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So let's go back to this thing that we just simplified here, right?

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We're going to talk about something called a truth table now, which is going to let us map out all

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the possible combinations of true and false we can have and whether it would lead to this result.

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So our goal is to find out whether this evaluates to true based on all the things that could be either

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true or false, whether this event evaluates to true based on all the things that C, D and E could

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be.

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So we're going to look at each variable and think of all the combinations of like, true, false, true,

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true, true, true for each one of these variables and see if it would get us to a conclusion where

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this is true and this is true, then it would lead to this.

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Piece of code right here where we say B is now false.

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Bob doesn't get the bananas right.

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Bob gets bananas is false.

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Because that would be was.

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So first thing, we're going to look at the not right, so we have, if not not up here, we're going

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to look at even a similar kind of situation of this.

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So a is our variable.

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And then so we have a here, and we're going to think about what happens when air is false, what will

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not be.

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You probably can figure out this out on your own.

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It's pretty straightforward, but basically when you say not something, all you do is flip it or take

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the inverse of it.

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It's the opposite.

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So.

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You can think of all these words are actually very important in computer science, so envious and flipping

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of a bit, you know, remember the zeros and ones are bits.

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If you go back to that lecture, where we were talking about with Patricia was talking about the numbers

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in computer science, the number of systems we have zeros and ones for binary right and their bits.

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So if we flip a bit, that means that we turn it into the opposite.

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So if it's a zero goes to a one, if it's a one, it goes to a zero.

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So thinking about it in that sense, if we take the inverse or flip this for not a it's going to be

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one.

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If a is one, then not a will be zero.

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Right?

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So now let's expand it, so our sentence or our little sub statement up here, which is also a condition

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because of this if.

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Says, if not, not a.

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So let's figure out what not not as in each one of these conditions in which a is zero and a is one.

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Right.

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So we've already figured out what not a would be, but what would not, not a B.

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And even more than that, we'll go further just for the heck of it and say what is not not not a which

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seems kind of crazy, but I just want to really like beat it into the mind that.

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You just keep flipping.

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You know, it just flips, flips, flips, no matter how many knots you have.

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So with that, what do you think will be right here if a is zero, not a is one?

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What is it?

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00:03:26,160 --> 00:03:31,860
Not, not a you can basically look right here and see what it would be if you just flipped it again.

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Right?

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So it is zero.

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A zero, not a was one.

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Now, we only have two options, zero one, so we put another notch in front of this, all it does is

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just flip it back to zero.

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Same thing for this, it's going to be a one, right?

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Because not a zero.

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So not not as one.

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With that, you should be able to figure out this as well, so what happens if we just throw another

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knot in front of this?

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It's just going to flip it back again.

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So not not not a will be one when not none is zero.

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Same thing with this flips again goes from one to zero.

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If you put another knot, it would be zero and one another.

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Not it would be one in zero.

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Another, not a zero and one just over and over again, you know, one zero zero one one zero zero one

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one zero zero one, you can just keep feeling this forever.

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So that is something I really want to point out, it is a big concept in computer science.

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You know, when you say something is not, it means that you are taking the inverse of it, you're flipping

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the bit, you're flipping it from false to true, true to false like that.

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All right.

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So let's move on to a bigger example.

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So we're really going through it now, this whole thing.

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So we have this, if not, not a now.

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We're moving on to this next portion right here.

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So if C and D or E, so this is where we get into a real truth table.

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So the truth tables are something where we want to think of every possible combination that each one

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of these variables could be.

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And then we are going to figure out what this statement is based on each one of these potential values.

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So each one of these rows, we will evaluate these D or E right here in C and D or E.

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So this specific side, we start out with this or which is an important topic in computer science,

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the OR means that if either side is true, then the result will be true.

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So an either or type of thing.

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So if this is true, then it doesn't even matter what it is because the whole thing, the Ori will be

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true because we said one or the other has to be true, right?

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So if this is true, then the story is true.

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If it is true, then the Ori is true.

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All right.

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00:06:17,420 --> 00:06:22,720
So this truth table, just to kind of point out again, you notice a we only had one variable to consider.

