this video I'm gonna share a bunch of tips on how to prove math theorems and met theorems can often be a little bit scary and a little bit challenging for students who are just starting out their journey on proving math theorems indeed it's often the case in first and second year math courses at university the students are expected to show a lot of procedural fluency with computations and so when it comes to transition to be able to prove things it can be a little bit challenging so in this video I really want to give you some tools some ways to scaffold your thinking so that you can at least have some opening moves to get started on whatever it is that you might want to prove and the first portion to be I'm gonna focus more on the logic behind different types of proofs and in the final half of the video I'm gonna talk more about how you come up with the idea is to perceive your mathematical proof all right so step one that I always like to do first is to identify the logical structure of whatever it is that I've tasked with proving the most common one that you're gonna see is going to be conditional probabilities P implies Q whenever you see this symbol with this arrow what it means is that you have some assumptions P and you're trying to prove some conclusion q its if then statement so as an example I could say if I love my wife then I'll do the dishes or something like this it's got an assumption and it's got a conclusion and this basic structure is there regardless of the actual details of the assumption of the details of the conclusion so for example consider this claim if X is an even number then x squared is an even number that is also a mathematical claim that is also in the structure of a conditional statement and often proofs aren't written quite so clearly as an if some assumptions then some conclusions we may have sort of turned the words around but if you can take whatever is given and interpreted in this way where you've got an assumption and a conclusion and say if the assumption then the conclusion and that's to be very useful for coming up with a strategy to actually prove this but now conditional statements are not the only type of statements I have to put up a list of a bunch of different types of possible logical structures that your claim has it might for example this I can be of biconditional my conditions are interesting because they're kind of like two different conditions at the same time you're saying if P then Q and if Q then P if a by conditional you have really two proofs that you have to do so in the one direction and the other sometimes however it's not structured as a statement like for example the claim there are infinitely many prime numbers sometimes you have a P and a Q or a P or Q and sometimes you have structures where what you're really trying to do is show that there exists this is what that backwards a means there exists some value that obeys some properties an example of this might be like there exists and oldest human on planet Earth okay so after you've analyzed the sort of logical structure and try to figure out what you're doing the next step would be to try to figure out well what method are you going to use to prove it there's actually a bunch of different methods and it depends on what this structure of the claim was so go into the most common one the proof methods for a conditional and if P then Q well it turns out there's a bunch of different methods that you can use to prove a conditional like this the most sort of direct one is well uninspiring to call a direct proof it says just assume the Assumption which is P and then you do some sort of manipulations until you have managed to conclude the conclusion q you just basically start the assumptions and move directly towards the conclusions and probably the most common proof structure you'll see but there's others for example there's the contrapositive kind of positive goes the other way around it says let me assume that the conclusion is false that is let me assume not cute this little squiggle here means not so let me assume the conclusion is false and then I'll continue to do some manipulations until I can finally conclude the negation of the Assumption as and that is I want to conclude not P indeed going the contrapositive for the claim that if X is even x squared is even would be that x squared is nullity that implies that X is not even and indeed both of these are logically equivalent it doesn't matter what you do but one of them might be easier for you than the other so both logically the same but you might have a preference for them based on the complications in the manipulation you need to do another method that's common is the method of contradiction and it says I have my Assumption P already but let me also assume that my conclusion is false and if you do some manipulation where you're taking this your initial assumption P and you're assuming the conclusion is false and you get some nonsense like 0 is equal to 1 or X is even and on at the same time or or something is literally independent and dependent at the same time if you get some contradiction like this then it must be your assumption the not Q could not have been the case and so that is another method to be able to prove base and then the final method is actually not what you want to prove something's when you want to prove a conditional is false and let's method a counter example so this is not me proving P implies Q it's me showing that P implies Q is false well for a counter example I just need to give you one example one instance where the assumption is satisfied but the conclusion is not and it's important to know the counter example is really quite fundamentally different from the other three this is because for the direct proof and khun positive and contradiction we're suing them in complete generality but for come for example we can just use one specific concrete example that's sufficient indeed one of the most common errors that I see is a professor who raised people trying to do proofs is where someone does a proof by example they kind of do a correct proof and they show what's true in one example but not show it is true with all examples and so that's very important stink now to be fair you might not know which of these methods you need to begin with until you really understand what proof is asking but we're just showing the stage what the possibilities might be now the reason I really like this is because the way I think about proofs is I start with some assumptions and then I continued doing manipulations until I get to my conclusion and what those different proof methods does is it tells you what you put in for the assumption what you put in for the conclusion if you're doing a direct proof or contrapositive the assumption the conclusion might differ the neck strategy might seem obvious but it's actually super important and for people who are the beginning of their proof journey and their easy one to miss and that's just write down the definitions of every word and your assumptions and in your conclusions this is going to give you your first trick to be able to proceed in your proof so remember about how you might have assumptions and conclusions well write down the definition of the words and the assumptions and write down the definition of the words and the conclusions their mathematical precise definition and then what is in the bill well the middle what I'll call the step three here is just the manipulations all the ways that you can connect the top to the bottom all the ways that you can go from your definition of your assumption down to the definition of your conclusion that's sort of the hard part but by scaffolding this way with assumption inclusion definition of assumption and definition of inclusion it will hopefully sort of make the gap between these where you have to do your manipulations a bit smaller okay so let's see an example in the case where we had if X was even that x squared wasn't then I can write down my assumption of my conclusion so my assumption is but X is even and my conclusion is that well x squared is even so I just write those down in the beginning and the end of my proof the next thing I have to do is what do I mean by even what is the definition of that and even has a very precise