Proving a Subset - $(A∪B)∩C⊆A∪(B∩C).$
It seems like you confuse the assumption and what you have to show.
You begin correct:
Let $x\in (A\cup B)\cap C)$.
We want to show, that $x\in A\cup (B\cap C)$.
Since $x\in (A\cup B)\cap C$. It is $x\in A$ or $x\in B$ and $x\in C$.
If $x\in A$, then $x\in A\cup (B\cap C)$.
If $x\notin A$, then $x\in B$. Hence $x\in B\cap C$. And therefore $x\in A\cup (B\cap C)$.
I'm not really sure why that "assume" line is there. It should probably be "by our assumption that $x \in \cdots$".
Anyhow, I feel like you get the general idea of how this proof is meant to go - if you want to prove $A \subseteq B$, you want to show $x \in A \implies x \in B$. However, these things are a bit more complicated than that when you have multiple sets and such on each side.
I like to think of this in two steps - "unraveling" the left-hand side to figure out what sets $x$ is in, and what it isn't in, and trying to "ravel it back up" to make the right-hand side.
Some of your wording obscures this idea but you get the idea, I believe. Rewriting it would help your clarity come through.
- Assumption: $x \in (A \cup B) \cap C$
- Thus: $x \in (A \cup B)$ and $x \in C$ (to be in the intersection, it must be in both)
- Thus: $x \in A$ or $x \in B$ (to be in the union, it must be in one or the other, possibly both)
So, we know for sure $x\in C$, and $x$ is in one of (or both) $A,B$. We have "unraveled" this half of the proof, so to speak.
At this point it gets a bit tricky. It's handy here to take this by "cases" where $x \in A$ or $x \in B$.
- Suppose $x \in A$. Then $x \in A \cup (B \cap C)$
- Suppose $x \in B$ instead. Then since $x \in C$, $x \in B \cap C$ and thus $x \in A \cup (B \cap C)$
Thus, $x \in (A \cup B) \cap C \implies x \in A \cup (B \cap C)$, showing $(A \cup B) \cap C \subseteq A \cup (B \cap C)$, completing the proof.
I feel like you get the idea of what's going on and the basic idea - your writing simply obscures that fact. It's important to keep in mind why everything follows from one step to the next; writing that explanation out would be very helpful, both for your professor to follow your proof, and for yourself to justify what's going on.