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So that's why we only have two rows because a can be either zero or one.

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But for this, C can be zero or one.

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D can be zero or one.

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And E can be zero one.

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So we've got to think of all the possible combinations that we could have between these three variables

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in the way and easy way to get all of the combinations filled out in your truth table, which is this

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whole table is to count in binary.

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So if you have, you know, three binary places here for a binary number, just remember that the fact

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that you have two choices for it.

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Notice how many rows we here have here.

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One two three four five six seven eight.

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The reason that we have eight is because we're actually having two to the third power row is because

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that's how many combinations we can have of zero on one, because we have two.

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The two is for the fact that there's zero in one.

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There's two choices.

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The three is because there's three variables, so you can put either a zero or one for three different

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variables, that's why we have to go to the third power, which is eight rows, because that's how many

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combinations we have.

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So I just want to point that out.

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So that's a good way to double check, to make sure that you're filling this to truth table correctly.

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You may or may not ever be using truth tables.

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If you are a student doing a computer science major, you will definitely encounter a truth table at

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some point, most likely in discrete structures, discrete mathematics or something like that.

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Maybe just a logic class in general.

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But this is a good way to double check on your truth tables regardless if you're going to end up doing

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this or using it.

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So, you know, how many rows there are a good way to fill it out to is just to start with zero and

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you notice I just kind of am like counting up here.

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So.

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Yeah, I just, you know, I have one.

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And then, you know, this is two zero one and then I have this two one is two and then I have three.

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And you know, I'm just I'm just counting up in binary here, basically, and that's how I filled all

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the rows.

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So you should remember that from the lecture where we went over a number of systems and in particular

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went over that.

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So if you're still confused on that and this is confusing you right now, you can go back to that and

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kind of brush up on how to count up in binary.

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So with that enough of me talking kind of on a tangent about this part, let's get started with this.

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So we talked about the or meaning either or one of these has to be true for this whole sub statement

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to be true.

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So let's tell you this first row, we're looking at Dear E. So we don't really, really need to look

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at Si column, right?

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Because we don't care about that.

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We only care about you.

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So let's look at these two columns here and go row by row this first row.

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What is going to be the answer here is zero and is zero zero means false, so we have false and false.

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What did we say about the order here?

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So before I say any more, see if you're going to figure out on your own.

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OK, so false or false, we said that one of them either or has to be true for the whole thing to be

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true.

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Right?

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So if either side is true, then result would be true, except neither.

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Neither side is true.

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Neither side is true, so it's false or false, that means that we have a zero, which is false.

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OK?

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Because neither side was true.

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Let's look at the next one.

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The next one is zero four D and E is one.

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So we have a zero, which is false.

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And then we have a one, which is true over here.

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Since if either side is true, then the result is true, that means that it will be true because it

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is true.

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And that's one side, right?

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Same thing, D or E is so D is one e is zero.

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00:10:49,930 --> 00:10:54,050
I'm just going to go ahead and fill this out, but if you want, it would be great for you to pause

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00:10:54,050 --> 00:10:56,270
the video and just see if you can fill it this whole column.

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I'm sure that at this point, you should be able to to figure that out.

153
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So it's going to be true for this one.

154
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I look at the next row, I notice they're both true.

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So since we're either or, it's going to be true.

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The next row we have another, you know, D and E is zero.

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It's pretty much the same as up here.

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He knows this all three one zero here, but this had a one.

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That's the difference.

160
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So we're only looking at these two.

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So it's false because neither side is true, then we have another one.

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And pretty much wherever you see a one on either one of these columns, it's going to be a one in this

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dear column.

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So we're in for the one in a one.

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OK, so let's move on to this part here.

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We already know what the order is and we already know about.

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But now we factor see into it.

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And so we're looking at not just this part.

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But we're actually looking at the whole shebang here.

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So this whole line, so we've kind of been progressing our way up.

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00:12:01,330 --> 00:12:09,370
To get to this column right since we did dear, we can use this column now and combine it with it and

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in this.

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So what is the and mean and means that both sides must be true for the result to be true?

