Finite Mathematics - Introduction to Logic

everyone so today we're gonna be talking about logic which is a start of chapter six in your textbook I believe so there's already a handout that's in d2l and that should have been emailed to you by the time that you see this video and this video is meant to be a supplement to that handout now logic is the study of statements and really at the very central heart of mathematics this is what we study in math most important think that mathematics is the study of numbers or functions or operation strings of numbers but that's really not as a mathematician what we do at all what we really do is study statements and very often of course these statements in full numbers or operations between numbers but they're not the central focus that's the statements are the focus so a statement there is a sentence that is either true or false and so some examples of statements would be something like 2 is greater than 1 3 plus 2 is equal to 6 and x squared is bigger than 5 so here are some examples of statements that use numbers that we might be used to so some of these things to greater than 1 for example is a true statement 3 plus 2 equals 6 as a folk statement and x squared is greater than 5 is either a statement that's either true or false depending on what X is but the point is that for any value X and would either be true or be able to be false that it couldn't possibly be both or neither in any given circumstance so a statement is any sort of sentence that is either true or false and like I mentioned we don't have to limit ourselves to just statements about numbers buildings is a city in Montana it's certainly a statement and it is certainly takes on the value of either true or false now what we're going to do in this section of course is we're going to be studying statements operations on statements and compositions of statements which are very similar to composition and operations on numbers so let's say as an analogue here we have numbers and here we have statements so in the typical math class you might have something like that deals with numbers like X plus y and what is this equal to well it depends on the value of x on the value of y right so X is a number Y is a number X plus y is also a number so what we mean by an operation statement is we'd have a statement P and then sum operation and then some number Q and that's gonna have some value as well but instead of being numbers P and Q and this thing are going to take on the value true or false right the numbers here could have infinitely many values as infinitely many values that could be here but depending on our operations here P could either be true or false Q can either be true or false and the resulting thing is then either true or false all right so let's suppose P is the statement I will eat breakfast today the first example of an operation is the not operation which we describe as not P so not P is the statement that is true whenever P is false and false whenever P is true and so an example here at P is the statement I would eat breakfast today then not P should naturally be what what do you think it is oh you've been paying attention to everything I've been saying here right and so my stepson is behind the camera here so right so these are sentence that since you ate breakfast today so what's not that what is not P being done piece so it's not P yeah so what is that that appears eating breakfast today then what is P not happening exactly I will not eat breakfast today okay you're just looking for a statement that's true whenever P is false and false whenever P is true so P is a statements that's true when you eat breakfast then the negation or the not of that statement as a statement that's exactly the opposite of that so not eating breakfast today if you eat breakfast then P is true and not P is false if you don't eat breakfast then P is false but not P is true so that's exactly the circumstances that determined that and what we would describe as a sort of table to sort of others drink that is something called a truth table so what we would say here is for both values of P that is either true or false the not peace values have to be false or true all right I'm gonna add a second operation - this whole mess our second statement sorry we're going to add one more or two more operations BQ I'll add to this is I will eat lunch today and now the second operation additive is the operation and and there's a statement we express and as a with this carrot symbol here always say P and Q is a statement that is only true if both P and Q are true alright so Byron yeah if P is our eat breakfast today and I and Q is our eat lunch today then what would P and Q so just right so so it's just a statement right yeah you don't need to include the description as part of the statement so it really is I will eat breakfast and our eat lunch today there's a lot of ways you can phrase the same statement in the same way that in algebra there's a lot of ways you can express the same thing right so like 2 times X plus 3 is 2 X plus 6 for example and so we're not when we make these compound sentences were not so concerned about the exact wording as we are as just expressed in a statement that is true only F P and Q are true so let's say you do say the sentence you say I will eat my dress and I'll eat lunch today well that's only true if you eat both breakfast and lunch if you decide to skip one of these then you're not telling the truth anymore if you skip both you're definitely not telling the truth anymore so the truth table for something like this it's a little more complex because there are two bodies for p and two values for Q and so there's four combinations in total they are both true P is true and Q is false P is false and Q is true they're both false and in this case P and Q it's true when P and Q are both true and false otherwise and so in the example of our lunch scenario or a meal scenario or it says alright if you make this statement a your breakfast an hour eat lunch if you end up eating both you told the truth any other circumstances means that you lied all right so that