Discrete math: What is the difference between false and inverse in conditional statemensts? Let's say there is this conditional statement:
If I am in Paris, then I am in France.
So, p = 'I am in Paris', and q = 'I am in France'
I do not understand when p and q are false, how would that translate to?
Would it not be the same as inverse of conditional statement?
I would be grateful if you guide me on my misinterpretation.
 A: Note: for the following we shall assume that "Paris" refers exclusively to the Capital of France rather than any of the many other places named Paris.
The Statement is: "If I am in Paris, then I am in France."   Given the above, this is definitely true.
$$p\to q$$
The Inverse of this is: "If I am not in Paris, then I am not in France."   This is not necessarily true, as I may be in Lyon, which is also in France.
$$\neg p\to\neg q$$
The Converse of the statement is: "If I am in France, then I am in Paris."   This is also not necessarily true, for the same reason.
$$q\to p$$
However, the Contrapositive of the statement is: "If I am not in France, then I am not in Paris."   This holds true (given our assumption).
$$\neg q\to \neg p$$
   The Contrapositive of a statement is equivalent to the statement.
$$\neg q\to \neg p \quad\iff\quad p\to q$$
A: I prefer to think in different terms:
It's much simpler to use if p then q.
if condition one is met than move on to condition two.
this is a capacitive decision unrelated to the outcome.
if p then q does not represent an equation.
