hey guys and welcome to this tutorial by the end of this video you'll be able to prove statements by contradiction using four really simple steps that work every time the prerequisite knowledge required for this video is that you're able to work with numbers in a really general sense so performing simple operations you're able to work with algebraic expressions so that includes manipulating and simplifying algebraic expressions and that you're able to understand different number types for example prime numbers composite numbers integers and so on and that you're comfortable expressing these number types in different ways so let's crack on with the tutorial and let's understand what proof by contradiction is and why we use it to prove statements just like many of the other proofs that you will have seen at this level such as proof by deduction exhaustion and counter example proof by contradiction is just another technique used to prove statements and it's used to prove statements that are very difficult to prove by just using the statements themselves so here's an example of one of those types of statements all numbers are positive this would be a really difficult statement to prove or disprove by just using the statement alone and that's because there are an infinite amount of positive numbers so using this method of proof by contradiction what we would do is use what's called a negation statement to prove or disprove the original statement okay so we're going to look more into negation statements shortly but let's have a look at a negation statement that we could derive from this original statement here it is so the original statement was that all numbers are positive and the negation is that there exists a number that is negative now we know of course that there exists many numbers that are negative minus 1 minus 20 minus a thousand and so on but by just finding one instance of a negative number we've used the negation to disprove the original statement really really quickly much quicker than it would have taken us to disprove the statement if we had used the statement itself let's have a look at another example here's a statement five times five is equal to twenty-five the negation of this statement is that five times five is not equal to twenty-five in this case the negation is clearly not correct or in other words there is no case where 5 times 5 is not equal to 25 and therefore the negation statement is false and therefore the original statement can be proven so we've just used these two examples to give you a general idea of how we use the negation statement in proof by contradiction the method is to use the negation to prove or disprove the original statement in proof by contradiction questions they'll generally follow this format where you need to prove the original statement by working out that the negation isn't true or gives you a contradiction so let's go ahead and look at the four simple steps that you need to use in order to prove statements by contradiction so we've called it the knack method because it's just easier to remember okay so the first letter n stands for negation and the first thing you do is write the negation statement and we're going to look at how to write the negation statement for different types of statements shortly the next step is to assume the negation statement that you've just written is true the next step c is to perform calculations now assuming that negation statement is true we need to do some logical calculations with the assumptions and the final step is the contradiction so based on your calculations you will come to a contradiction which will then allow you to conclude the proof of your statement now this is all going to make much more sense when we go through a couple of examples however based off of what i've seen students struggling with here i'm going to focus on the negation statement how to derive the negation statement and as it's the first part of this method it is in some sense the most important part so let's have a look at some examples of negation statements now what is the negation statement now in most cases it's just literally the opposite of the original statement or perhaps a better way to think about it that it's a statement you could form that if it were true it would actually negate or nullify the original statement so looking at these examples we're going to see that the negation statement that we use isn't always just the exact opposite of the original statement so we're going to start off with a couple of easy examples and then we'll move on to some of those harder examples where it's not an exact opposite here the statement is that n is odd now the negation is that n is not odd of course another way to express this is that n is even another statement is that the square root of 20 equals five the negation would be the square root of 20 does not equal five let's have a look at this one all tall guys get girls what do you think would be the negation of this statement well an easy negation would be that no tour guys get girls right but if we remember from the earlier slide the main reason for using the negation is to actually make our life easier it's actually to make the proof much quicker so if we go from saying that all tall guys get girls too the negation is that no tall guys get girls then proving that no tall guys get girls is a lot harder than proving that there exists one tall guy that doesn't get girls okay and this would be a much more suitable negation statement in this case so when you see the keywords all an associated keyword for the negation should be that there exists okay and that's why i've highlighted these in bold for you okay let's have a look at another example no pigs eat mud well a negation that you could use is that there exists one pig that eats mud so notice how we're using the keyword there exists one instance of something quite often for these negation statements looking at the last example we have that if n squared is even then n is even the negation for this statement would be that if n squared is even then there exists an n that is odd okay so we've used these examples to show you that of course you should be thinking about what a negation statement is but this gives you a kind of more scientific way of coming up with a negation statement by using key words such as if then then there exists none goes to there exists all goes to there exists okay and do you try practicing this with other examples in your textbooks okay so now we hopefully understand how to form a negation statement let's go ahead and do examples to complete a proof by contradiction so the first example asks us to prove by contradiction that if n squared is even then n must be even so using the knack method the first thing we need to do is write the negation statement and from the previous slide we saw that the negation statement would be that if n squared is even then there exists an n that is odd the next thing we need to do is write an assumption assuming that the negation is true so we're going to assume that if n squared is even then n is odd for the next part we need to perform some calculations based off of our assumption okay so based off of our assumption well we've assumed that n is odd and therefore n can be written as equal to two k plus one where k is an integer and as we have an expression for n we can write an expression for n squared therefore n squared is equal to two k plus one multiplied by two k plus one if we multiply the brackets this is equal to four k squared plus four k plus one and we can factorize both of these terms to get that this is equal to two times by two k squared plus two k plus one now you often need to manipulate algebraic expressions in order to be able to make conclusions about what you found now looking at this we have two multiplied by two k squared plus two k now two k squared plus 2k can just be seen as some integer right so that's two multiplied by some integer plus one and by our definition of an odd number well that means that this term must be odd n squared is odd and this is where we come to our contradiction because if n squared is odd this contradicts the assumption that we made here that n squared is even and therefore we cannot go with the negation we must go with the original statement that therefore if n squared is even then n is even okay let's have a look at this example prove by contradiction that the square root of two is irrational so the first thing we need to do is come up with a negation statement because we see the key word is for the original statement the natural negation that you could use is that the square root of 2 is not irrational another way to say that is that the square root of 2 is rational so we're just gonna use this negation okay the next part we need to assume the negation is true so now we're going to assume that the square root of 2 is a rational number now if the square root of 2 is a rational number it means that the square root of 2 can be written as a fraction a over b where a and b are integers and a over b is a fraction in its simplest form now the idea of it being in its simplest form does seem a little bit random at the moment but what we do know is that any fraction can be written in its simplest form okay now as we've written the square root of 2 is equal to a over b let's do some logical calculations on this so what we could do is multiply both sides of the equation by b to get that the square root of 2 times b is equal to a we could get rid of the square root by squaring both sides of the equation giving us that 2b squared is equal to a squared and if we look at this we can see that a squared is equal to 2 multiplied by b squared and since b is an integer it means a b squared is an integer now if a squared is equal to two times by an integer it means that a squared must be even and from the previous question we learned that if a squared is even then a would be even and therefore we can write a as equal to 2n where n is an integer so now we have an expression for a we can substitute it into this equation to get the following that 2b squared is equal to 2n or squared expanding the brackets on the right hand side we get that 2b squared is equal to 4n squared and if we divide both sides of the equation by two we get the b squared is equal to two n squared so what does that tell us well b squared is equal to two times by an integer because n squared would be an integer and that means that b squared is also even and therefore b is even now this is where we come to the final part of this proof the contradiction because by performing some calculations assuming the negation was true we found that actually a was even and b was even but if a and b are both even then a over b cannot be in its simplest form as we assume to be true this contradicts the negation statement and therefore we can conclude that the square root of two is irrational so hopefully this tutorial gave you the tools you need to do proof by contradiction questions keep up the good work and i'll see you soon if you like this video please give us a thumbs up leave your comments down below and subscribe to this channel so you'll be the first to know when we release our next videos