definition it means it's twice an integer so for example six is twice three and eight is twice 4 and so forth so I don't write down what my definitions are my definition is that X is 2 P where P is an editor and my definition of my conclusion which is that x squared is even is that x squared is equal to 2 Q where Q is an integer one of the things that happens in this stage is you want want to introduce variable names I mean in both cases of the assumption of the conclusion I'm introducing an integer but I don't wanna repeat myself because they're representing both X or x squared respectively so I use P for the X and Q for the x squared so part of your proof will involve defining some notation like this and this case I'm defining where the P lens is an integer and where the Q Lidz it's also an editor okay so now we have the manipulations and that again is the hard part but I mean if you want to you could but now we have manipulation you wish you can always pause the video and try to see it before I'm going to spoil it for you okay next step I want to try to figure out those manipulations but how do I do this well I really want to have a goal of going towards whatever that conclusion is I know on that the assumption that's my sure to start a spot and I want to hang the direction of the conclusion so what we were just looking at with this X even implies x squared even because the conclusion has something to do with x squared there's really a very natural manipulation to do I have an X I want to get to an x squared so if I'm looking at the conclusion and trying to make what I have look like what I want then that's one thing to do is to square my X and so that's what I do for manipulation I square it and I noticed that this is gonna give me 4p squared and since I want to write that x squared is twice something so I just factor 2 out and write this is 2 times 2p squared and then that's exactly a perform two times something where that something is this 2p squared and I'll just call that Q so to find my Q and that L is my proof this is a complete proof of this particular claim so the point here was that I found a direction for what manipulations I should do by saying I want my assumptions to look like the conclusions which but I had to square write and write it is twice something and that gave me all the ideas that I needed for my manipulations so that was some of this scaffolding these sort of logical structures that I like to think about when coming up my proof and now we actually have to proceed to something we actually have to proceed to those manipulations and maybe even if you aim for manipulations you don't see a path for and in fact everything that you've done previously could almost be done without having any idea about what the statement meant at all which is sort of a bit bizarre but I do want to try to understand the statement first maybe that should be step one but I put it down here because this is when it really transitions to not just following this cookie cutter method that I've outlined but actually using your own intuition your own understanding so I really try to read the statement and try to understand what is it asking what is the Assumption mean what does it mathematical mean what is the conclusion mean what does this connection sort of feel like her look like I try to really have a nice sort of large-scale understanding of what this is actually asking before I try to actually go and prove it the next thing I really like to do is to well it's just basically draw a picture and we can place a bunch of proof it's gonna be this business would be X squares well there wasn't really a nice natural G picture-book nevla linear algebra there's all kinds of beautiful pictures that you want to draw so does your statement that you're trying to prove could you represent it geometrically could you draw a picture that interprets the assumptions or interprets the conclusion and trying to see how those might be connected if you can draw a picture a lot of the time mathematics is just running down the picture that you're able to draw so really encourage you to try to come up with a beautiful geometric picture if you can the next thing I really like to do is to come up with some specific examples I mean I already talked about how you cannot use a concrete example as your proof well it doesn't mean concrete examples are pointless D a concrete example is extremely valuable for giving you intuition so I would put in some numbers I would choose a specific instance and I would try and compute it out in that specific instance and see why might the conclusion be connected to the assumption can I understand that in one concrete example if I can do it once I might be able to then generalize and prove it arbitrarily indeed this sort of connects to the point of drawing a geometric picture maybe you want to draw the geometric picture in a specific example something that you know about indeed any time I'm learning new mathematics field if there's some new definitions I always try to write down as many little examples as I can that explore a specific definition so get the idea of a basis that he's trying to draw as many little examples of the basis till I understand what is a basis one is not a basis I feel that sort of doing that exploring through canonical examples is an extremely important part of doing mathematics the next thing they often look for I'm not getting any ideas is what are the relevant theorems about the kind of mathematical objects that I'm talking about so when I look at these sums when when I look at the conclusions I don't know what are the major theorems that we see so if you're in a course you'll often get these from your textbook it'll be taught about in class these are the major ideas that you need indeed a lot of proofs are basically just write down the definitions see what theorems applies or to connect those together to get to the assumptions and I sort of the core math is just linking one or two theorems to the definitions that you actually have and so I really encourage you to sort of see what's available here and if you want you can even go a little bit further this doesn't always work but sometimes if you read the proofs of the relevant theorems that that can give you ideas for the types of manipulations people do with the objects that are in the theorem that you're talking about and then the final thing I'm going to talk about really just to sort of play around I noticed that when I was younger and I would try to prove something that I got challenging I got stuck on well it would have to feel a little bit discouraging I might not want to continue the obvious first thing I tried it didn't work well I'd really encourage you to treat proving things is a more iterative process where you can keep on playing around trying new examples trying new ideas see if you can connect a new theorem in see if there's some way you can visualize it all these things we've talked about just keep playing around and hopefully you're gonna get some insight is how you can proceed forward and often you find that you go down a weird rabbit hole and you've been doing all of this trying to do a direct proof but perhaps the kind of positive it felt really easy so just because one method is really not proving fruitful doesn't necessarily mean that this theorem is impossible or you're not in any way capable of doing it just mean maybe you had to play around to get that sort of key insight to look at this problem a little bit differently I'll say for myself that I originally started University as a physicist that's what I thought I wanted to go into I want to go into physics but it was through exposure to mathematical proofs and I really just fell in love with the sort of way that two people can communicate through completely logical rigorous arguments and one could convince the other this by the proof I just love that whole idea and it really drew me towards mathematics where I eventually went and became a math professor so I would encourage you to stick with your proving because it really can be very rewarding now you enjoyed this video please do give a like for the usual algorithm if you have any questions about the video leave them down in the comments below and we'll do some more math in the next video