in this first video of the discrete mathematics playlist we are going to look at propositions what they are and then look at negations conjunctions and disjunctions and we'll also take a look at the truth tables that go along with those connectives the connect toasts are the negations conjunctions and disjunctions so our first new terminology here is a proposition and a proposition is just a declarative statement that is either true or false so notice I have five statements written down three are in pink two are in green the statements written in pink are all considered propositions because they are declarative statements that are either true or false the sky is blue is a declarative statement generally what we will do is we will take this statement and we will say let's let P represent that statement or the moon is made of cheese also a declarative statement again that could be either true or false so I'm going to say false on that one and I'm going to say that is proposition Q the moon is made of cheats again it is considered a proposition because it is an eclair ative statement that is either true or false same thing with Luke I am your father this is a declarative statement I am your father either true or false I'm gonna say that one's false and we'll say that's R now take a look at the difference of the statements that I've written in green D says sit down sit down is a statement but it is neither true nor false it I can't say true you have sat down that statement would have to be you sat down that's either true or false but sit down is just telling you what to do and that is not a proposition a lot of people struggle with E because they say well that could be true or false and while you're thinking is correct the way that it is written right now is not a proposition if I replaced X with some value or if I said where x equals 5 well now it's a proposition because I can say 5 plus 1 equals 2 is false or if I said where X equals 1 then 1 plus 1 equals 2 is true but if I don't assign a value for X and I just leave it as X then this is not a proposition because it's not true or false so propositions themselves are fairly straightforward again we're using a lower case letter PQRS etc to represent a proposition and then what we're going to do is we're going to end up making a compound proposition using these connectives so I want you to think of these connectives like operators as I would take 1 plus 2 that's an operator so these are just operators for propositions and I've put them all on one page together just to have one page that you could refer back to they're all called connectives so we're going to go each of through each of these in detail in the following slides but I did want to put them all here so let's talk about them very quickly we have the negation and again this is how we would use that negation so I would say not P and again I would say it as not but that is the symbol that I would use conjunction is and and again we'll talk about each of these in detail so obviously I would be taking P and Q a disjunction is an or so that would be P or Q an implication is an if-then so this is an if P then Q and then a by conditional notice it has an arrow on each end here says if and only if so P if and only if Q and that means they both have to share the same truth value so again we're using p q RS etc using those lowercase letters to represent each proposition so let's get into each of these connectives in further detail the first connective is a negation and again the negation is not and this is the symbol that we would use so the negation of the proposition P is not P so example if I say P denotes the grass is green then not P denotes it's not the case that the grass is green now it's silly to write it that way so we don't instead we say the grass is not green so let's take a look at these few examples I have here and then I want to look at some truth tables with you to make sure that this all makes sense so the first one says my dog is the cutest dog which is a true proposition by the way so my dog is the cutest dog is my proposition P if I want to write not P then it would be my dog is not the cutest dog so that is not P again we could say it is not the case that my dog is the cutest dog but it's easier to write it the way we would normally say it in the English language the door is not open is P now again if the door is not open is P then if I'm negating something that already seems like it's negated remember in mathematics is the only place that two wrongs do make a right so the door is not not open which means the door is open again we could say it is not the case that the door is not open but in real life we would just say the door is open last one are we there yet be careful with this because this is not a proposition it's not a declarative statement that is either true or false and because of that I can't negate it so let's talk now about truth tables and how a truth table works so if we have a truth essentially what we have is a row for each possibility of the truth values of our propositions so I want you to think of this as having two parts so this is a very very basic truth table that we're going to do but the left side of our truth table has all of the combinations of the truth values for our propositions so in this case if I just have one proposition P that P can either be true or it can be false so let's say P again represented my dog is the cutest dog then not P represents my dog is not the cutest dog so how does the truth table work well again on the left side we're going to give all of the combinations which is very easy for one proposition and on the right side we're going to use whatever our connectives are and so we might have several columns on each side in this case we just have one column on each side but we might have several depending on how complicated our truth values or our truth table is going to get in this case let's look at our proposition P and here's how a true table works let's say P is true so here I am P is true my dog is the cutest dog is a true statement which means not P my dog is not the cutest dog would have to be false because my dog can't be the cutest dog and not the cutest dog at the same time let's say instead that my proposition was false I said my dog is the cutest dog and that is an incorrect statement then not P which says my dog is not the cutest dog would have to be a true statement so that's how a truth table works is on the left side we have all of the different combinations on the right side we have whatever connectives are going to use truth tables were going to be very important to us so that's why I wanted to introduce them to you in this video so that you had a good foundation when we get to our next video where we're going to use them in more detail before I continue on to our next proposition I just want to remind you that in this case I only gave you one proposition we're going to have several in fact our very next