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So you basically have to have a true and true for the whole thing to be true.

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If you had if you had, let's say you had a and B, both A and B would have to be true for the statement

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A and B to be true.

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So let's go ahead and look at this.

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So we're going to be looking at this column.

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Of course you have.

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00:12:45,030 --> 00:12:50,100
You're like o d or e i need to resolve or do you even know you don't actually, because you've already

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00:12:50,190 --> 00:12:53,490
solved for all of the degrees right here in this column.

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So this whole section circling right here is taken care of.

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00:12:59,010 --> 00:13:06,990
So now all you have to do is pretend this degree is one thing you can almost like replace D or E with

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00:13:06,990 --> 00:13:08,610
a new variable like F.

185
00:13:09,060 --> 00:13:13,230
You could just say C and F and F is going to be all of these things in here.

186
00:13:14,130 --> 00:13:15,360
You don't have to imagine like that.

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00:13:15,360 --> 00:13:19,170
If it's confusing, just know that there is this whole thing that I'm circling with.

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00:13:19,170 --> 00:13:24,500
My mouse is already taken care of here, so you'll be able to replace it with a zero.

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00:13:24,510 --> 00:13:30,450
Like, for example, when we start on this first row here, we're going to take a zero and replace this

190
00:13:30,450 --> 00:13:31,140
whole thing.

191
00:13:31,150 --> 00:13:32,520
I'm circling with a zero.

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00:13:33,550 --> 00:13:38,350
And then we're going to say C and zero for this first one.

193
00:13:38,530 --> 00:13:39,460
So what is that?

194
00:13:39,720 --> 00:13:44,440
C zero, if we have a zero and zero, that's false and false.

195
00:13:44,440 --> 00:13:47,200
We said that and means both sides must be true.

196
00:13:47,640 --> 00:13:48,610
The result to be true.

197
00:13:48,820 --> 00:13:50,740
So therefore, neither of them are true.

198
00:13:51,040 --> 00:13:51,940
That's false, right?

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00:13:53,020 --> 00:13:57,040
So if you think you already got a grip on this, go ahead and pause the video and see if you can fill

200
00:13:57,040 --> 00:13:58,630
out this column by yourself.

201
00:13:59,970 --> 00:14:01,590
All right, so I'm going to go through it.

202
00:14:02,520 --> 00:14:07,600
The next one we have dear, he is one and C is zero.

203
00:14:08,340 --> 00:14:10,410
But both sides got to be true, so that's false.

204
00:14:11,410 --> 00:14:12,700
Same thing on the next one.

205
00:14:12,910 --> 00:14:19,090
Both sides have to be true, which they're not only one side's true false false again.

206
00:14:20,080 --> 00:14:25,030
This next one, we have a false over here in a two here, so that's false.

207
00:14:25,870 --> 00:14:28,590
This one, however, both are true.

208
00:14:28,820 --> 00:14:31,480
We have see as one and deary as one.

209
00:14:31,930 --> 00:14:32,770
So that's true.

210
00:14:33,920 --> 00:14:38,350
Then the next one is the same, I see a one and a one here, so that's true and the last one of the

211
00:14:38,360 --> 00:14:40,070
all ones all the way across the board.

212
00:14:40,080 --> 00:14:41,210
So that is true as well.

213
00:14:43,670 --> 00:14:48,710
OK, so hopefully that was pretty straightforward for you.

214
00:14:48,890 --> 00:14:56,420
Now we're kind of going to look at the last little thing, which is the fact that this is a if something.

215
00:14:57,550 --> 00:15:01,360
And then inside of it kind of here we have another if something.

216
00:15:02,560 --> 00:15:04,230
But really, if you think about it.

217
00:15:05,150 --> 00:15:09,650
These are two separate conditions that need to be fulfilled.

218
00:15:11,300 --> 00:15:12,560
For this to happen.

219
00:15:14,310 --> 00:15:22,380
So technically, we don't need to just put this if this whole thing here, if C and E and I'm circling.

220
00:15:23,310 --> 00:15:27,720
That doesn't necessarily need to be like indented inside of this if.

