Proof verification: Prove $A\cup B = B\cup A$ and $A\cap B=B\cap A$. Can someone please verify whether my following proofs are logically correct? :) I am not sure how to go about these proofs, so I just guessed.
$A\cup B=B\cup A$
Proof: Let $x\in A\cup B$. Then $x\in A$ or $x\in B$. Then $x\in B$ or $x\in A$. Then $x\in B \cup A$. Then $A\cup B \subset B\cup A$.
Let $x\in B\cup A$. Then $x\in B$ or $x\in A$. Then $x\in A$ or $x\in B$. Then $x\in A \cup B$. Then $B\cup A \subset A\cup B$. Therefore, $A\cup B=B\cup A$. $\square$
$A\cap B=B\cap A$
Proof: Let $x\in A\cap B$. Then $x\in A$ and $x\in B$. Then $x\in B$ and $x\in A$. Then $x\in B \cap A$. Then $A\cap B \subset B\cap A$.
Let $x\in B\cap A$. Then $x\in B$ and $x\in A$. Then $x\in A$ and $x\in B$. Then $x\in A \cap B$. Then $B\cap A \subset A\cap B$. Therefore, $A\cap B=B\cap A$. $\square$
 A: Your proof is correct and well-presented. Good job!
The one nitpick I have would be in the style; it feels a little repetitive/monotonous to keep saying "then [...]. Then [...]. Then [...]." I would either do this proof as a sort of bulleted list, where the implications are obvious ...


*

*Assume $x \in A \cup B$.

*Thus, $x \in A$ or $x \in B$.

*Equivalently, $x \in B$ or $x \in A$.

*Therefore, $x \in B \cup A$.

*Therefore, $A \cup B \subseteq B \cup A$.


... or perhaps purely symbolically, as

$$\begin{align}
A \cup B = B \cup A &\implies A \cup B \subseteq B \cup A \\
x \in A \cup B &\iff x \in A \text{ or } x \in B \\
&\iff x \in B \text{ or } x \in A \\
&\iff x \in B \cup A \\
\end{align}$$
$$\therefore A \cup B \subseteq B \cup A$$

Both are a bit less grinding to read. Personally, I prefer the first, since words are useful and important in proofs. The second would be okay if everything is clear and unambiguous, but I can see it being problematic for a novice reader of proofs to see what the purpose of the above is.

Mostly posting this so this question can be finally considered to have an answer and thus be removed from the unanswered queue. Made it Community Wiki since I have nothing much of substance to add.
