What is the purpose of implication in discrete mathematics? I would be obliged if you can show me an example of a truth table for implication where there is a also a real life aspect to it. (i.e., where would someone use the scenario to make F->F = T and also the same for the remaining 3 cases).
However, this one scenario should be able to be adjusted to fit all three. Hopefully that makes sense.
I am just trying to understand the concept of implication.
 A: Consider the implication "if it rains, then I take an umbrella."
(1) If it rains and I take an umbrella, then this implication is true.
(2) If it rains and I don't take an umbrella, then this implication is false.
Hopefully these first two are not controversial. Now consider the other two:
(3) If it doesn't rain and I take an umbrella, then this implication is true.
(4) If it doesn't rain and I don't take an umbrella, then this implication is true.
One way to think about this is that the implication "if it rains, then I take an umbrella" says that under any circumstances I will do whatever it takes, umbrella-wise, so as to stay dry.  The only way that I will get wet is (2): it rains and I don't take an umbrella.  If it doesn't rain, then the implication is trivially (or "vacuously") true: I will stay dry regardless of whether I take an umbrella.
A: Here's another example:
(A) If I hit my thumb with a hammer, then my thumb hurts.
If I'm in a reality where I don't hit my thumb with a hammer and my thumb doesn't hurt, I'd still consider (A) to be true. 
