Problem in understanding p implies q

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?

Solution 1:

This is a simple matter answered by the truth table of $\Rightarrow$:

$$\begin{array}{ c | c || c | } P & Q & P\Rightarrow Q \\ \hline \text T & \text T & \text T \\ \text T & \text F & \text F \\ \text F & \text T & \text T \\ \text F & \text F & \text T \end{array}$$

This shows that when $P$ is false, the implication is true. Note that this is the definition of the table, there is no need to prove it. This is how $\Rightarrow$ is defined to work.

As an example, here is one:

$$\textbf{If it is raining then there are clouds in the sky}$$

In this case $P=$It is raining, and $Q=$There are clouds in the sky. Note that $P$ is sufficient to conclude $Q$, and $Q$ is necessary for $P$. There is no rain without clouds, and if there are no clouds then there cannot be any rain.

However, note that $P$ is not necessary for $Q$. There could be light clouds without any rain, and there could be clouds of snow and blizzard (which is technically not rain).

Solution 2:

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.