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in this video we are going to prove a theorem it's a relatively straightforward theorem it says that if you have two different even integers and you multiply them together then what you get out is going to be an even integer as well but the real point of this video is not this particular theorem but looking at how do we prove things what is their process what mental steps do I go through and then even if I manage to conclude for myself that my theorem is true how do I present it in the right way in it a way that is accepted and convincing and compelling for everyone else to read and follow along my proof now step number one I've got some theorem and notice that it's got a bunch of words that it's got even integer even integer even integer those are the main thing then as well as everything else sort of filler and connective tissue but the main word that we need a definition for is the idea of an even integer and we might be able to come up with an even integer definition relatively quickly but for more complicated theorems focusing very precisely on what it means for each of these words is going to be an almost necessary first step so let's go back and and before we even get to the theorem at all let's investigate the idea of an even integer so my sort of informal definition of an even integer is that it's an even integer if it can be written as twice some other integer so for example like six is twice three or twelve is twice six and so on so if I can write my integer is twice some other integer but I'm going to be even a little bit more mathematically precise with this particular definition I want to use all of the fancy terminology that we'd be developing in this course that has a lot of precision to it and will be very amenable to manipulating as we go on in our proof so I want to come up with a formal definition now notice that in my informal definition what I have is n is an even integer if there's an existence claim snuck in here if there exists some number such that twice that number is what I began with so that's what I want to note is that that in side of this informal definition is an existence claim you are claiming it can be written in some way and I want to express that in my formal definition so here's how I'm going to translate it I'm going to say that n that's my integer under consideration and I'm going to say that n is an even integer that's what I'm trying to claim and note that I'm now going to put the biconditional symbol here and the reason I write this by conditional arrow here is that all of these definitions work two different ways the definition of even is going to be this stuff I'm going to write down and then if you have this stuff I'm going to write down you're going to get even so all definitions are if and only up anyways I had any event and what do I want to claim I want to clean it there exists something there exists some other integer such that your your n can be written as twice this integer that we're describing so I'm going to give a name for it I'm going to say that there exists a P and I'm going to write it like this there exists a P inside of the integer so there's this some other integer and then so that my n is equal to twice that other integer P and that is going to be my formal definition note that you have some leeway here I chose to use the backwards efore there exists in the the symbol here for element of and the integers to have this sort of real shorthand for the larger English phrase there exists a number P in the integers if you prefer to write down the larger English phrase there exists at P in the integers instead of my mathematical shorthand I have no objection to that I just like be efficient on the board alright so back to our theorem so now we know what an even integer is and if we need to go in forward we can always go back and recall that but my next point is that my theorem that I have written down here is written a little bit informally and I want to I want to clean it up a little bit I want to say it in a little bit more of a precise matter so I want to describe what my theorem says formally the first thing I want you to note is that there's sort of a hidden Universal in my informal presentation of this theorem it says an even integer times an even integer is another even integer but implicit in the way I phrased it if the claim every single time I find those pairs of integers that are both even every single time then they're products going to be an integer so hidden inside of this is a for all claim and so I'm going to write that down first I'm going to claim that for all my upside down a for every and notice it's an all claim about two different numbers is every time I have two different numbers that are even so for all M and n they're going to begin adjust generic integers then I'm going to have that if my m and n are even that this is going to lead me to conclude that their product are even as well then the product and times n is also going to be even so the key thing in my formal statement of my theorem here is is to note that it says for every time I take a pair of integers if those pairs of integers that I get are both even then their product is even as well alright so what we've done so far was step one we formally defined the different terms in our theorem and then step two we took our theorem and we formally wrote it with all of our notation little short hands of the for all but but most importantly have have formally written it in this very precise way that is going to be amenable to manipulating and deducing our theorem I also want to note that this this theorem that we stated here is in a very common form that a large swath of theorems are going to be in particular the format of it effectively is for all things in some domain in this particular case the things were pairs of numbers both in the integers but for all things from domain if you have an initial property there even then you get some other property in this case that their product is even as well so this format that we have for setting this theorem is going to be one of the major sort of classes of how theorems are going to be presented of course the domain and the initial predicate and the final predicate all of those are going to change from theorem the theorem but this logical structure is pretty common all right so what's next so let's go back to our theorem and I'm going to leave it actually in its colloquial form it up here I can bring in its formal definition when and if I need it but I like it phrased in this convenient way and remember what we want to do is we want to write down some formal proof of this that is going to convince everybody that yes indeed this theorem is valid however notice that I write this as step four because being able to jump immediately to a crystal clear perfect proof is for most cases a step too far in fact what we want to do is step three which is the most important step them all and that's a step that I like to call playing around and this is where we try to get some scent for ourselves not a formal proof but why do we think this theorem is true because I write down a few examples can I do some algebraic manipulations can I get some sense of why it is that this thing is actually true once I have an idea for myself then I actually can go back and fill it in precisely so it's important that I'm able to do that that's important that I'm able to come up with some sort of intuitive idea all right so let me just choose a couple different numbers here how about four times eight just to sort of get the ball going and then I think about what's going on here we're going to have some particular product in this case is going to be 32 and I notice that four was even an 8 was even in 32 was even so at least I've just prove myself right off the bat but then if I think about the for our definition is an even integer why do I think four is an even integer well it's because it's twice some other number in this case it's twice two and the reason why I think that a tear is an even integer that's twice four so that's the reason why I think that both 4 & 8 are an even integer over they're divisible by two works that can be written in this manner two times this times this so then if I go down to my 32 which is the product of two these two things it's like okay it's twice two multiplied by twice for all right and then if I look at this the way I think about this you know there's like a 2 there and that's what we want it for it to be even right you want to be two times a bunch of other stuff