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okay in this video we're gonna look at four basic proof techniques used in mathematics and what we're gonna do is talk about direct proof proof by contradiction proof by induction and proof by contradiction we're gonna do that the result that we're proving is going to be something very basic you might even think do we really need to prove that but again what we're trying to do is focus on these techniques and how they're used and some other videos I'll do some more direct proof proof by contradiction induction and contrapositive examples and then we'll do some that are more complicated so again just trying to focus on the techniques here so if you're new again to writing proofs hopefully this'll steer you in the right direction so we're going to prove one of these binstock statements if P then Q and we're going to use the four approaches we just mentioned those different types of proof techniques so what we're going to prove in each case is we're going to prove the following it says the sum of any two consecutive numbers is odd and again not a mind-blowing result by any stretch so write three plus four is odd four plus five is odd five plus six is odd etc etc etc and to do that well we're going to use some definitions so definition one so an integer number in is even if and only if there exists an integer K so that we can write that number as N equals two times K definition two so an integer number n is going to be odd if and only if there exists an integer K such that we can write n as equal to 2k plus one that's going to be the definition of an odd number and definition three two integers a and B are going to be consecutive if and only if B equals a plus one so one number one of the numbers is is one larger than the other one alright that's what you're saying okay so this statement the sum of any two consecutive numbers is odd it may not at first glance feel like one of these if P then Q statements but of course you could rephrase this slightly and we could put the F of N in there and say well if a and B are consecutive numbers then the sum of a is odd so that's gonna be our evidence statement just to make it really crystal clear so okay so let's actually start doing these I think a few minutes to get through all four shouldn't take too terribly long though so let's look at the direct proof first so we zoom out here a little bit to get everything in there okay so direct proof so this is going to be our first proof technique so the basic idea of a direct proof of if P then Q is we're gonna assume that this statement P is true so we're not even going to you know work on justifying that we're gonna assume that we have some statement that's true and we're going to use that that the true statement P to show that Q is true and this is you know typically and of proof that's going to be the hard part so that's the part that's gonna require a little bit of legwork so one little remark here you know I say learn to read we read understand and love your definitions a lot of it especially very fundamental things that you prove at the very beginning you're gonna try to share that some form or some definition is being satisfied so in my case we're gonna show that you know this N equals to k plus 1 we're gonna keep referring back to this form where K is an integer over and over all right so let's prove let's prove this a a and B or consecutive then a plus B is odd so let's assume this is gonna be our proof now and I'm gonna be a little sloppy here my handwriting so we're gonna assume that a and B are consecutive so we're assuming that those are consecutive integers well that means we know that we can write well we can write B as a plus 1 we can write one of the numbers as the other number plus 1 that's what we're given because again that's what it means for them to be consecutive so here's our here's our argument here's the big justification well we could write a plus B that's what we're trying to show is odd we could write a plus B simply as well we could write it as a plus and we said that B is equal to a plus 1 so I'm just inserting that there so we're writing a plus B is a plus a plus 1 well I can simplify this and say that that is equal to 2a plus 1 and this was the definition of a number being odd so a plus B equals 2a plus 1 we said we had to have this form of 2 k plus 1 where K is an integer and again I'm gonna you know we're assuming again that a and B are integers they're consecutive integers I said I'm gonna be a little sloppy and not write everything out so we're now done proof complete so we've written the sum a plus B in this form 2 a plus 1 where a is an integer and we said that's the definition of a number to be odd as if we can write it as 2 times some integer plus 1 that means that values odd so boom we're done proof complete sometimes you'll see a little boxes at the end of your proof I used to always like writing those because it made me feel like I've done some major accomplishment here so let's see so that one again you know simple results so we're not gonna have to do a ton of work in this case let's look next at proof by contradiction so just a brief outline of what's going on here so the idea is write that a proposition is either true or false but not both so we get a contradiction where we can show a statement as both true and false showing our initial assumptions or inconsistent if we get a statement that's both if we've got something that's true and false something is wrong somewhere right so we use this to show that if P then Q is true by summing that assuming both P and not Q are simultaneously true and then we derive a contradiction so when we reach this contradiction it means that one of our