TITLE: Problem in understanding p implies q
QUESTION [19 upvotes]: 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?

REPLY [4 votes]: Once I read a classic example which I would like to share with you. 
A politician said.........
p(If I Will win) q(I will cut down taxes by half)
People will feel cheated only in the case when the politician actually won but taxes remained the same. 
In this case, q is FALSE but p is true so p->q become false.
 We are okay with the rest of cases/situation, so rest of combination of p and q this statement is TRUE.
Lets examine "p is not necessary but sufficient for q"................
There can be a case when this politician lose but taxes get reduced so for the reduction of taxes p is not necessary but sufficient...........
regarding "q is a necessary condition for p" 
we can say reduction in taxes is necessary after the victory of the politician else the entire statement of the politician will be false or we can say p->q will be false.