this is like two times a bunch of other stuff in this case two times two times four so I think that that works 32 is an even integer because it's written as twice times blah and the BLA all came about because it sort of broke up by four and eight and I applied their definition when I put them together so this has sort of my intuitive picture here I could have imagined that the four and the eight don't really matter it could be any M and n here and I should get any values out of this and then when you multiply them together I think it should work I'm 100% sure yet but I think it should work that you just sort of get the one two that came maybe from the first one everything else gets lumped together all right so that's my tentative intuitive picture why I think maybe this is going to be true but let's see whether we can formalize this properly with all of our precise definitions and fill out an actual proof all right so here we are step 4 we're going to be trying to prove this claim and I want to note that the proof that we're going to have here is going to fit a relatively standard format in fact we don't really necessarily care about this particular theorem although that's what we're going to use to illustrate the point but the format is going to be quite constant indeed what is the first thing that we're going to want to do is we want to write down whatever our assumptions are going to be that's how we're going to start our proof we say well what do we know for sure in this case we know that we have two different even integers that's what we know and then we want to apply our definition okay you've got two even integers so what what does that mean and then we have that really formal definition of what it meant to have two different even integers and then in the middle this is the part that really changes and varies from prove to proof is we've got to do our plane around this is where we use algebra where we use logical implications where we use facts that we've proven previously or that we've looked up somewhere this is where we do all of our manipulations and we try to take these starting assumptions that have been precisely written with their definitions and keep on massaging them using all these different tools until it looks like our conclusion all right so that's the basic structure that I'm going to be trying to put in place when I do my proofs I also want to note that I can bring over at any point my formal definition or I could bring over at any point my formal statement of my theorems that I have these here if and when I need them alright so first up was to speak what our assumptions are going to be and in this case the skinny assumptions is that we are going to begin with m and n being an even integer so this is my first line of my proof suppose m and n are even integers maybe the one thing I want to note about this line is that that this line that I has you're supposed Emin and or even integers is I'm choosing my M&N completely arbitrarily in fact this is the same thing as saying for all M and n if I just choose any arbitrary pair and prove it for an arbitrary pair I proved it for everything so that's what I'm doing I'm supposing that I got two even integers and I'm giving them some labels M&M now we mixed up we want to figure out how we apply our definition so even integers meant specific things so in this case because of your bullets even it meant that we got two other numbers that I'm going to call these other numbers R and s so I'm going to say there exists an R and an S so that the M is twice R and the N is twice s so this statement here is me applying my definition of an even integer to the two even integers that I have the M and the N next up I want to do some algebraic manipulations we're trying to get to M times n right that's what our goal is we want to show that M times n is an even integer so maybe my first step is just to say well look I've got my M times n let's substitute in the 2r and the 2's that I have for the M in the N and I can write it down this way so I substituted in the values that I have for my definition I related M + n - twice R times twice s next up is a little bit of algebraic manipulation that we saw previously when here just sort of playing around and see what might be true I've got one two inside of here so let's pull that out the front and just leave everything else sort of hanging behind it so I'm writing this as two of - RS this property by the way is referred to as associativity I also move one of the twos around but these are properties that we know to be true from numbers that we can change our brackets around and we can reorder things that's perfectly fine alright so what do we got we got that the M times n is indeed going to be written as twice something but I need to really lock out my little proof here that twice something that something should be given a specific named and noted that it is an integer so that's what I'm going to do next I'm going to say let T equal the two RS and all this stuff over here be given your name and it is an integer and the reason why this is important is that the the T that we have that was our existence claim remember when we talked about our definition of it being an even integer there existed something in the integers so that whatever you had was twice it that's precisely that statement there exists a T so that the product is twice that in the definition of this is that my MN is an even integer and so I write thus MN is an even integer and then I finished off my proof by loaded putting this little squared symbol here stands for for QED we put that at the end of our proves to be like voila we have successfully proven what we set out to prove now in this formal proof which I believe is convincing and now that people who read this they can apply it and agree that that yes indeed the product to even integers is an even integer and this structure that we have has a couple different key components that I want to identify first would you look at the first and last lines well notice if I read them together suppose m and n or an even integer thus M times n is even so in other words what I have by reading the first and the last lines together is the statement of my theorem and that's the way that is supposed to work you start by assuming the the P of X and you get out the Q of X you start by your assumption and you get your conclusion then if you read the beginning and end of your proof together it should be this statement of your theorem all right next up I want to talk about definitions we use this a couple times notice says the first one is right here in the second line and down in the second the last line I'm taking the formal definition of even and I'm applying out of both ends and it makes sense that I'm applying it both end because in my statement I begin with a claim about two even integers and then I deduce a claim about even integers so if I split that up to the start of the I sort of go in one level of applying the definition right at the beginning and right before the end and then the stuff in the middle here this is going to be my manipulation but in this in general could be all sorts different things in this case I little substitution a little bit of a suit if attea a little bit of algebra to manipulate what I had in my assumptions until a form that looked like my conclusions so this is the the general strategy that I use when trying to prove things and maybe we can summarize it like this step one define our terms last we did we defined our even terms step two state our theorem formally so we know exactly what it is that we're trying to say then this is the most important part step three this is by playing around this is me trying to figure out how do I know this theorem is true like what's the key step what am I going to put into my formal proof but what you could do step three even earlier than this maybe you don't want to formally define things want to do your plane around ahead of time that's also okay although I will caution it's often helpful at least when we're starting out here too to go through this process of really formally defining what it is that I'm trying to state that will help inform my playing around but if you want to do some playing around at the beginning and then some precise statements then maybe a little bit more playing around not yourself out that sounds wonderful and then finally we're going to go and do our formal proof where we start with our assumptions we apply some definitions we do our manipulations and our quoting of prior theorems and finally we get to our conclusion