Help me understand the logical operator "IMPLIES" in a clear manner

First, you don't actually need to use "implies" for anything.   In math, "p implies q"  is just shorthand for "q or not p".    So if it continues to bother you, just treat it as a concept completely different than the normal English word "implies".

That said, a common way to explain the rationale to people is to think about statements like:  "If it rains, I always take an umbrella".        If it turns out to not rain and I don't take an umbrella, have I lied?
You can think of P -> Q as saying "whenever P is true, Q is also true." This is vacuously satisfied if P is never true, regardless of if Q is true or not.

If P -> Q and P are true, then Q is too, but P -> Q and ~P don't imply anything about Q.
There are several ways how to look at it:

0. There is no undefined value in binary logic. Every statement is either true or false (if we are not diving into self referencing paradoxes)

1. In the topic of binary logic, this is just a definition of an operation IMPLIES (vice versa IMPLIES is the name for the give truth table). There is no need to overthink it. Sometimes operations follow the language (AND) sometimes they don't (OR).

2. In maths, truthfulness of the operation IMPLIES is the assessment of if you have applied the rules of mathematics correctly. Not what initial statements you took. Imagine a true IMPLIES rule:

> if a=b, then ac = bc

then you can apply it to correct initial statement:

> if 2=2, then 2×3 = 2×3

or to a false initial statement to get either false or true statements:

> if 1=2, then 1×3 = 2×3

> if 1=2, then 1×0 = 2×0

the fact that you have applied it to a false initial statement, doesn't mean that the rule is false.

4.Philosophically, A=>B means that set of all events making A true S(A) is contained in the set of events making B true S(B). Because of this the relation is not symmetric (which is reflected in assymetrical sign =>). If you take an event making A false, it can be in S(B) or not be in in S(B). The relation is not reversible. True general statements imply that specific statements are true as well. (All people breath. I am a person, therefore I breath). However, true specific statements don't make general statements necessarily true (I can swim. I am person, therefore all people can swim too). The relation IMPLIES shows which of two statements is general and which is specific
You can think of it as a promise.   "If P, then Q."   It might be true, it might not be true.

If someone makes you promise that e.g. if you get all A's you can get a new cellphone, then if you don't get all A's, then promise wasn't false.   It just didn't apply.
implies is like a promise
"when it rains i will pick you up from band practice"

if on days its raining i pick you up from band practice then I have not broken my promise

if it is raining and i do not pick you up from band practice i have broken my promise

if it is not raining and i do not pick you up from band practice i have kept my promise

if its not raining and i pick you up from band practice then i have also kept my promise
by
In propositional logic, every statement must have a truth value.  For the sake of illustration (and making a connection with natural language), let's consider an example where time is a factor, so that when a statement is made we do not yet know its truth value.

You friend says "If you get an A on your test, I will buy you dinner."  If you then get a B on your test, maybe your friend buys you dinner anyway, and maybe they don't.  But you can't say that the statement was false, because they only said what their actions would have been if you had gotten an A. And if their statement isn't false, then it is true.

This is very closely related to statements being "vacuously true."  Suppose, for the sake of example, that unicorns do not exist.  Your 5 year old neighbor insists that every unicorn can fly.  Her brother insists that black unicorns cannot fly.  Their disagreement turns violent and you leave before you get dragged into it.  But you ask yourself "Well, are either of them *wrong*?" And the answer, at least in propositional logic, is no! The statement is "vacuously true" because there are no cases to disprove it.  There are no unicorns that don't fly, so you can't say that the sister is wrong.  But there are also no black unicorns that fly, so you can't say the brother is wrong either.  Note that the word "vacuous" in this context comes from the meaning of "empty", as in "the set of all unicorns is the empty set."

But with the semantics of implication, we can get vacuous truth by turning the "every" statement into an implication statement.  "Every unicorn can fly" can be rewritten as "If x is a unicorn, then x can fly."  By giving implication the semantics that we do, these two statements are equivalent.
Logical implication is somewhat different than implication in language. Just memorize the truth table.
I like to think of it in terms of universal statements, where it is absolutely needed that P=>Q is true if P is false.

Consider the sentence "Every leaf is green". It is a universal statement about leaves, which is only true if the statement "This object is green" is true whenever "this object" refers to a leaf. If there is even one non-true statement of the form "This object is green" referring to a leaf, then the universal statement "Every leaf is green" is false.

Now let's consider more complicated universal statements, like the following: "Every dog wags its tail if it's happy". The statement we make universal here is an implication: if happy, then tail wagging. The universal statement is *only* true if the statement "If [dog's name] is happy, then it wags its tail" is true for *every* dog. Even the unhappy ones! So when Odie is unhappy, the statement "If Odie is happy, Odie wags his tail" must still be true, otherwise the universal statement about dogs is false!
Consider this example from Susanna Epp's Discrete Mathematics:

If 0=1 then 1=2 (by +1 to either side)

This logical statement is true. If it were that 0=1, then 1 would equal 2.

It turns out of course that both 0=/=1 and 1=/=2, and we have a clear example of F->F=T

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For F->T=T consider this: there are some results in mathematics that have been proved via Riemann hypothesis, as in, we have presented a valid argument that IF the Riemann hypothesis is true, THEN theorem x is also true. It may happen that the Riemann hypothesis is false, but we could find some alternative proof for theorem x, so we'd know that it is true despite that. Then we'd have F->T=T
we are asserting that "p implies q" if p is not true then it cannot disprove our assertion.

take  s = {1,2,3,4,5,6,7,8,9,10}
let p be numbers in s divisible by 4, p = {4,8}
let q be numbers in s divisible by 2, q = {2,4,6,8}
if a number is divisible by 4 it is also divisible by 2
here p -> q (p is a subset of q)
in other words we are asserting that if a number is in set p, then it is set q.
now take 1, it is not in p or q so it cannot disprove our assertion. or take 2, it is in q but not in p so our assertion still stands.

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