Question

I'm having a total mental block today...

If A contains B, prove (C\B) contains (C\A). Prove that the converse is true or provide a counter example.

Answer 1

If I unterstand correctly, you want to prove:\[A \subseteq B \subseteq C => C\;\backslash B \subseteq C\;\backslash A \] I'll write the proof in logical formulas. First you should be aware of the equivalence of\[(1) \;A \subseteq B \Leftrightarrow (x \notin B \Rightarrow x \notin A)\]Now I'll start with the proof. Let x be an arbitrary element of C\B, then the following holds:\[x \in C\backslash B \implies x \in C \;\wedge \; x \notin B\] Because of (1) we know that\[x \in C \wedge \; x \notin B \Rightarrow x \in C \wedge \; x \notin A \Leftrightarrow x \in C\backslash A\]Because all our conclusions are valid for any x as in C\B as x was arbitrary, it follows that
\[C\backslash B \subseteq C\backslash A\]