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.

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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.

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.