assumptions cannot be correct either P's not our assumption about P is not correct or not Q's not correct and typically we'll assume that P is true just we all just oftentimes assume that it is true so that by default the inconsistency must be that not to you is false and if not Q is false that means that Q is true and that's what we want to do okay so let's look at our example here let's let's prove again that if a and B are consecutive integers then the sum a plus B is odd so again we're gonna assume so here's our if part we're going to assume the a and B are consecutive integers so I'll put integers in here I felt I feel bad that I didn't write it in the last one so let's assume that a and B are consecutive integers and now let's assume though let's assume that some a plus B is not odd so let's assume that it's not odd okay well if it's not odd and again when I say if if it's not odd the sum if a plus B is not odd then what happens well there's no integer K there's no inter integer K such that we can write the sum a plus B equals 2k plus 1 if the sum a plus B is not odd again we set of K is an integer 2k plus 1 is going to be odd so there's no K so that this happens this is it does not hold true but we've already seen as in the previous example that we can write the sum a plus B we can write that as a plus a plus 1 because a and B are consecutive well that gives us 2 a plus 1 so we've got we've got a contradiction here right now we've shown that a plus B does not equal to K plus 1 where K is some integer but we also have shown that a plus B does equal to a plus one and again a is an integer so this is our contradiction these two results these contradict each other so we're assuming the a a and B are consecutive so it must be incorrect that our second assumption that a plus B is not odd that's not correct at all that's our that's our inconsistency here so by default it must mean that in fact a plus B the sum a plus B is odd and that would be again our proof by contradiction okay so we assume that they're not odd the sum is not odd so we can't find this integer K so that a plus B equals 2k plus 1 but hey we certainly can because a plus B we can write that as 2 a plus 1 says we can never do this where K is an integer but we know that a is an integer so that's going to be our contradiction and we've got this result that a plus B must be on some people don't like proof by contradiction because a lot of times a direct proof for some mathematicians a direct proof is nicer because usually there's more information that is given you do something a little more constructive and a lot of times with the proof by contradiction you're just showing that a result doesn't hold but oftentimes you know sometimes it doesn't give you any any insight as to why something is true so some people don't like proof by contradiction I was always a big fan of it I seem to have good results with it all right proof by induction so you may have seen proof by induction again in a precalculus class a calculus class I've definitely seen people you know run into these and in high school and they're not certainly not taking some advanced math course so proof by induction so proof by induction is a method to show an infinite number of facts by showing some specific case holds and then using the assumption that the proposition is true for some value of n we assume that that the proposition is also true for n plus 1 so the idea is when you're showing that it's true for some value of n and that it's also true for n plus 1 you're kind of showing that well if it holds for one case it holds for the next one but then again because of that generality it would hold for the case after that as well and the one after that and one after that and one after that etc this first part showing that it holds for a specific case kind of gets the whole ball rolling it's kind of like if you have a bunch of dominoes it's kind of you know yeah dominoes right you know so if one knocks over the other will knock over and if that one knocks over the oven will knock over and if that one knocks over the other one will knock over well okay that means that they'll all knock over as long as we can get it started and that's what this specific case does it kind of gets it started and then this general result will shows that it will keep going on and on indefinitely so okay so let's repeat this we're gonna show that a propositional form P of X is true for some basis case so this is just kind of terminology gobbly we're gonna assuming that P of n is true for some in we're gonna show that this implies that P of n plus one is true so again we're showing it holds for some men then it holds for n plus one as well and then again just by what we said by the principle of induction it's gonna follow that this propositional form is true for all N greater than or equal to the bases case okay so let's write our proof out here for induction okay so we're gonna let so we'll let the propositional form we'll let the propositional form f of X be true when the sum of X and the number following it its successor is odd okay that's when our propositional form is going to be true so let's consider the our base case here so our basis either our base case or our basis so we'll call this the base case here let's consider the propositional form f of 1 well f of 1 is going to be so so notice the sum 1 plus 2 equals 3 is odd because we can show we can show there exists some K that belongs to the integers so from the previous video remember this little backward z remembers there's some there exists some k belonging to the set of integers maybe you remember from your again hopefully your math class is the set of integers we can write that as a capital Z so we can show that