All right, part three of my truth tables tutorials. I guess this is like a series now. Way back in 2016, I made this video on truth tables. People asked a bunch of questions about it.
I just ignored them because I was lazy. Sorry about that. And now finally, four years later, I'm getting around to answering some of those questions.
So what I'm doing is I'm answering those in two different videos. I made a part two where I answered a lot of those questions. Namely, what I did here is I introduced different notation. for answering the same questions that were asked in the part one video.
So for example, P and Q can be written this way. Who knew, right? Maybe if you saw a question asked this way, you wouldn't know what it was asking.
I told you in that second video that it meant P and Q. And also in that second video, I introduced some less common operators. This one that's called the biconditional, and this one that's called the exclusive or. And I showed how to fill out the truth tables for those. I left those out of the first one.
Because in the first one, I wasn't trying to help everybody. I was just trying to help the students in my class. And in my class, we used a book that asked the questions this way and didn't even ask the biconditional or the exclusive or.
So I left them off. And I wrote things in English because that's what their book did. But other students started using this and other students use different books.
And so their questions might be asked differently. So I want to make this as applicable to different people as possible. So that was the idea of the second video. What I want to do in the third video is get to the real expert level of this stuff.
And the real expert level of truth tables is when you have three, as opposed to two, prepositions on which the questions are based. So instead of just a P, a Q, there's a P, a Q, and an R. And typically, these will already be filled out for you.
The teacher will give you the P, the Q, and the R. The Ts and the Fs underneath these columns will be provided for you. But sometimes they're not. It's possible a teacher could ask you to list all the possible scenarios for trues and false with three different prepositions p's q's and r's and so if you were doing that i want to show you how to fill it out you could fill it out in any order you want but there's a standard way that these are listed and the standard way the idea is to take the t's and the f's as we wrote them before so true true true false false true false false that same order and write that in the right most columns so what i mean is right true true and then the second one was true false and then the third one was false true and the fourth one was false false so i'm just copying exactly what we saw in the previous video this exact circle right here i'm copying that right here and what you're going to do is you're going to copy that twice and the reason you have to copy that twice is because to list all the possible scenarios these used to be all the possible scenarios when you only had two propositions but if you have a third one Well, you need all of these with a third one true, and then you need all of these with a third one false. And typically the order these are written are, as you see here, T's here, and then this exact same circle with F's out in front.
And again, you don't have to write them in that order. And again, typically, this will already be given to you, the student. But sometimes it's not.
And in case your teacher doesn't give it to you, I want to show you how you come up with it. And you can convince yourself, it might take you a little while, to convince yourself that I really did list all the different possibilities for T's and F's. by going through this way.
So now that I have these three different prepositions, I can combine any of the three together using the different operations that we learned in the previous videos. So for example, I could say, I don't know, P or Q. And again, as we saw in the second video, this can be written this way or this way or this way, it all means the exact same thing, P or Q. As we learned in this video, in the first video, I guess, P or Q was asking you to look at the P's and the Q's and ask yourself the question, Is there at least one T?
Yeah, there's at least one. There's two. That's at least one.
Yeah, there's at least one. It's right here. Yeah, there's at least one. It's right here. Nah, there's not at least one.
That's why there was an F in this fourth row, but T's in the remaining rows when I was asked P or Q. Now let's go over there, go into this new video and apply that P or Q. Note, it doesn't reference R, so I'm never gonna look at this column. I'm just looking at the P and the Q.
And the question is, do I see At least one T in these two columns. Well, yeah, there's two of them. So I'll put a T right here. Yep, there's two of them.
Yep, there's one. It's right here. Yep, there's one.
It's right here. What about down here? Yeah, there's one. It's right here.
Yep, there's one here. And finally down here. Nah, I don't see any T's here. So P or Q gets an F.
Similarly on this last row, it gets an F. Whoa, there's probably some shortcuts I can do where I kind of memorize the order of these based on the orders that I have in the last video. Yeah, you can. But I wouldn't recommend memorizing any of these orders because they're easier to just understand. P or Q is saying look at the P and the Q column and ask yourself the question that you learned in the previous video.
