 So we're going to try some practice exercises using truth tables. The first exercise is just to determine whether a proposition is true using a particular row on the truth table. You're given four options, yes, no. But you have to be able to spot the right reason for whether the proposition is true or false. So following our rules we've got Rule 5, we've got two variables in our formula P or Q. So following Rule 5 that's four rows with our following Rule 6 we've got our distribution, every possible combination of truth assignments for the atomic propositions. And we have a formula in here and then following Rule 7 we copy and paste the truth assignments for P and the truth assignments for Q. Again I want to emphasize how easy it is to do this. If you already have a table loaded in a spreadsheet and you simply paste a bunch of these tables as kind of a template, you just copy and paste a bunch of these tables and just use them for your work here. It'll just make your life so much easier. So we've got our truth assignments. Now we just look towards row 2. Look at row 2 and we've got T is true and Q is false. But since it's at least one district is true, this proposition is true. Rule 8, we enclose it in parentheses and we look for our options. We've got two options for yes. At least one of the districts is true. That's one option there at the top and the third one is the consequence is false. Q might be listed second but this isn't a conditional so the Q isn't a consequent there. You've got to be careful. I'm going to list a variety of different reasons that look plausible and some even look true or in some sense of the word. But only one is actually the answer to the question. So for instance there, that second one looking says at least one district is false. Well that's true, at least one district is false. But that doesn't make the complex proposition false. Remember for a disjunction for the truth Q addition for a disjunction just at least one has to be true. So be careful. I'm going to be a little sneaky there with some other reasoning. Okay so we looked at one. Let's try another one. And we're just going to go ahead and zip right to our row. We've got our formula not P and not Q. So following rule 5 we've got our four rows. Following rule 6 we have our distribution of truth conditions. Following row 7 we simply copy and paste the truth assignments under P and then under Q. So we have the exact same ones. Alright so now we keep in mind overall, even those two negations in here, overall this is a conjunction. I recommend doing the negate, well you can not recommend right, you should follow, give the truth assignments for the negations first. So if P is true then the negation of P is false. So we lose that and if P is false the negation of P is true. So it kind of gives the opposite truth value. Same thing for Q. So we just list those, same thing for Q. And then we look at row 4 to determine whether this proposition is true. And sure enough both of these conjuncts is true. Both of these conjuncts are true. So yes the conjunction is true. Now we look over our options. We've got three yeses and one no. All atomic propositions are true. Both conjuncts are true. The consequent is true. Well that consequent is kind of an easy one to leave aside because again we don't have a consequent here since this is a conditional, not a excuse, since this is a conjunction not a conditional. So we can leave that off. Now all atomic propositions are true. That's actually false. Because they're both false. But even if it were true that all atomic propositions, that's not what makes a conjunction true. What makes a conjunction true is that both conjuncts are true. And that's the second option there. Both conjuncts are true. Let's take another look at another one. Now we've got three variables. P, Q, and R. So following our rule 5 we've got eight rows. And again just to kind of spell out the application of rule 6. The first four rows of P are true. The last four are false. Then we just look at Q and we're just dealing with that first half there. We've got half of those are true and half of those are false. So copy and paste that all the way down. And then R just gets true false. And we just copy and paste that all the way down. Now we put our formula in. Okay so it's a bit more complex this time of this formula. We've got a parenthesis. We've got a parenthesis so keep that in mind. So the next step following rule 7, we copy and paste our truth assignments for the atomic propositions. And then we take a look at the complex formula. Now you might be tempted just to go ahead and shoot right on over to R because it's simple. And in this case it would be harmless if you did that. But I recommend just get yourself into always taking care of the innermost parenthesis first. So we only got one set of parenthesis here. But if we had two, take care of the innermost parenthesis first. So for P, that's just kind of a simple matter of giving the opposite truth value there. If P is true then the negation of P is false. Same thing for Q. We get the opposite truth value. And now we find the truth value for the complex proposition. So here's just a quick little shortcut. Don't bother checking both at the same time because you drive yourself nuts. So the truth condition for a disjunction remember is that at least one disjunct is true. So just go on over and look on the Q side because there's a bunch. Or if you look on the P side you can start whichever one. But we've got our truth just hanging right over there. That true disjunction just hanging out right there. So it's just easy just to slap those T's in. And then you do the same thing over for P. And you notice there's some overlap. But you quickly kind of get those truths in real fast. And now we have our false disjuncts here. And so we find out that there are two rows with a false disjunct. And that's kind of easy breezy simple to do. Now we take care of our truth assignment for the negations. And so here's a little trick. So what I did is since it's a negation for one that just completely alternates like that. I copied the first two rows there. And just moved it out and pasted it. And just pasted again and again. And it has that nice alternating pattern really fast. It just kind of works out. And then you manually fill in the other two rows. Okay. Now we look on our row two. And we see that at least one disjunct is for overall for this disjunction. Because overall this is a disjunction. At least one disjunct, the negation of that is true. Even though the disjunct on the left hand side of the disjunction of the negations of p and q. But that's a mouthful. Even though that's false. At least one disjunct is true for our disjunction here. And so that that gets our truth assignment for t. And so we look we know that it's true. And so we look for our yeses as a yes. Both conjunct are true. And yes at least one disjunct is true. Well that both conjuncts don't get tripped up there. Conjuncts are for conjuncts. But this is a disjunction. This is a disjunction. So you know it's at least one disjunct is true. That's our conditions for whether a disjunction is true. And you look at that no, that no, you know I have that no there. At least one disjunct is false. And it's true that at least one disjunct is false. But I'm trying to mess with you. It doesn't matter if a disjunct is false. At least one disjunct is false. A disjunction is false only if both disjuncts are false. Okay. But that's not the case here. We have at least one disjunct is true. Similar comments apply with you know the antecedent is false. It's you know the first proposition