 Welcome to the screencast provocatively titled, How do we know if a statement is true? So in the last video we defined mathematical statements to be declarative sentences that have a definite truth value. Some statements are obviously true, like two is even, and some are obviously false, like three is even. But what about statements whose truth values is not so easy to determine? How do we determine truth? Well let's look at a case study here, and we're going to define an important kind of number here first. We're going to say that a positive whole number p is a prime number. If p is bigger than or equal to two, and if it only has two divisors, namely one and itself. So the numbers two, three, five, seven, eleven, one hundred and thirty-nine. My favorite prime number, eight, six, seven, five, three, oh nine, which was immortalized in a catchy tune from the eighties, are all prime numbers because they are whole numbers, they are all bigger than one, bigger than or equal to two, that is, and can only be evenly divided by one and themselves. And non-prime numbers include numbers like four and twelve, which have divisors more than just one and themselves. For example, twelve could be divided by one, two, three, four, and six and twelve. So here's a statement to think about, whenever p is a prime number, the number two to the p power minus one is also a prime number. Now this is a statement because it's a declarative sentence, it's declaring that something is true and the statement's truth value doesn't depend on opinion or on the particular value of p. It's saying that as long as p is a prime number, whenever p is a prime number, so is the number two to the p minus one. So this is a statement, it's a claim. Being a statement, it's either true or false and the question is, which one? Well, it's very hard to tell just by looking. Finding out the truth value of this statement is a mathematical problem and a solution to that problem will consist of two things. First, we have to clearly state whether the statement is true or false. That's gotta be part of the solution. And then, very importantly, we have to explain why the statement is true or false. And to understand the solution of this problem, as with the solution to any problem, we need to first experiment with the problem and play with it. So let's pick some prime numbers and just compute two to the p minus one and see what happens. And we'll keep our results in a table here. So when p is equal to two, that's the smallest prime number, then two to the p minus one is two to the two minus one. And that's four minus one, and that's three, and that's a prime number. When p is equal to three, then two to the p minus one is two cubed minus one. That's eight minus one, which is seven, and that's another prime number. When p equals four, oops, I can't choose that. I can't choose p equals four, now why not? Well, four isn't a prime number because it can be divided by two. And so it doesn't fit the description of the statement. The statement is claiming something is true if p is a prime number. It says nothing about non-primes. It makes no guarantees as to what happens if p is not prime. So there's no point in playing with this problem to choose p to be anything other than a prime number. So let's move on to the next prime, which is p equals five. Now two to the fifth minus one, two to the fifth is 32. 32 minus one is 31, and it so happens that's a prime number two. An x prime number up the list is p equals seven. Two to the seventh minus one is, let's see, two to the seventh is 128. 128 minus one is 127, and we can check and see that that's prime. So this is fantastic. All of our examples are working, so what does this mean? Let's take a quick concept check here to see what you think it means. This is a very simple concept check, but I have to think about the answer here. So the fact that the two to the p statement was true, for p equals two, three, five, and seven means what? That the statement is definitely true, or that the statement might be true, but we don't know yet. Now think about that question and pause the video and unpause it when you think you got the answer. So the correct answer here is b, the statement might be true, but we can't say that the statement is true for sure yet. Why not? Well, recall what the statement actually says. It says whenever p is a prime number, two to the p minus one is also prime. That word whenever is all important. It is saying that every example of a prime number that we or anybody else could ever produce should work. Four examples, which is what we've done here, actually five examples, p equals two, three, five, and seven, I guess it is four examples. That's just not enough because what about all the examples that we haven't tried? And in fact, speaking of those examples we haven't tried, let's go to the next prime number. The next prime number after seven is 11. And we can calculate that two to the 11th minus one, two to the 11th is 2048, minus one is 2047, and unfortunately 2047 is not prime. It's equal to 23 times 89. So it actually turns out that this statement that I made here is false because it claimed that every time p is prime, two to the p minus one is also prime, and that's simply not the case. The statement is sometimes true, but because it is not always true, we're gonna say that it is false. So it seems like we failed somehow, but we've actually been successful in solving our problem. We've determined that the statement that whenever p is prime, two to the p minus one is prime is actually false because it made a general statement about prime numbers. But we found a specific instance where that claim didn't work, a prime number p such that two to the p minus one is not prime. So one example was enough to prove the statement is false, but notice that no amount of successful examples would ever prove this statement true because even if we had a million examples that worked, what about that million and first one? So there's an important factor to take away about statements like this. We cannot prove a general statement to be true using a list of examples. Now those examples are useful because they help us to make good guesses as to whether a statement is true or not. They help us to believe whether a statement is true. But it doesn't constitute a proof. So let's recap what we've seen. Given the statement that has a definite truth value, but you don't know what it is, the main way to get a feel for its truth value is to play with the problem rather than try to solve it immediately. This has two positive effects. First, you'll understand the problem better. Second, you may stumble across something that actually answers your questions, as we stumbled across the fact that two to the 11th minus one isn't a prime number in the first problem. In either case, you'll be on a more sure footing to proceed further into the problem whether you believe that the statement you're trying to work with is true or not. We've seen that examples help us give a sense of the truth value of a statement. But examples don't really prove anything unless the example shoots down a general statement. That's all for now. Thanks for watching.