circ r)} \circ \color{blue}{(r \circ x)}$.
• Using (ii), $x = r​$, which is a contradiction.
• Therefor $\circ$ is not commutative.

$\circ$ has no identity element

• Prove by contradiction. Assume $e$ is the identity element of $\circ$.
• From (ii), if $x \circ y = r$, then $x = y$.
• Since $e \circ r = r$, $e = r$.
• Hence $x \circ r = r \circ x = x$, which will lead to contradiction.