Let's say you have sets $A,\, B,$ and $C.$
How would you show that $[(A-B) - C]\subseteq (A-C)$ using a venn diagram or logical translations?
How can this even be done when you don't know the members of $A,\, B,$ or $C$?
You'd need to use logical translations, implicitly (no need for logic symbols though - words work fine), in the sense that you need to use the definition of set-minus where $\;x \in X - Y\;$ means that $\;x \in X\;$ AND $\,x \notin Y.$
Showing that set-membership in $\,X\,$ implies set-membership in $\,Y,\,$ proves that $\,X\subseteq Y$. Logically, this is establishing that $\,x \in [(A - B) - C] \implies x \in (A - C).$
In this case, start by assuming $\,x \in (A - B) - C,\,$ and then unpack what this means using the definition of set-minus. Using this you can argue that it must follow that $\,x \in A-C.\,$ This is equivalent to proving that $\,(A-B) -C \subseteq A - C$.
Let $x\in(A-B)-C$. By the definition of set difference we know that $x\in A-B$ and $x\notin C$. Since $x\in A-B$ we know that $x\in A$ and $x\notin B$. Thus we have $x\in A$ and $x\notin C$ which implies that $x\in A-C$ and we can conclude that $(A-B)-C\subseteq A-C$.