I want to prove $(A\cup B) \uplus C \subseteq (A\uplus C)\cup (B \uplus C)$, but I have not learnt how I could start with disjoint sets. Here is what I did:
Let $(A\cup B) \uplus C = \{(x,0): x\in A \lor x \in B \} \cup \{(x,1): x\in C \} $.
Then I want to ideally show that for all $k \in (A\cup B) \uplus C \implies k \in (A\uplus C)\cup (B \uplus C)$
I want to say that all $(x,0)$ should lie in either A or B since they are on the left hand side of the disjoint union. But I don't know if this is the mathematical way to do it and it seems like I'm just reiterating the original statement.
How should I start for disjoint sets?
I do not see anything specific of disjoint sets. You simply consider an element $x$ in $(A\cup B) \uplus C$. It is necessarily of the form $x=(y,n)$ where $n=1\land y\in C$ or $n=0\land y\in A\cup B$. You consider the two cases séparately, and so on. In the end, for each case you had to consider (one case will again divide in two), you simply see that $x$ has to be in $A\uplus C$ or $B \uplus C$ (and it can even be in both). Hence the inclusion.