I am trying to understand what “$p$ implies $q$” means. I read that $p$ is a sufficient condition for $q$, and $q$ is a necessary condition for $p$. Further from Wikipedia,
A necessary condition of a statement must be satisfied for the statement to be true. Formally, a statement $P$ is a necessary condition of a statement $Q$ if $Q$ implies $P,\quad (Q \Rightarrow P)$.
A sufficient condition is one that, if satisfied, assures the statement's truth. Formally, a statement $P$ is a sufficient condition of a statement $Q$ if $P$ implies $Q,\quad (P \Rightarrow Q)$.
Now what I am stuck with is that if $P$ is not satisfied will the condition still always be true?