statement is true if you do both both otherwise then the sort of analog and I'm going to keep this because we're studies that is our example but is or or is a lot more forgiving than and right or p4r with this be like wedge here and then we say P or R there's a statement that is true if sorry P or Q if either P or Q or both are true so P is true then it's true Q is true then the true and they're both true it's definitely true so again a P is out eat breakfast today and Q is out eat lunch today then P or Q what would be a good statement for the P or Q I would eat their breakfast or lunch today good I will eat either breakfast or lunch today and so let's write out the truth table for that so P Q P for Q we have our same or combinations so I'll eat either breakfast or lunch today if you ate them both is that true true if you ate breakfast and skipped lunch sure is that true yeah if you skip breakfast but ate lunch true if you skip breakfast and lunch false false all right so given the statements P and Q we can compose them with and we can compose them with ORS and we can also apply nots to either statement so let me test Byron here little with we'll start with a statement here maybe I will skip breakfast and have lunch all right what is what would using P and Q and the operations we had described so far today what would make up this sentence here we would definitely be using and of course okay in there P would be I will skip breakfast well we're stuck with these piecing things yeah so instead of our skip breakfast what could we use so we changing P that we express skipping breakfast in terms of P changing like the wording of that yes I was about to say I will not eat all right Soames right he is eating breakfast still not P a skipping breakfast right and you're gonna have lunch so we'll read course so it does seem like just Q is fine yeah right so appears having breakfast Q's hugging lunch not having breakfast and having lunch it's not P not breakfast ad Q lunch so let's see when this thing is true or false as you become more comfortable with these logical propositions you don't have to write all these intermediary steps but it helps for now but we're still getting used to this so here we have p and q and are therefore combinations again I not P is false when P is true all right and it's true and P is false now false and true is folks right that and statement is only true when both things are true so false and false is both true and true is true and false or true and false is false so here we have a statement that's only true in this role and false in every other role now let's see if that makes sense just intuitively if you say I will skip breakfast and have lunch you should definitely skip breakfast you should definitely add lunch if you don't have lunch your folks if you eat breakfast you're false because you said you would skip breakfast so the only circumstance under which this statement could be true is exactly when you've skipped breakfast and had lunch like you said you would all right so when we worked through the logic and when we look through this example they should coincide they should both make sense now at a certain point we will be working with just these statements independent of some representation in the same way in algebra we can work with variables and numbers without having some concrete thing attached to them but if it helps you get used to it whenever you see some collection of statements it is certainly helpful to make up values or pnq make up statements for these things and then see if what you've produced makes sense in relation to that okay then we will do one last sort of example for this video and that is we're just going to introduce a third variable R certainly we could say that our is I wouldn't dinner or something like that but again at this point we should be moving past the need to attach a concrete value to our variables so that's like at what is the truth table for not P or Q and R so P could be two things Q could be two things our new thing r could be two things so that's eight possible combinations after time P is true half the time PS bows half the time q is true at the time Q is false half the time R is true half the time R is false and you follow this pattern down we see we have every possible combination of true and false between P Q and R so start not P is going to be false and then true okay so false false false false true true true true all right Q and R let's take a look at that well Q and R is only true in both q and r true so true here false false false true here false false false alright so then we write false in for all of these so not P or Q and R we will look at not P we will look at Q and R and it's true whenever either of them are true so because this one's true true they're both false so folks they're both false so folks they're both false so folks they're both truths are true this one's true this one's true this one's true so this would be the truth table for not P or Q and R so in the context of our meal examples we could say that this is I'm going to better make trim stone the range okay I think I'm still in there I'm either going to skip breakfast or I'm gonna have lunch and dinner right so you have lunch and dinner that's true right so if you have lunch and dinner that's true and if you skip breakfast that's also true right but let's say you eat breakfast and you eat lunch but you don't have dinner oh you ate breakfast so that's not true right you ate lunch but you didn't have dinner so that's not true either right these are both false you didn't skip breakfast you didn't have both lunch and dinner so your statement ultimately was false all right so I hope this was instructive informative and that this will help you with your studies again you have any questions I'm happy to address them in video form if you want or you can always send me emails in the usual way and I'll send that back to you Biron could you kill the recording yep thank you very much

Channel: Tien Chih

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Id: XHRRZId8Dm4

Length: 24min 22sec (1462 seconds)

Published: Tue Feb 27 2018