 Every branch of human endeavor has its way of establishing truth. For example, consider the statement, the product of even numbers is an even number. A scientist might run an experiment. 2 times 2 is 4. 4 times 8 is 32. 6 times 2 is 12. These are all even. A politician may say something like this. 3 times 5 is 35. It's even. 35 is a very even number. And the mathematician would say something like, well, what do you mean exactly? Since our concern here is with mathematical proof, let's talk more about that. A mathematical proof begins with some assumptions, which we call the premises. It uses the rules of inference to derive other results and ends with a conclusion. Provided the premises are true and the rules of inference are valid, the conclusion is guaranteed to be true as well. This, however, begs the question, what is truth? Well, that's actually a hard question to answer, so we don't try. Instead, we approach the problem this way. A statement is a complete sentence that is either true or false. For example, 5 is equal to 2 plus 3. This is a true statement. This is a true statement. 2 plus 2 equals 5. This is a complete sentence. It happens to be false, so this is a false statement. Is it hard to prove things? This is a complete sentence, but it is not something that evaluates to either true or false, and so this is not a statement. We say that a statement has a truth value. This is true. This is false. A statement in mind is that it's not necessary that we know whether the statement is true or false, so the 500th digit of pi is 8. Well, this is definitely a statement, but I don't happen to know whether this is true or false. Likewise, you will get an A in this course. Again, definitely a statement, but at this point we don't know whether this is true or false, and in some sense mathematical proof is all about deciding whether statements like this are true or false. A simple statement is one that cannot be broken into statements. For example, the sky is blue. If we take any part of this, we don't get a complete sentence, and so no part of this can be broken into a complete sentence that has a truth value. This is a simple statement. Wow, this is a hard course unless you study. So if we take this sentence, we can actually break it apart into a couple of sub-sentences. For example, this is a hard course, or you study. And the thing to recognize here is that both of these portions can be evaluated as true or false, and so this can be broken into simple statements. This is a hard course. You study. That's no moon. It might seem that this is a simple statement, but in fact, we can take part of this sentence and give it a truth value. And we can do that if we drop the word no and do a little bit of grammatical correction, and so this statement can be broken into a simple statement. That's a moon. And this last one is important. There's a moral here. Yes, you can take things out of context. So for example, let's identify the simple statements in the sentence, what's a glurb of a grass it is quirty if and only if all of its fortechi are not squiggly? What's a glurb, grass it quirty, fortechi, and squiggly? Who knows? It doesn't matter. We can still identify the simple statements. And again, these simple statements are the shortest sentence that could be true or false, whether or not we know that they are true or false. So if we look at our sentence structure, it seems that this first part, the glurb of a grass it is quirty, that seems to be a complete sentence that could be true or false. And so it's a simple statement. There's all this other verbiage and then we get to the it's fortechi are not squiggly and that also seems to be a sentence that could be true or false and so it's fortechi are not squiggly seems to be a simple statement. But remember, not is one of those words that if we omit, we still get a sentence and we still get a statement. So in fact our simple statement should be it's fortechi are squiggly. We often join statements together to form compound statements and there are four basic types of compound statements. Suppose x and y are statements. Not x is a type of statement known as a negation. Both x and y is a type of statement known as a conjunction. Either x or y is a statement known as a disjunction and if x then y is a conditional. So for example let x be the statement 5 is equal to 2 plus 3 and y be the statement 2 plus 2 is equal to 5. Let's write the negation of x, the conjunction of x and y and the conditional if x then y. So remember definitions are the whole of mathematics. All else is commentary. If we want to write the negation, conjunction and conditional it helps if we know what these mean so we'll pull in those definitions. So the negation of x, well that's not x. So we might write that as 5 is not equal to 2 plus 3. The conjunction of x and y, well the conjunction is both x and y and so we might write this as 5 is equal to 2 plus 3 and 2 plus 2 is equal to 5. The conditional if x then y if 5 is equal to 2 plus 3 then 2 plus 2 is equal to 5.