Quality of proof of elementary set theorem A very simple theorem in set theory, and my attempt at a proof. I've always struggled with proofs, and as such am going back to basics and working up from there. How would you rate the  quality of the proof?


*

*Does the argument follow? 

*Is it presented OK (i.e. understand, and aesthetically mathematical?


Theorem.
$\ A \cup (B \cup C) = (A \cup B) \cup C$.
Proof: Let$\ x \in  A \cup (B \cup C) $. Use the definition of union, namely that if$\ x \in A \cup B$, then$\ x \in A$ or$\ x \in B$. So$\ x \in A \cup (B \cup C)$:


*

*implies$\ x \in A$ or$\ x \in (B \cup C)$,

*which implies$\ x \in A$ or $\ x \in B$ or $\ x \in C$,

*which implies$\ x \in (A \cup B)$ or$\ x \in C$ , 

*which implies$\ x \in (A \cup B) \cup C$, as required.


First, you've only showed one direction of the inclusion. You still technically need to state why $(A \cup B) \cup C \subset A \cup (B \cup C)$.
The argument is analogous completely though.
To answer your questions:


*

*Yes the first inclusion follows.

*I wouldn't use indexed lined to present implications, rather use $ \implies$ between each line:)
