I need to show that $(A \cup B) \subseteq (A \cup B \cup C)$
My Work So Far:
What I really need to show is that $x \in (A \cup B)$ implies $x \in (A \cup B \cup C)$
So I translated my sets into their logical expressions
$x \in A \vee x \in B \longrightarrow x \in A \vee x \in B \vee x \in C$
This is where I’m stuck. How do I show membership of $A \cup B$ implies membership of $A \cup B \cup C$?
We need to show that following holds:
Proof: Since by definition the above statement means that $(x∈A∨x∈B) \rightarrow (x∈A∨x∈B∨x∈C)$, we now have a new goal. Since our new goal has the form of a conditional, we start by assuming that the antecedent is true, that is, we assume that $(x∈A∨x∈B)$ is true. Then if $(x∈A∨x∈B)$ is true so is $(x∈A∨x∈B)∨x∈C$ (it is formalized in the natural deduction and known as the disjunction introduction rule). This shows that $(x∈A∨x∈B) \rightarrow ((x∈A∨x∈B)∨x∈C)$ and, therefore, that $(A∪B)⊆(A∪B∪C)$ holds as required.
In my personal opinion, is always a good practice not omitting parenthesis when your statement starts getting longer (it avoids ambiguity and makes explicit the unique parsing of a formula).
Let’s take for instance the formula you just stated (parenthesis are now explicit):
$(x∈A∨x∈B) \rightarrow (x∈A∨x∈B∨x∈C)$
this has an easier reading than the one omitting them:
$x∈A∨x∈B \rightarrow x∈A∨x∈B∨x∈C$