I need to prove that any convex property for graphs can always be expressed as the intersection of an increasing property and decreasing property for graph, specifically that:
$\forall A\subset B\subset C ~~$ and a property $~~Q~~$ if $~~Q\in A \land Q\in C \implies Q\in B $.
I did the following using logic: $\forall A\subset B\subset C$ $$Q\in A \land Q\in C \iff Q\in A$$ so $$Q\in A \implies Q\in B $$ by the same argument: $$Q \in A \land Q\in C \iff Q\in C$$ therefore $$Q\in C \implies Q \in B$$
That is for all $A, B, C$ such that $A\subset B \subset C$. My question arises in how can i combine the two previous results to get the the desired intersection or something like $\forall A\subset B \subset C$ $(Q\in A \implies Q\in B) \land (Q\in C \implies Q \in B)$ which will result in the intersection of an increasing property and decreasing prrperty by definition.
Thank you!