Okay, 8.5 truth tables. Truth tables look much harder in Hawks because they give, I'll show you, I'll pull up the homework so you can see, but let's just talk a little bit about, we're going to learn how to build a truth table. In Hawks, they just give you truth tables all filled out and you have to say if it's.
right or wrong. It looks overwhelming. So let's get a few definitions down.
We talked about truth values. A statement is either true or false, but not both. They say a tautology is a statement that is true in all possible circumstances. So mostly we'll be looking at the truth table.
And if every line says true, true, true, true, true, true, truth, then it's a tautology. I'll show you an example of that. Okay, so we've talked about conjunctions, disjunctions, and conditional statements.
Now we're going to talk about their truth values, a conjunction. So it's A and B. The notation is A and B. And it's only true when both statements are true.
All otherwise false. Well, only true. Okay, so we'll get statements A and B, but if you think of an and, think of it as it pertains to your life.
So if I said I'm a math 167 student, it's statement A. Statement B is I'm taking graduate level German. Okay, so if you said I'm a math 167 student and I'm taking graduate level journal. graduate level German, the and would probably be false because I bet you are a student in math 167 or you wouldn't be watching this, but you're not. So this whole and statement would be false because both have to be true.
I am a freshman and I am a freshman. Statement A, statement B is I'm taking math 167. So if you said I'm a freshman and I'm taking math. 167 both would have to be true for the and to be true okay all right so a disjunction and it i just want you to keep those things in your mind because when we get to the truth tables i'm gonna say this is an and we just look at the truth values it's true and true it's the only time and it's true all otherwise it's false okay all right so a disjunction a or b you And our notation is A or B. Okay.
So an or statement is I'm a math 167 student or I'm taking graduate level German. Well, that would be a true statement just because one of them is true, right? Or is like union.
Okay. So an or statement is always true unless both statements are false. So if we said... a student at Mississippi State or I am playing in the roller derby. So that would probably be false because I bet you're not in the roller derby and I bet you're not in the United States.
So the only time an or is false is when both statements are false. So let's write that. So or is always true unless both statements are false. Okay, so if we just started.
I'm not going to get to the message statement and statement be a not a not be a and be let's say a. or not B. Okay, so this is what a truth table is going to look like. And you'll be given this, all the possible truth values. So false, true, false, false.
Okay, so these are all the possible truth values for A and B. So when we say not A, what is not A? Well, that's the negation of A. So what does that do?
It just changes the truth value of the column A. So we're not looking at the column B. We're just changing the truth value of A. So false, false, true, true.
And then what about not B? So if not B, I'm going to get all my pencils in a row. All we're doing is looking at B and we're changing the truth value. So true becomes false.
False becomes true. True becomes false and false becomes true. So we're getting there.
Okay. All right. So if we say not A and B.
So here's what I want you to think about. An and, okay, an and is true only when both statements are true. All otherwise it's false. Okay.
So when we go look at our and, we look at not A. So we don't need. Not A. We don't need that.
And we don't need not B. Okay. So an and statement, we're looking at not A and B. And they say, what do they say for an and?
A conjunction. It's only true when both statements are true. So false and true.
False and false. False. True and true is true. So an and is only true when both statements are true.
Okay. And then let's look at my or. So we're just going to look at A or not B. So what else do we not mean here?
So we're not looking at these things. So the truth table can look overwhelming, but I'm just looking at A. Here's A or not B.
I'm looking at. These two columns. And so what does it say about an or?
An or is always true. OK, unless both statements are false. So let's look at this. So an or is always true unless both statements are false. True or true.
False or false. And then that's how to build you. So you have to think.
And is. Always false unless both statements are true. So if we did not A and B. So always false unless both statements are true.
And the or is always true, true, true, unless both statements are false. Okay. That's how to build our quick table.
And they'll just keep on asking all kinds of crazy things. And we just look at the two columns that matter. Okay. All right, so that's the start of a truth table, but we didn't talk about conditioning, which is also the implication.
A implies B. And the notation. is a arrow b okay and um it's red if a b okay so let's talk about the truth value of a conditional i think of it as a promise um So let's see. let's let statement a be um i win the lottery b is i give you a thousand dollars okay so to write a implies b it always has it if and then you put your a and then and you put your b okay so statement a is if i win the lottery then i give you a thousand dollars okay and that's great okay that's a conditional if then think of it as a promise okay but all right so let's talk about the truth values okay so if you think of this as a promise is if i won the lottery then i give you a thousand dollars and so um if statement a with all the possibilities and these you'll be given um the true true you True, false.
These are all the possibilities. False, true. False, false. Okay. And let's talk about A implies B.
