TITLE: Showing one set is a subset of another
QUESTION [5 upvotes]: 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 $C$?

REPLY [2 votes]: 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 $(A-B)-C\subseteq A-C$.