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okay in this video I want to start talking about an introduction to some basic math proof techniques and whether you're just simply interested in mathematics as a hobby maybe or thinking about being a math major maybe you're taking classes like computer science or linear algebra and you're kind of finally encountering some some math some questions where you have to actually prove things mathematically this is just going to be a quick little overview it's by no means comprehensive I mean there's there's entire books dedicated to this stuff so I'm certainly just gonna hit the highlights over the course of the next few videos so this one's gonna be just you know unfortunately me reading a bit I'm not going to go through a lot of stuff I'm just going to talk about some terminology here at the beginning so well we're going to talk about basically in this video we're going to talk about what are known as statements and logical operations and then we're also going to briefly discuss this idea of quantifiers in the next video I'm going to look at sort of four fundamental proof techniques there's what's known as a direct proof also called a proof by construction there's proof by contradiction there's proof by induction and proof by controversy in a proof by induction even like in a pre-algebra class or a calculus class of free probably teaching probably not pre algebra but precalculus you may have seen some some induction I've definitely got videos on proof by induction that we're geared towards high school students so in the next video I'm gonna prove a very simple statement but I'm gonna prove the same statement using all four different techniques just to give you a flavor and then in some other videos I'll do some variations I'll do some you know direct proof proof by contradiction induction contrapositive just some different random examples again they're not necessarily going to be the most complicated problems but again just to give you a feel and a quick little introduction so so let's talk about statements and logical operations so a mathematical statement as it is a statement that's either true or false but not both so let's look at these next five statements and let's consider which of these are considered mathematical statements so six is an even integer well that has a truth value and it well it's in fact it's true so we would say yes this is a mathematical statement ten is an odd integer well again that has a truth value and if it turns out right it's false ten is action even integer but that is still considered a mathematical statement because it is it's a statement that's either true or false well it happens to be happens to be false and again so just to highlight three and four they don't have to necessarily deal with mathy things it's just again a statement that has a truth value to it so Austin is the capital of Texas that is considered a mathematical statement because well it is true Austin is the capital of Texas number four Austin is the best city in the world well we wouldn't consider that one a mathematical statement because well it doesn't really have a truth value right you think you could argue about that all day long personally Austin is certainly one of my favorite cities in the world and let's see another statement two plus two equals five hey-ya again that is a mathematical statement because it has a truth value again it turns out to be false but it would still be considered a mathematical statement so a lot of times you will often see the seems like you always see P's and Q's it's kind of like X&Y when you do algebra you see P's and Q's to denote statements so for example the lowercase letter P could denote the statement that two plus two equals five you could summarize that you could say that's the statement P and for example ten is an odd integer you could call that the statement Q so those those letters that you see P's and Q's we're going to be talking about statements so just like these mathematical operations you can add subtract multiply divide there are also what are known as logical operations and those are ways to combine or modify statements so we'll talk about what are known as and statements or statements not statements and if then statements we're also going to talk about a couple of these two I think I also mentioned here we're going to talk about if and only if statements and contrapositive but those are really just sort of special cases using some of these other logical operations so all of these two have an Associated truth value so again if you've done truth tables or maybe you've seen taking a logic class you certainly will encounter this type of stuff and in other settings as well besides a math class again I'm thinking specifically about sale logic class or something when you're doing computer science maybe you're making a truth tables or something like that okay so let's talk about not statements first so if P is a statement then not P is defined to be so again we're going to talk about truth value not P is gonna be true if the original statement P is false so not P is true P is false and vice versa not P will be false when P is true so the statement not P is often called the negation of P and it's denoted so I've seen a couple of different notations there's almost like this little uh this little bar with the arm on it that would be read not P I've even seen like a little a little twist or a little tilde so not P but the one that I've most commonly seen is the one with a little the little bar here and that's the one that I'm going to use so a couple other here a couple others here real quick we've got and statements so P and Q are two statements than the statement P and Q is defined to be true when both of the original statements P and Q are true and it's false if either statement P or Q is false or if they're both false and again there's different notation so a lot of times the notation you'll see is the little it looks like a little to it the way I kind of remember this notation so P and Q this little this little P R it almost looks like we added the the you know the extra bar it would look like the letter A so P and Q right and starts with the letter A so that's how I remember it for statements so if P and Q are two statements and the statement P R Q is defined to be it's true when there P is true or q is true or both P and Q are true and it's false only when both P and Q are false so a little just remark here when I