Yeah, there's more rows, but don't let that throw you off. You're just asking eight questions instead of four. No big deal. That's how you do P or Q. How can this be harder?
Well, one thing you could do. is i can say p or q or r right now i'm referencing all three variables p q and r well that seems like that's going to be really really hard how am i ever going to figure that out it's really not that bad the way to think about it is p or q is a thing a column that we already know right it's this guy right here this is p or q so all this question is asking you is that column that I just listed here and the R column. You're only referencing two different columns, right?
One of them is the P or Q column and the other one is the R column. Right, sure, if you want extraneous arrows, maybe that makes it better, maybe it makes it worse, I don't know. All I'm saying is just like we figured out P or Q by looking at these two columns, we could figure out something or something by looking at these two columns. So I look at these two columns and I ask myself the exact same OR question.
Remember the or question is, is there at least one T in those columns? So look at these two columns. Do you see at least one T? Yeah, you see two of them.
Put a T there. Is there at least one T here? Yeah, there's one right here.
About here? Yeah, there's two of them. About here? Yeah, it's right there. Here, there's two of them.
Here, it's right here. Here, it's right here. And then finally down here. No, I don't see any Ts at all.
So I put an F in there. P or Q or R ends up looking like this. It's worth pointing out that you could ask this question without the parentheses.
If somebody just wrote P or Q or R, you could ask the question this way and it would mean this. How in the world would you ever know that? Well, the way you can think about it, and maybe this is getting too advanced for this, is P or Q.
One way you could write that is P plus Q. All right, so you can kind of think about this as P plus Q plus R. Remember order of operations, PEMDAS? Please excuse my dear Aunt Sally if that means anything to you. Where you do multiplication and division before you do addition and subtraction.
And when you have ties, you break them by going left to right. All right, somebody asked you, what's 3 plus 5 plus 2? What they're really asking you is, what's 3 plus 5? That's 8. And then plus 2 more, that's 10. And because multiple... or addition is commutative you can do this in different orders and get the same answer but some operations don't commute and sometimes it's really important what order you do these in and if the operations are the same you always do them from left to right to avoid any confusion it's the same idea here right because these have pluses here you can kind of think about them as oh they're the same operation and when i have the same operation i do them from left to right so first i'd have to figure out p or q which is exactly what I did here.
And then I'd have to figure out that thing or R. Anyways, that's a lot just to say that if I wrote this thing. Without parentheses, the answer would be the exact same.
But I think it would be a purposefully confusing question if it were written without parentheses. But there you go. Now you've seen it.
Similarly, let's throw some ands around. What if I said P or Q and R? This would be a really confusing question. A really hard question if a teacher asked you this.
But they could. P or Q and R. How would you answer this? Well, again.
Ors can be written this way and ands can be written this way. And means multiplication or means addition as we learned in this video. And multiplication or addition. And if somebody asked you what is 2 plus 3 times 5. These are one of those you see them like on Facebook. 90% of people can't get this right or whatever because it's kind of a trick question.
You might think this is 25 because 2 and 3 is 5. 5 times 5 is 25, but it's not. What you're supposed to do, order of operations, you do the multiplication before the addition. So this is 2 plus 15. This is really 17 if someone asks you 2 plus 3 times 5. Similarly, when you have ands and ors, it's important that you do the and first and the or second.
It's important that you first figure out Q and R and then figure out P. And I don't think that that's obvious at all. if it were written this way it'd be tempting to first figure out p or q, because we already have it right here, and then add on r.
But if this is written without any parentheses at all, what this means is this. If it's written without parentheses it means this one. It means these parentheses. And if you want to do the p or q first, you have to put the parentheses in there. No parentheses, it would mean this, although it would be a really mean question for a teacher to ask you.