there exists some integer such that you can read the colon as the word such that 2k plus 1 equals 3 because again that's what it means to be odd right we have to be able to show that there's some integer K so that 2k plus 1 equals whatever number we're claiming to be odd and namely that's when K equals 1 when K equals 1 what 2 times 1 plus 1 well that's 3 so that certainly satisfies that definition of being odd so we've now shown the base case ok so now our step 2 so we're going to assume that so now let's assume that f of X is true for some value of x ok so this is now we're going into the realm of generality well if it's true for some X well we know that X plus its successor which would be X plus 1 know that that's on because we're assuming that to be true so X plus its successor is odd okay so next what we're gonna do notice that if we add one to X and then also to X plus one so I'm going to add one so notice if we add so I'm gonna take X plus one plus X plus one and we're gonna add an extra one in there as well well what's that going to give us that's gonna give us X plus one plus X plus two notice I'm not simplifying it all the way down and this would be our statement f of X plus one this would be our propositional statement f of X plus one because we said our propositional statement we take whatever number that's next to our F that's inside of our F and we take its successor and we add those together so this is going to be the propositional statement f of X plus one and we can certainly claim so our claim here is that since X plus X plus one is alright would we do well we really just added two right we added two we added one plus one so since X plus X plus 1 is odd adding two to this value again gives an odd number this again gives an odd number and that would be proof complete so since X plus X plus 1 is odd well if we add 2 to that it's still going to be odd but when we write X plus 1 plus X plus 2 that's going to be our next propositional statement so assuming that f of X is true we've now justified and true in the sense that it's going to be the statement is odd we've shown that its successor is also I okay so that would be a by induction so that would be our general step here last but not least let's look at proof by contradiction if P then Q the contrapositive of that is if not you then not P and these propositions are equivalent and what that means is that is that if you can prove the implication F P then Q you've also proved the implication if not Q then not P and vice versa so just as I've got written here proof by contradicting the statement around a little bit it sort of gives you a different a different way to think about the problem so again we wanted to show if a and B or consecutive enter integers the sum a plus B is odd so think about that for a second what would the contrapositive of that be well okay so this first if that's our statement P so if a and B are consecutive integers then the sum a plus B is odd so that's our statement Q so now we flip them around and we negate so our Q goes first and we put a knot in front of it we're negating it so if the sum of a plus B is not odd then a and B are not consecutive integers that's gonna be our contrapositive and that's what we're gonna try to we're gonna try to prove this if-then statement okay so this is going to be our contrapositive statement now and i've got not underlined there once let's underline not in there again okay so okay so let's prove this so we're gonna assume so we're going to assume that the sum of a plus B so the son a plus B is not odd that's our assumption so that means there does not exist so there's our little exist statement and we put a line through that through that that means there does not exist an integer K so there does not exist an integer K such that we can write a plus B as 2 K plus 1 this integer K T does not exist ok so because if it did write a plus B would be odd so this this does not exist so well we can again rewrite this I could write 2 K I plus 1 I could write that as k plus k plus 1 so this statement a plus B equals k plus k plus 1 does not hold for any integer k and but notice you know since but since k plus 1 is the successor of k this implies this implies that a and B cannot be consecutive all right so since we can't write a plus B as some integer k plus k plus 1 right if we could do this that would mean that there's that there those numbers are consecutive but since we can't do that so since this statement does not hold for any integer k we've now shown that a plus B are not consecutive integers so we've now proved the contrapositive statement which again so we've shown not Q implies not P and therefore we've also shown P implies Q our original statement so okay so I went through those a little fast because I didn't want this video to drag on too long so for basic little proof techniques again I'll make some more videos where I do each one of those techniques and we'll look at some some hopefully some slightly more complicated and more interesting variations some more proofs but again you know this is kind of some basic proof techniques there's some other you know sort of tricky things you can do as well from time to time but these are definitely the fundamentals so if you're new to proving things I hope this kind of puts you in the right direction again and I shouldn't say here it's not like this stuff is mechanical there's still some subtle reasoning going on in all of these problems for sure so if there's anything where it seems like maybe I went through the little fast definitely go back and look over and take your time and really think about why those statements you know what I said is true or correspondingly not true depending on the situation