[NOISE] Hi this is Michael. And in this lesson we're going to study examples, counter examples, and the basic constructions of mathematical logic. And we'll start with examples. Sometimes to prove a mathematical statement just one example is enough. For example, if want to prove the statement that white horses exist, it is sufficient to just show one white horse. We don't need to go and show many white horses, we don't need to show that all the horses are white. We just need to show one white horse, and that proves the statement. However such examples are not always easy to come up with. And now we'll consider some more problems which can be solved by examples. First problem is, is it possible that for three positive integer's, a squared plus b squared is equal to c squared? And here we can get an example with the three small numbers for a is 3, b is 4, and 3 squared plus 4 squared is 9, plus 16, which is 25. Which is the same as 5 squared. So we take c equal to 5. So this example solves the whole problem. How to come up with this example, well you can for example remember the Pythagorean Theorem from school. And that there exists a right triangle with sides 3, 4, and 5. And so the sum of squares of sides is equal to the biggest side in the right triangle. And so this is what this example is about. Now let's look at more statements like that. So in the previous problem we had a statement about squares of positive integer numbers. And now we want to ask the same about cubes of positive integer numbers. So a to the power of three is also called, a cubed. So is it possible that for three positive integers a, b, and c, a cubed + b cubed = c cubed. It would be also a solution to just one example to this problem, and then the answer would be positive. However in fact this is impossible, and so there is no example. Fermat's last theorem, one of the most famous mathematical conjectures, states that for any integer n > 2. There are no such integers, a, b, and c that a to the power of n plus b to the power of n, equals c to the power of n. Mathematicians were trying to prove this unsuccessfully for hundreds of years, there were thousands of attempts. And some of these attempts were pretty silly. But many of these attempts led to a lot of useful mathematical theories. And finally, in 1995, Andrew Wiles famously proved it. In it's particular case for n equals 3, it states that there are no such positive integers a, b, and c that a cubed plus b cubed is equal to c cubed. So this answers our problem with a negative, and it means that there is no example in this particular case. Another similar problem, is it possible that for positive integers a, b, c, and d, sum of their 4th powers of a, b, and c is equal to the 4th power of d? Well this problem, one could think that it is also impossible due to Fermat's Last Theorem. You just need applies for n equals 4. However, this theorem only says something about equations of the form a to the power of n, plus b to the power of n, is equal to c to the power of n. So it is not applicable because in this example we have three on the left, a to the power of 4, plus b to the power of 4, plus c to the power of 4. And Fermat's last theorem is about the cases when we have only two on the left so it is not applicable. Actually in fact the answer for this problem, again yes this is possible. However, the smallest possible example is huge, so it's 95,000 something to the 4th power, and 200,000 something to the 4th power, and 400,000 to the 4th power equal to 400,000 something to the 4th power. So this example's huge and I don't imagine that you could come up with this example and then your reasonably tiring yourself. Computers are often used to generate such huge examples however, the number of possibilities in many mathematical problems can be so big, that it is hard to find an example, even with the help of a computer and it would take a whole lot of time, even if we take all of the computers on Earth. But here an example was found, it was even proven that this is the smallest possible example. There are no numbers a, b and c which are less than these three. For which a to the 4, plus b to the 4, plus c to the 4 can be equal to some integer to the 4 power. And the last problem for which we're going to consider an example in this video is the following. Is there a power of 2 that starts with the digits 6 and 5. And here, the example, you probably, some of you know this example, that 2 to the power of 16 is equal to 65,536. And this is the whole solution, we just show this one example and we solve the whole problem. In fact, we can show that for any integer, positive integer n, there is a power of 2 that starts exactly with this integer, but this problem is much harder to prove. I should also note that you've met with examples as proofs in the previous lectures, and the previous weeks of this course. And for example, there was a problem is it possible to cover a chess board with dominoes, and it was solved using a particular example of how we can cover a chess board with dominoes. And there were several examples by Alexander, but it was sufficient to show just one example to solve that problem. So examples are a common way of solving with medical problems. And in the next lecture, we are going to talk about counter-examples. [MUSIC]