221
00:15:29,110 --> 00:15:36,010
Up here in that, if not, not a it could instead all be in the same continuous line like one statement,

222
00:15:36,010 --> 00:15:42,490
because you really have to do this has to be true and this has to be true.

223
00:15:43,730 --> 00:15:53,750
For Beagles falls to happen, notice the and I said there, this has to be true and this has to be true.

224
00:15:54,950 --> 00:16:00,830
For this to happen, that means that we could just instead put it all on one line and put an end in

225
00:16:00,830 --> 00:16:03,010
between like this.

226
00:16:04,290 --> 00:16:07,110
So if not, not a and.

227
00:16:07,980 --> 00:16:12,660
If C and D or E then B is false.

228
00:16:13,500 --> 00:16:15,220
So I want to point that out as well.

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That is some things, something that you can do a lot of times to condense your code and simplify it.

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So a nice thing to know is kind of look for these kind of logical simplifications that you can do.

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You don't necessarily need to put some type of if condition inside of another condition for it to work

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out.

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If you notice that they could, if they both need to be fulfilled or something happen, you can to put

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an end to their.

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So something to think about.

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And now that we have all of this kind of in one line, we don't necessarily we didn't have to put it

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on one line to really think about it as one line, but now it's easier for us to see we actually could

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like majorly expand this truth table and and really go all out.

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You notice I put a red arrow here because we would need more rows for this.

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What you can do is add a into the equation kind of literal equation, but you factor that into this

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whole table here.

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So you have a column and then you can do the have a not not a.

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And then the whole thing over here in this column, you know, not not A and C and D or E.

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So you could fill this out.

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You're going to have more rows, of course, because you just added another variable in here.

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So that's actually going to be two to the four amount of combinations you can do, right, because you

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have four things now.

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So I'm not going to fill this out, but if you would like to, this would be a good exercise to fill

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out if truth tables are something that you are interested in practicing.

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You will not need to use truth tables so much in the future.

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I am guessing a lot of times you just kind of handle this logic on the spot mentally in your brain.

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Although they are useful, I'm sure some people have to use them sometimes.

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So you never know when it'll come up, so it might be might come in handy.

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I'm not going to fill it out, but if you'd like to, you definitely can.

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So we just learned a decent amount of stuff, but I don't want to stuff your brains too much with this

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logic and go straight into the cold like immediately.

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So maybe a review kind of what you've looked over here, but in the next lecture, we're going to discuss

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how to implement all this stuff we just talked about in code.

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A big thing I'm kind of jumping down to here is the if I didn't explain that a ton, but you might have

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already heard that if something, then something else has a lot to do with code.

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And so we are able to use this if that we saw in those statements in code as well, and we're going

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to cover that.

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They are called conditional statements.

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And it's not just in if we have some other things that go along with it like else, if and else stuff

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like that.

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We're also going to go over operators, so today we talked about the and the OR and and not.

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But we've got more than that that we can put in code and I'm going to go over that as well in the next

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lecture.

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So hopefully this was all understandable for you and try to break it down.

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The truth tables really good to practice.

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And also just remembering the whole thing about each variable that we are talking about.

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There was something that was either true or false.

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This logic we're talking about is Boolean logic.

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You can get deeper into this topic and look up stuff on Boolean algebra, and that gives a little more

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theoretical.

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I kind of just combined a lot of things here into one lecture where I was doing it in kind of a casual

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pseudo code way, using the word, not in the word or in a word.

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And but if you look in a Boolean algebra, they have symbols to represent all these just like we will

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in code, but a little bit different.

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You know, there's a whole theoretical representation to this Boolean true or false algebra stuff.

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So if you're interested and you'd like to learn more about that instead of just enough to be able to

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start writing code with it, then you can go ahead and look up a Boolean algebra.

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Of course, if you're on a computer science major, you're most likely going to have to do a lot of

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this in discrete mathematics or discrete structures or whatever.

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The corresponding class that relates to that is in your major.

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So with that, I'm going to let you go and I will see you in the next lecture where we kind of turn

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this all into code.