I want to show that some problem $P_1$ is NP-hard. I have a problem $P_2$ that is NP-complete. From an instance of $P_2$ I created in polynomial time an instance of the problem $P_1$.My question is: Should I verify both direction ($\Leftrightarrow$) or only one direction ($\Rightarrow$)? More precisely, which one to show from these two:Solve $P_1\;\Leftrightarrow$ solve  $P_2$Solve $P_1\;\Rightarrow$ solve  $P_2$
There is equivalence in an NP-hardness proof or not?
complexity theory;np complete;reductions;np hard
You only need to show that solving $P_1$ allows you to solve $P_2$ (and, hence, every other problem in NP).$P_1$ might not be in NP so it's not necessarily the case that solving $P_2$ allows you to solve $P_1$.  On the other hand, if $P_1$ is in NP, then you already know that solving $P_2$ allows you to solve $P_1$, by definition of NP-completeness.