Help with Subset Problem So I'm "supposed to show that":
$(A \cup B)  \subseteq (A \cup B \cup C)$.
For $(A \cup B)$, $x \in A \cup B$ and therefore $x \in A \lor x \in B$. For
$(A \cup B \cup C )$ , $x \in A \cup B \cup C$ and therefore $x \in A \lor x \in B \lor x \in C$. I'm not quite sure how to explain my results.  The best I can come up with is because $x$ is an element of set $A$ or set $B$ or set $C$,  $(A \cup B)$ is a subset of $(A \cup B \cup C)$. How can I explain this better?
Follow up question:
Thanks for the explanations and I'd like to get help on another one.
$(A \cap B \cap C) \subseteq (A \cap B)$.
So for $(A \cap B)$, $x \in A \land x \in B$. And $x \in A$ and $ x \in B$ must be true. Can I assume that  $(A \cap B \cap C)$ must be true as well if it is a subset of $(A \cap B)$ and that $x \in C$ must be true?

To show that $P\subset Q$, you suppose that $x$ is an element of $P$, and show that it must also be an element of $Q$.
Here you suppose $x \in A\cup B$, so, as you said, you know either $x\in A$ or $x\in B$.  You don't know which, so you proceed by cases.


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*Suppose $x\in A$. Then $x \in A\cup B\cup C$ because... ?

*Suppose $x\in B$. Then $x \in A\cup B\cup C$ because... ?