And if there are parentheses, I'm going to do it both ways and show you how you get different answers. P or Q, that's a column I already have, that'll make life easy. I just want to do that column and R. So I want to ask myself the question, do I see a T both in the P or Q column and in the R column? Yeah, I do.
There's a T in both of those. So I put a T right here. Do I see a T in this column and this column? No, it's not. It's an F over here.
Do I see a T in this column and this column? Yeah, they're both T's again. So I get T over here. Do I see a T in this column and this column?
No, there's one of them, but the other one's not. So I put an F. Do I see a T in this column and this column?
Yep. This column and this column? Nope. This column and this column? Nope.
And finally, this column and this column? No. If you were asked P or Q in parentheses and R, this would be the way you'd fill out your truth table.
And note that that is different than what I'm about to do here, P or Q and R. If you were to ask this question, it would be pretty hard because you don't have a Q and R column made. So if I were given this like on a test, what I would do is off to the side maybe, create my own Q and R column, Q and R, right?
Because I'm going to want to reference that to figure out this answer. So Q and R, what do I remember about and? Oh yeah, yeah, yeah, we just did and.
And is I have to look at these two columns and say, do I see a T in both of them in the Q and the R column? Well, here I do, but here I do not. Here I do not, and here I do not.
Here I do, here I do not, here I do not, and here I do not. The only places where there's a T in both of them is on this first row and on this fifth row. So the only T's in this new column are in the first row and on the fifth row.
And now that I have a Q and R column, here's the annoying arrow thing that I do sometimes, I can answer P or this column by looking at the squeeze. Yeah, sure. Those two guys.
All right, P or this Q and R column. Remember, or means do you see at least one T? Yeah, I see at least one T. I see two of them, but I don't even need two of them.
I just need one of them. Do I see at least one T? Yep, it's right here.
Do I see at least one T? Yeah, it's right here. Do I see at least one T?
Yeah, it's right here. Do I see at least one T? Yep, now it's over here.
Do I see at least one T? Nah, they're both F's. So I put an F right here. Do I see at least one T? Nah, they're both F's.
So I put an F right here. Again, do I see at least one T? Nah, they're both F's. So I put an F right here. Note that this green is different than this red.
And that's confusing because they all say P, they all say OR, they all say Q, they all say AND, they all say R. The only difference is the order of operations. I think the teacher should always give you these parentheses in here so you know what order to do them in.
But as a reminder, if you were given P or Q and R, that would mean this one and not this one. Because just like you do multiplication before addition, you do and before or. You can see how these can get really confusing. Maybe a more blanket statement on order of operations is I talked about how and, which you can think of as multiplication, has to come before or, which you can think of as addition. And that's true.
That's right. And comes before or. But what about the rest of them, right? What about like not and if then and all those kind of things?
Well, it turns out that not goes first and then if then. And then if and only if. And some of these kind of make sense. Some of these you probably do without even thinking about it.
In fact, I did one of these in a previous video without really even thinking about it. Doing not before stuff like and and or kind of feels natural. Here, for example, we had not P or Q. So we looked at the P and we negated it first. We looked at the not P column and the Q column.
These were the two columns here. that we used to create this column. But you could argue that there's a different way you could do it.
You could look at the P or Q column, this guy right here, and then negate this column. So just change each of these false, false, false, true, and write that over here. Whoa, that would have given me a different answer.
Yeah, because that, we would have been doing the operations in a different order. I think it's more natural to negate the P to do the not first. and then do the OR second, which is exactly what we did, which is the right way to do these.
But if you really think about it, you might argue that there's a second way you can do it until you learn these order of operations. So in the previous example, I talked about how AND comes before OR, which is why if this is written without parentheses, it means to do the AND first and then the OR second. And that also holds true with NOTs and IF, THEN, and IF and ONLY IF, all those.
Alright, one last example. Let's try to make a big mess here. What if somebody said if q and r, then I don't know, p or q?
