This is not homework. I'm just studying for my Discrete Mathematics course. I'd like to know how to prove the following using element wise proofs:
Given a proper subset:
$$A \subset B \iff (A \subseteq B) \land (\exists x
\in B, \space x \notin A).$$
Prove that $(A \subset B) \rightarrow \lnot(B \subseteq A).$
I know that a proper subset $A$ is not equal but is a subset of $B$, i.e. $A=\{1,2,3\}$ and $B=\{1,2,3,4\}$ I just don't know how that proves the right hand side of the above equation.
Best Answer
You already have that $B$ includes an element that is not in $A$. Therefore $B$ is not a subset of $A$.
(To be a subset of $A$, every element of $B$ would have to also be an element of $A$.)