I am horribly stuck on $3$ proofs for my discrete math class. Any help would be greatly appreciated.
Prove that if $R$ is a partial order, then $R^{-1}$ is a partial order
Prove that if $R_1$ and $R_2$ are equivalence relations on the set $A$, then $R_1\cap R_2$ is an equivalence relation.
Thank You!
First thing is that you absolutely must know the relevant definitions: what is meant by partial order, inverse, equivalence relation, intersection, reflexive, symmetric, transitive, antisymmetric. If you can't write these definitions down instantly then you need to work on learning them thoroughly.
Hint. Here is part of (1). The rest of (1) is similar, so is (2). Problem (3) is really a completely different topic, I suggest you delete it and ask a separate question.
Let $R$ be a partial order: therefore $R$ is reflexive, transitive and antisymmetric. We prove that $R^{-1}$ is transitive.
So, suppose that $x\,R^{-1}\,y$ and $y\,R^{-1}\,z$. By definition of inverse this means that $y\,R\,x$ and $z\,R\,y$. Since $R$ is transitive we have $z\,R\,x$, and using the definition of inverse again, $x\,R^{-1}\,z$. We have proved that if $x\,R^{-1}\,y$ and $y\,R^{-1}\,z$ then $x\,R^{-1}\,z$; by definition, $R^{-1}$ is transitive.
Observe that this proof really uses pretty much nothing except various definitions. So I hope this underlines the importance of knowing the definitions properly.
Have a try at the rest for yourself. Good luck!