Sure. If you're asked this, if q and r, then p or q, if you're written without any parentheses at all, we could figure out the order that is implied with our order of operations here. But I think the way this should be asked is if and then q and r in parentheses then and then p or q in parentheses this means this how do you know that well if you think about our order of operations there are no knots in here so we can ignore that so next we'd look at ands do you see the word and written anywhere yeah right here so i should put parentheses around this i should do this first q and r should be the very first thing that i do all right what about or do you see any ors written yeah there's an or right here So second thing I should do is P or Q, which are these parentheses right here. And then finally, I do my if then. So if these first parentheses, then these second parentheses.
And maybe you can see how there's different ways you could do this. Like you could do if Q and R, then P. You can figure that thing out.
If Q and R, then P or Q. If you had parentheses written different, you... come up with a different answer than what's given here but the parentheses are not given there if you had no parentheses at all it would mean these i feel like i'm making this more confusing than it has to be so let's do this and then i'll call this video good and hopefully this is helpful and not just driving people crazy this question it's really an if then question right if something then something else so remember as we saw in a previous video that the if then ones are the ones that are kind of confusing and the logic is hard for people to follow. But I think if you're just looking for the answer, all you gotta do is look at the first thing. And if you see a T under that first thing, then you got some more work to do to earn the T in this column.
So if you see a T under the first thing, P in this case, then you got some work to do. Then you got to go over to the Q column and look for a T. So if P, then Q. If you see a T in the P column, I do. Okay, then you got some work to do.
You got to look at Q. Well, I see a T here. So I did the work. So I got a T right here.
But for the second row, if you see a T in the P column, yeah, I do. Then you got work to do. You got to look at your Q column.
Well, I didn't do that work over here. So that's why there's an F here. What's confusing to a lot of people is putting a T in this third and fourth row for this if P then Q. Because the idea here is you look over at your P column, and if you don't see a T in the P column, you don't have any work to do at all.
You don't even look at the Q column. It doesn't matter at all. You get your T for free. If there is not a T in the P column, and you get your Ts for free here and here, why is that? That's a great question.
that I don't want to spend a lot of time in this video in. I try to answer it in the comments of the first video. But just know that that's the case. That's how you deal with if and then.
If you understand how to deal with if and then, you can do it even in a huge composition like this guy. If, one of these columns, the Q and R column. Yeah, I got that guy. Then, another column, the P or Q column.
So for this one, I'm going to reference two different columns. I'm going to first reference the Q and R column. Sure. And then I'm going to reference the P or Q column, which is this guy. So I know it's hard to do, but try your best to ignore all the other T's and F's on this page and just look at these green ones and these red ones right here.
If you see a T in the Q and R column, then you got work to do. This first row, do I see a T here? Yeah, I do. So I got work to do. What the work is, is looking at the P or Q column and seeing if there's a T there.
There is. So I put a T over here. Again, if you see a T in this Q&R column, then you have work to do.
I don't see a T here. Well, then you don't got any work to do. And you get your T for free. Similarly, I don't see a T.
You got nothing to do. You get your T for free. I don't see a T here. That's great news. You got no work.
You get your T for free. But if you do see a T in this Q&R column, then you got some work to do. Then you have to come over to the P or Q column.
and see if you see a t here. I do, so I get a t here. If you don't see a t here, you need a t for free over here. I don't see a t, I get my t for free.
I don't see a t, I get my t for free. It turns out that if q and r, then p or q, that's a statement that's always true. It's called a tautology.
You don't have to worry about that. Don't worry about when these are all true or when these are all false. Just learn how to deal with all the parts individually, and then you can put them together however you want.
Q and R, I can figure that out. P or Q, I can figure that out. If something, then something else, I can figure that out. You can do any column that you need. and again the order of operations is maybe this should have been like a fourth video but maybe you're getting sick of hearing my voice and you don't want a fourth video these are really hard and i would argue that a good teacher should always put the parentheses in for you you shouldn't have to figure out the order of operations on your own but in your in case your teacher some crazy hard teacher asking these ridiculous questions if these questions are asked without parentheses put in your own parentheses according to this order do the knots first then the ends then the oars then the if thens then the if and only ifs