And this is where students get confused with the implication. It's just not as straightforward. Okay. So if you think of this as a promise. So if A is true, if I win the lottery and give you $1,000.
So I. I win the lottery. It's true. I give you a thousand dollars.
Did I keep my promise? Yes. So the whole implication is true. Okay. So here, this is true.
I win the lottery. I don't give you a thousand dollars. Did I break my promise? Yes. The only time an implication is false is when you break your promise.
Okay. It's only false. True implies false. Because what if I don't win the lottery?
What if A is false? Did I break my promise? No. So if I don't win the lottery, but I give you $1,000, it's still a true statement because I didn't break my promise.
If I don't win the lottery and I don't give you $1,000, the implication is still true because I didn't break my promise. So you have to think of it as a broken promise. Okay.
It's a little trickier. Okay. So.
Let's do a couple of truth tables that I already have ready, and we'll just see what some bigger ones look like. Okay. So do you see, this is how you build your truth table. All the values for A and B will be given.
Okay. And so when you look at it as multiple choice, they're going to, it's going to all be filled in. So you don't really know what you are given and what you need to know, but let's talk about and what about A and B. So we just need to know A and B. is true only when both statements are true, okay?
All otherwise, it's false. So it's not that hard to fill out. What about A or B? A or B is only false when both are false, okay?
So if true or true is true, true or false is true because I'm taking math 167 or I'm taking German 757. Well, it's a true statement or it's true if you're just doing one of them. Okay, false or true is true, false or false is false. So the or is only false when both statements are false. Okay, so what about A implies B?
And remember, we're only looking at A implies B. So if. True implies true.
So I win the lottery. I give you $1,000. That's true. I win the lottery.
I don't give you $1,000. That's false. I don't win the lottery. I haven't broken my promise. If I don't do the A part, the implication is still true because I didn't break my promise.
Think of it as a promise. And then they just want you to make sure you know this notation, not A, just means what? Change the truth value of A. and not B just means change the truth value of B.
Okay, all right, so that's building a truth table. Okay, and I'm here to tell you that I'm gonna look at some more. Some of them can same. you probably came and read this over here. Yeah, it's okay.
All right, so here we go. We're going to fill in the truth value. So let me just this is given. When we say what is not A, we're just changing the truth value of A.
Right. And then not B just changes the truth value of B. So we're looking there. False becomes true.
True becomes true. All right. So not A and B.
OK, so we're not looking at A. We're not looking at not B. Okay, so and means what? Before I even get started, I think and, in order for an and to be true, they both have to be true. So this is going to be false.
This is going to be false. This is true. And this is false.
So an and is only true when both statements are true. Okay, what about a or not be? And this is and, and this is or. Okay, so A or not B.
What does or mean? It's always true unless both are false. Always true. Always true unless both are false.
Here they're both false. So that's false. Here they're both.
One's true, one's false. So it's true. Okay, now what about A implies B? What do we know about A implies B?
I'm not looking at any of this. It's right here. A implies B. I win the lottery. I give you $1,000.
I win the lottery. I don't give you $1,000. That's the only time an implication is false is when you break the promise.
True implies false. It's false. All otherwise, it's true.
I don't win the lottery. It doesn't matter what I do. I didn't break my promise.
Okay. So if the A part is false, it doesn't matter. You didn't break your promise. Okay. So let's do, this says B implies not A.
All right. So true implies false is false, right? False implies false is true.
True, true, false, true. Okay. So that's how that. Did I do it right? B implies not A.
True implies false is false. All otherwise is true. OK. All right. So the last one we're going to do is three variable.
Looks a little harder. OK. But I want you to know that these are given.
So when you look at the multiple choice, all the possible truth values for A, B and C are given. And so at any one time, we're only looking at two columns. So what is not B?
I'm just changing the truth value of B. Right. So that becomes false, false, true, true, false, false, true, true.
OK. A implies C. OK, so A implies C. What does that mean?
The only time it can be false is when you have true implies false. Let me get this. We're not looking at. So A implies C.
True implies true. True implies false. I won lottery.
I didn't give you any money. You broke your promise. All otherwise, that's going to be true implies false.
False implies true. It doesn't matter. False. False. If the condition part is false, then you didn't break your promise.
So A and not B. So we cover these and and means what and means they both have to be true for everything to be true. So false, false, true, true, false, false, false.
OK. And so and if they then be. This is or.
So we just look when it's or is always true unless both are false. So or is not B and C. OK.
Or is always true unless both are false. True, true, true. Both are false.
True, false, true. OK, so that's how to fill out. And usually on hooks are asking for this last. Okay, we'll pull that up. All right.
