Simple proofs for these are pretty straight forward such as proving if two sets are equal then they are subsets of each other or if you want to show one set is a subset of the other just show that there exists an element in one set and show that they are in the other but when you add in cross and implications it gets kind of tricky.
For example,prove or disprove if the statement is true or false.
if $A\cap C \subseteq B \cap D \implies A \subseteq B$
My proof, Given $x\in A\cap C$
$\implies x\in A$ and $x \in C$ if $x \in A$ then $x \in b \cap D$ since $A\cap C \subseteq B \cap D $
then $x\in B\cap D $ then $x \in B$ and D since x is in A and B then they could be subsets but it also means that A=B therefore this statement should be false?
Can someone critique my proof or tell me if this isn't the way to go about this proof?
Thanks in advance!
Best Answer
Something more concrete.
Let $A=[0,1]$,
$C=[0.5,1.5]$,
$B=D=[0.4,1]$.
Then $A\cap C=[0.5,1]$ and $B\cap D=[0.4,1]$. So $A\cap C \subset B\cap D$. But $A$ is certainly not a subset of $B$.