I am currently taking a class entitled "Discrete Mathematics for Computer Science." Part of my assigned reading is an introduction to logic. I have understood everything until I reached a part called "Conditional Statements." The statement p is "Maria learns discrete mathematics." The statement q is "Maria will find a good job." so p --> q is "If Maria learns discrete mathematics, then she will find a good job." The truth table in the book is as follows p q p-->q T T T T F F F T T F F T I can see why if Maria learns discrete mathematics, she will find a good job. I can also understand that if she learns mathematics, but does not find a good job, then the statement "If Maria learns discrete mathemetics, then she will find a good job" is false. However, how is it that if she does not learn discrete mathematics that p-->q is true? To me it would seem that it would be false because she never learned the necessary materials to make the conditional statement true. The final one, where both p and q are false, makes no sense to me at all. How is it that she doesn't learn the math, doesn't get the job, yet the statement "If Maria learns discrete mathematics, then she will find a good job" is considered true? Any help with this would be greatly appreciated. Thanks.
That is a though one to get your mind around, no? I remember when I took an Introduction to Mathematical Reasoning class we covered this, and it took me forever to figure out why that was the right definition. Try thinking of it this way: We want to decide whether or not the following statement is true: If a Maria learns discrete mathematics, then she will find a good job. Now how would we every prove this statement is false? Well, the easiest way that I can think of would be to find a counter-example. That is to say, we need to find a situation where (p,q) = (True, False). If we can find the situation to be that she learned discrete mathematics but did not get a good job, then p -> q is obviously false. But what about (p,q) = (False, True) or (p,q) = (False, True)? In either of these situations, we can't say that the p -> q is a false statement because, so as far as we know, p -> q is still likely to be a true statement (like you said, she never learned the right skills for us to tell) . So the only way to be absolutely certain that p -> q is False is when you have (p,q) = (True, False). Other than that, we say that it is True. Now I know that argument isn't air-tight, but it's not really supposed to be. In the end, that's the definition that works so we keep it, but hopefully my very qualitative/heuristic explanation at least makes it plausible and someone else with some more formal training can give you a more quantitative argument. - Jason
I think I figured out the very bottom row. Ok, if Maria has not yet learned discrete mathematics, she has not gotten a good job. She is still at "square one" so to speak, so she still has the potential to learn the math and, eventually, get a good job. Is this a correct explanation?
i think i also figured out the 3rd row. If Maria already has a good paying job, then there's still the possibility of her learning the math and getting a good paying job as a direct result of her learning mathematics. Therefore, the statement "If Maria learns discrete mathematics, then she will find a good job" is also true.
Think of it as "innocent until proven guilty". The statement says "If Maria learns discrete mathematics then she will get a good paying job" (obviously this is a discrete mathematics course!). It says nothing about what will happen if she does not learn discrete mathematics and so the situation in which Maria does not learn discrete mathematics tells us nothing about the truth or falseness of the statement. Since it hasn't be "proven" false, call it true.
At the bottom line, the single arrow just means that - if the precedent is true while the antecedent is false, the arrow sentence will be false - otherwise it'll be true. Don't let the weird intuitions of natural language confuse you