example is a proposition where I have to excuse me a connective that requires us to use two propositions so if I have two propositions I'm going to have 2 to the N rows which is 2 squared rows or 4 rows and really this is all about just how many combinations are there let's say I'm using P and Q and it doesn't matter what's the connective is in between there so we're going to do P and Q we're going to do P or Q and we're going to do these in just a little bit but just so we understand the left side would have to have all of the values for P and all of the values for Q all of the combinations of those two so here's the best way to go about this if I've got P and Q here then P could be true and Q could be true P could be true and Q could be false then it's also a possibility that P is false and Q is true or P is false and Q is false so notice on the left side which has two columns these are my combinations now do your professor a favor and write them in this way I have a lot of students who do true true and then false true and then false false true false and it gets very hard for me to check your work when you're all willy-nilly like that so do me a favor keep these groups together and then these will alternate and we'll continue to work on that together our next connective is called a conjunction and a conjunction of propositions P and Q is denoted P with the little arrow head basically Q and it's read P and Q so a conjunction is an and and one way to help you remember this is this kind of looks like a capital A if you added that little marking in the middle now the reason I bring this up is because our next one is going to be the arrow pointing down instead of pointing up so it's good to be able to keep them straight for a conjunction to be true both propositions must be true so we're going to create the truth table together but it's also important to think about the fact that p and q represent statements so let's say p is it is raining q is i am home so if i'm creating my truth table remember there are two sides to this and on the left side is where we just give all of the different possible truth table combinations and since there are true two propositions that means it's going to be 2 squared or 4 different rows so I'm going to this is not going to be a row this is just going to be where I put my values P and Q so I need 4 rows so 1 2 3 4 rows and of course a column for each each of my propositions so I've got P Q and I'm going to list all of the combinations so P could be true with Q true or true false or false true or false false so that's all of the combinations on the right side is always going to be whatever it is that you are doing a connective of so in this case we're only doing P and Q or the conjunction of P and Q and that's the only thing I will need on the right side of my truth table now before we start filling it in let's think about what this means P says it's raining Q says I'm home for this conjunction to be true both propositions must be true so it must be raining and I must be home so this says P is true Q is true it is raining and I am home the only way for P and Q to be true is for both P and Q to be true and that is the case here true true so P and Q is true for the rest of the rows this row represents it is raining I am NOT home so that's false because they're not both true because I'm not home this row represents it is not raining I am home and I am home so it's not raining and I am home of course would still be false because it's not raining and this row represents it's not raining I am NOT home again false because they both have to be true for it to be true so it doesn't matter if you put these propositions of meaning to the propositions but keep in mind that that's what you'll be doing quite often is you'll be using those propositions using those statements to write them as letters and then go from there to make a truth table another connective is the disjunction the disjunction of propositions P and Q is denoted P with the opposite facing arrow so basically a V and read P or Q and for a disjunction to be true I either proposition must be true so when we were talking about a conjunction both had to be true for the conjunction to be true for a disjunction either proposition must be true so again when I create my truth table I'm always going to start on that left side where I'm giving all of the different combinations and on the left side again I've got true true true false false true false false so that's just the combination side I haven't done anything yet on the right side is where I'm going to write whatever my results are in this case the connective is the disjunction so I'm going to write P or Q and again I think of a disjunction or the oar as being a cup so if you can put anything in the cup if either one is true then the result is true so if P is true and Q is true then P or Q is true because either one of them is true if P is true but Q is false P or Q is still true because P was true only one of them needs to be true they can both be true but only one of them needs to be true if P is false but Q is true P or Q is still true because Q is true so the only false value I'm going to have is where both P and Q are false because neither one is true and for the disjunction to be true either proposition has to be true so that's my solution the ORS can get a little bit tricky most of the time in mathematics we're using that inclusive or that we just talked about that's the P or Q and that is saying for instance the prerequisite for ma 420 is either ma 315 or ma 335 and by that I mean you could have passed ma 315 you could have passed ma 335 or you could have passed both of them and you can still get into ma 420 that's the inclusive or that's the one we use most often there's also the connective or in English which is called xor and that is something like you get soup or salad with your entree now if I'm buying an entree that means I can get su or I can get salad but I can't get both I mean I can I can just pay extra but I can get one or the other but I cannot get both so while we just finished talking about the inclusive or where we said if either is true then it's true and that was true true true false the exclusive-or is only true if one is true or the other is true but not both so the difference here is because this one had two truths then the exclusive order is going to return a false value because I can't have both soup and salad I can have soup or I can have salad and then of course false false is not going is still waiting to be false so hopefully you can understand the difference between those now notice the different notation I'm going to use now this is the XOR notation which is basically just a circle with a plus sign in it up next we're going to continue our study of the connectives by setting implications and by conditionals and we're of course also going to look at the converse inverse and contrapositive of the implications I hope you can join me