So I'm going to make sure that all my conditions are due. So let's go look at the homework. Okay, so.
So A and B are two statements such that A and B is a compound statement, which is true only when both statements are true. Otherwise, it's false. That's a conditional and. Right. Oops.
A and B. Oh, it's a conditional. I'm at conjunction C. It's easy. It's easy.
All right. Two statements A or B or is a disjunction. Just get through these.
We've already talked about these. A table that has a row for each possible combination of truth values. That's just a truth table. OK. All right.
So use the variables to rewrite the given compound statement. I've done all this in your homework. I've worked out your homework. But if I am not tired. And so.
All right, so if I'm not tired would be not W. It's if then. So these are if. These are the only two it could be. If then, right, the arrow.
And so if I'm not tired, then I will stay up late. So this would be it. Because W is tired.
This is I'm not tired. Then I will stay up late. I will stay up late. So this is not W.
I'm tired is W. So that's I'm not tired. It's not W. And if then means the arrow.
I will stay up late. I will stay up late. So that's the same.
It's not litigation. OK. All right. So the truth tables, like I said, the M and N are always the same.
They're not tricking you here. OK. All right. So the not N, let's make sure. All of those are the same.
True, false, false, true, true, false. So we're just looking at this right here. So M and not N.
So what do we know about an and? It's always true. Now an and is never true unless both are true.
So that's not that's not right. Right. Right.
So we're going to look here and and is they both have to be true. And and is they. Both have, that's false.
That would be false. So let's look at this. Am, and, not in. So, oh, I was looking at the wrong one. True and false is false.
True and true. True and true is true. False and false is false. False and true is true. I think this is it.
I was looking at the wrong one. Am and in. So we're going to pick that one.
And that's the answer. So that's a small one, but sometimes they get kind of big. All right, so we can do the and. We already did the and. Let's try similar, see if they do.
This looks like it's an or. All right. And we're just looking at these two columns. Okay. And tell me about an or.
An or is always true unless both are false. So that's wrong. Right.
False or false should be false. False or false should be false. False or false should be false. This one could be it. False and true is true.
True and false. True, true, and true, true. So that's it. I'm just looking at these two columns.
This is the or. Remember, an or is always true unless both statements are false. So an or is always true unless both statements are false.
So this is the one that we're doing. OK. All right.
C implies A and B. All right. So A and B is true. think these this is the only one we're looking at okay and so um c implies a and b okay so we're looking at c so true true is true true false is false true false is false true false is false false true is true false false is true these are all true this was the one let's see why this is So true, if true, then true, true, false. They said that's true.
So we know an implication. If it's true and false, you broke your promise. So true and false is false. True and false is false.
True and false is false. They said false and true, if false. All right, so if the if part. If you didn't break your promise, it can't be false.
So we've got our answer. Okay, and it looks just horribly ugly. But we're just going to look at this column, am, and, in, and, not in.
So what do we know about an and? The only time an and is true is when both segments are true. This is accurate.
Let's go make sure. And it's not a tautology because if this column would all true, then it'd be a tautology. So let's see. All right.
So M or N and this. So it's always true unless both statements are true. This looks right to M and N.
M and N. M and N is true. No, or.
Okay, so this is the only one we're looking at. And. So that's false.
That's true. That's false. That's false. This looks right too.
And. False. True. False.
False. It's not a tautology. Okay.
This one is the same thing, but it's a tautology. Only if everything is true. Okay. That's what I was missing here.
Okay. So maybe they'll give us. And so again, you don't have to start building it from here. You're just going to go, okay, P or Q and not R.
So we're just looking right there. This is just telling us what two rows to use. So an or what?
That's an and. and is all true only when both are true. So that's already wrong, right? So this is false, true, false, true, false, true.
Nope, that's false. So let's look at this. False, true, true, true, false, true, false, true, true, false, true, false, true, true, false. Okay, so this is it.
Okay, so you see, I'm just looking at this is the and so it's this column and and means they both have to be true for the whole thing for the and to be true. So false, true, false, true, false, false. Okay, it just takes a little bit of getting used to.
Okay. All right. So use variables to rewrite the given compound statement.
If it is running outside. Then Jennifer is inside and playing a video game. So it is a W implies. So it is raining. Then Jennifer is playing a video game.
Is inside playing a video game. Not X. Jennifer is inside and playing a video game. Jennifer is inside.
and playing a video. This is it. If it is raining outside, then Jennifer is inside and playing a video game.
Okay. And that's it. All right. Y'all let me know if you have any questions.
Thanks so much. I'm going to quit sharing and good luck.