you know the first complex proposition there. True it's first and true it's false. But it's not an antecedent. Antecedents are for conditionals not disjunctions. So be careful when you're looking for these options. Now there might be a little late to mention this. But there is a shortcut way to do this. You could do the long way because I wanted to take you step by step for you know for filling out truth tables. And I think it's especially at this beginning I think it's a really good idea just to take a step by step. And by the way don't delete any of your truth tables. Just keep them. You did all that work. Don't throw them away. Just keep them. But there is actually a shortcut to this. We know our truth assignments for our common propositions because it's given to us by the row. So you can't just simply put that one row in there for these options. I've given you. You've seen them in the text. We've gone over them several times. You know the truth assignments just for that row. So following rule seven we put our truth assignments for the atomic propositions in the formula. Then we apply the truth conditions for negation within the parentheses. Then we determine the truth value of that complex proposition you know that one side of the you know the disjunction that's the left hand at this chunk right. The disjunction within the disjunction. Now we give the truth value of the negation on the right hand side and now we figure out the overall truth value for the proposition and we look and find our option just like before right just like before. So that is a shortcut. I don't recommend doing it at the beginning because you're going to need the practice of filling out truth tables. But there might come a time when you want to use shortcuts. It might be an advantage to it. So here's another kind of question. Sorry to cut off the instructions there but what you're supposed to do is to determine which rows are false for this proposition. So you're going to need to construct the entire truth table this way alright. Notice there's that option none. It's possible that none of these for the truth that sounds that none of the rows are false. It's possible but you know you need to check and make sure. So following rule five we got our P and Q that gives us four rows. We following rule six we have our truth assignments. We put our formula into the truth table. Following rule seven we put our truth assignment you know just copy and paste the truth assignments for the time of propositions in there. Okay. Now once again use you know solve the you know find the excuse me find the truth value for the complex proposition in the innermost parentheses first. You know work from the innermost outwards. Okay innermost outwards. So we've got a conditional here and remember the only way a conditional is false is when the antecedent is true and the consequent is false. So for this one that's only row two and the rest of the rows are true for this conditional within the parentheses. So we've got true, false, true, true. Okay now we move to the outermost parentheses and again we have it's a conditional the outermost parentheses outside the parentheses because overall this is a conditional. So we know that a conditional is false only if the antecedent is true and the consequent is false. So you don't even need to look at rows three and four. We can just skip those just like a rows one and two. So those are because those are both rows where the antecedent is true. So looking at row one we've got a true antecedent and a true consequent and let's not row one because we're looking for the false ones. So but row two we have a true antecedent and a true consequent. So row two is false and we mark that one and only that one for our problem here. Now keep in mind with some of these it might be more than one and it's set to take, you can mark all of them if you want to but you're going to get the problem wrong. And if you're marking none and one of the number then something went wrong. You made a mistake there. It can be more than one of these rows and the computer is set to accept more than one response for all of them, for all of them. So don't think that you just click and see, do computer tricks that way it's not going to work. But some cases it's going to be none some cases it's only going to be one row, some cases it's going to be multiple rows. So just be alert with that. Okay so let's try another one. Alright so we've got our formula here and again sorry it's cut off, the instructions are cut off but it's going to be a long treat table. It's the same sort of instructions except now we have three variables. So we've got rows one through eight and possibly none. So following rule five we have eight rows because we've got three variables two raised to the power of three that's eight rows and we put in our truth assignments for P, Q and R. Hopefully this should be real familiar by now if you're still lost with this you better go back and review that part in the video. Now we put our formula into our truth table and now following rule seven we copy and paste those truth assignments for P, Q and R into the truth table just like so. Now we've got two sets of parentheses here work from the innermost parentheses outward. By the way overall this is a negation. Okay so innermost parentheses and again just do yourself a favor. You've got alternating T's and F's here. You're going to drive yourself nuts if you try to look at both to determine. Just take with the Q first because we know the truth conditions for disjunction. We know that at least disjunction is true if at least one disjunct is true. So just go in there and look for the T's. They're grouped. Rows one and two, rows five and six just hit T for the disjunction there and then for R you just look over and you find the T's there and the rest are going to be false. So it's pretty simple little setup. Now we have the conjunction. So the innermost parentheses is a disjunction. Working outwards for that parentheses is a conjunction and one conjunct is P and the other conjunct is the disjunction Q or R. Now conjunctions also can be there's kind of a fast way to do this too especially with leftmost atomic proposition because you notice five, six, seven, eight P is assigned false. Well a conjunction is false when at least one conjunct is false. Well that makes that part faster. Six, seven, eight, that's false real fast. Now we just need to take just look at the disjunction and we got one false there for the disjunction. We will put that in and that means that the rest are true. That means the rest are true because all the P's are true and the disjuncts are true right there for rows one, two and three. Now overall this complex proposition is a negation is a negation and we're looking for the false rows. So for that we need to look for those rows where the conjunction is true. Well that's just the first three and that's just the first three. So rows one, two, three are false and we go into the computer and we mark one, two and three as our false rows. Okay so those are the kinds of problems that we got for this time around. I suggest keeping up our practice with your truth tables and getting real good at filling them out. It can be, it's not complicated if all the rules are not complicated. You can have a long truth table real fast. You have a really big truth table. But if you follow these rules just one at a time you can tick off what you need to do. Just get it done. Okay if you have any questions make sure you send me an email. You can come by my office. No problem with that. Good luck on these practice examples and keep thinking.