say don't confuse this with the typical usage for the word or in the English language because a lot of times when you when we use the word or in English we kind of assumed that maybe one statements true but the other ones not and here the or statement can be true if they're both true so for example if I said you know Patrick got an A on the math test or Sally got an A on the math test most people would think of that as saying well one of either Patrick or Sally got an A on the math test but but not in both of them but when we talk about it in a mathematical setting if I said Patrick got an A on the math test or Sally got an A on the math test it could be true that both people got an A on the math test so that's just one thing to remember when you think about truth values of these statements just a little distinction with distinction between how we use it in everyday English I think so just be careful about that other very common very common oh and I should say one last little thing here so the little symbol that we use for the word or we just flip right we have the upper carrot or whatever you want to call it for a we flip that over and that would be read the statement P or Q okay if then statements so again a P and Q statements then the statement if P then Q is defined to be it's true when both P and Q are both true or if the initial statement P is false and it's false when P is true and Q is false so it's an it's an implication and we denote it we put a little a little arrow if P then Q and just a little more terminology if the statement P is false we say that P implies Q is vacuously true so an example of a vacuously true statement again it's when you're starting with a false statement you could say if the moon is made of cheese then Patrick is the president of you know it states that would be a true statement of vacuously true statement there's kind of most no substance to it right the original statement is nonsense I guess is kind of how I think about it another remark we can talk about the converse of two statements so the converse of P implies Q or if P then Q we just flip them around is Q implies P so notice even though if the implication P implies Q is true it's it's not necessarily true that Q implies P you can't just flip them around and say oh the original one's true and so is the other one so just kind of one example I was thinking off the top of my head so if Patrick loves all teams in basketball then Patrick loves the Boston Celtics right okay I think most of you probably are watching this in USA the Boston Celtics right or a basketball team so if Patrick likes all basketball teams then Patrick likes the Boston Celtics okay that it would be assumed that's true the flip of that necessarily is not true right if Patrick likes the Boston Celtics then Patrick likes all basketball teams well may be the only team I like is the Boston Celtics it could be true but it's a you can't just conflate the truth value of the team all right a couple more things here again I know we're having fun this is just kind of the basics just because we're gonna be using this stuff and some of the other videos so okay if an only if statements these are basically implications that point in both ways so if P and Q are statements from the statement P if and only if Q is defined to be it's true when both P and Q are both true or both false so this compound statement P if and only if Q that has a true value if both statements are true or both are false and this compound statement is is false when one appear Q is true and the other one is false so basically when they don't have the same truth values P and Q then this if and only if statement is considered false so if this if and only if statement P if and only if Q is true we say that P and Q or what are known as equivalent equivalent in the sense that their truth values are the same so another very important proof technique that we'll use and we saw that at the beginning is going to be proof by contributions and some implications so I know I've done it some other videos we've looked at you can you can prove how statements have the same truth value and I can certainly direct you to one of those if you want to see that but the contrapositive of a statement P implies Q or if P then Q is we say not Q implies not P or if not Q then not P what's important is that these two statements are logically equivalent which means if the first statement is true so is the other one and vice versa so if you can show that P implies Q you have also proved that not Q implies not P and this is going to be important because a lot of times what you're going to do is you maybe you want to show that if P then something else follows a lot of times it's easier to flip that statement around and show the contrapositive instead and we'll see some examples of that as well so the last thing really briefly here just terminology I know this is probably not the most exciting thing but again just some background because we'll be using this so quantifiers so suppose we consider this following sentence X is even this is not a statement because it doesn't have a truth value we don't know what X is so we can't claim anything about it being even or not so this is not a mathematical statement but we could modify this to make a statement we could basically say something like oh when X is 10 X is even this is now a mathematical statement or we could say something for like for every integer X X is even well okay so maybe that's not quite true but still a mathematical statement so lastly we could say there exists X such that X is even that's another way to to make the these these are all in mathematical statements so when we talk about these phrases for every or there exists those are what are known as quantifiers and you'll often see the notation you'll see like a little upside-down a that denotes for all then you'll see a little backward e that means that there exists so just shorthand for for all and there exists so I may use some of that at some other videos okay so that's it for the boring terminology stay tuned for some actual proof videos again the the examples I'm going to do are going to be very simple again just to give you a flavor on how to use these different those for proof techniques that we talked about at the very beginning those four proof techniques again direct proof contradiction induction contrapositive and then we'll also go on to do some that are slightly more complicated and hopefully more interesting to you