Suppose that $A$, $B$ and $C$ are sets, and $A\setminus B\subseteq C$. Show that $A\setminus C\subseteq B$.
$A\setminus B\subseteq C$ means that there are $x\in A, x\notin B$ and $x\in C$.
Using contradiction, we say that $C \not\supset A\setminus B$ therefore
$x\in A, x\notin B$ and $x\notin C$, however this contradicts my hypothesis.
Is this right?
It looks kind of weird.
Best Answer
Let $x\in A\setminus C$; suppose $x\notin B$, then $x\in A\setminus B$, then by hypothesis we have $x\in C$. A contradiction.
We have that if $x\in A\setminus C$, then $x\in B$, or which is equivalent $A\setminus C\subseteq B$