00:01
For this problem, we are asked to find a truth table for p and not q implies r.
00:09
So i've decided to do it on a spreadsheet because then i can make some conditional formatting that will make things easier for everyone to see.
00:18
Okay, so let's get started.
00:20
The p and not q implies r, that's kind of the main thing that we're going for.
00:27
So that's going to be our rightmost column.
00:29
And i might need to add more columns in between here.
00:33
But to begin with, we are going to start off with a column for p, q, and r, since those are three simple statements or three letters that you see.
00:44
So with the p, q, and r, we'll start off with some truths and falses, so that we can have every possible combination there is.
00:53
I'm going to start with the p's.
00:56
I do four t's, four fs.
01:01
And how do i know that there is this many? well, one trick is that you can do two to the power of however many letters you have here.
01:12
And i'll tell you how many rows you need in your table.
01:15
So since i have pqr, i have three of them.
01:18
Two to the third power equals eight rows.
01:22
So just keep that in mind if you're interested.
01:29
Okay.
01:30
For the q, i'm going to do t.
01:32
I get all capital t, t f f t t f t f t f t f.
01:39
Notice how i'm changing the alternating pattern here.
01:43
I'm trying to get every possible combination of truths and falses.
01:48
Then i have for my r column, t f, t f, and i'm just going to continue that pattern.
01:56
So right here, this is a standard.
01:59
I'm starting for all of these statements.
02:02
I have three simple statements.
02:04
Now to build, i'm looking at what we have here.
02:09
Well, we have in parentheses, is not q.
02:12
So i think i'm going to need that.
02:16
I'll do not q next.
02:21
And then once i have the not q, i'll do the p and not q.
02:31
So again, i'm trying to build up to our final statement.
02:39
And how do you know which ones to do first? usually you do the knots first.
02:44
And then the ands ors.
02:46
And then finally, it's, i'm going to put this here because this is what i'm assuming.
02:51
I'm assuming that this whole left -hand side here implies r.
02:57
So i need that whole left -hand side first.
03:01
And that's where this column comes into play.
03:03
And then i finally stick in some information about r.
03:08
Okay, well, let's get started.
03:11
So with filling in the rest of these, it gets really overwhelming because there's so many, so much things going on.
03:18
I like to kind of focus on certain columns at a time.
03:26
I'm going to maybe highlight the ones that i'm looking at, so you know exactly where i'm looking.
03:33
I'm going to focus on working on this not q column, and in order to fill that one out, i'm going to be looking at the q column.
03:42
Now, not q is the opposite of q.
03:45
So whatever i had for q in that same row, i'm going to put the opposite.
03:50
So, for example, in the first row of q, i have t.
03:53
Not q is going to be f.
03:57
Second one, same thing, t.
03:59
So i get f.
04:02
Third row i have f, so i'm going to have t and so on and so forth.
04:06
So the pattern just becomes opposite of each other here.
04:15
Done with that one.
04:18
Let's work on the next one.
04:21
This column, p and not q.
04:25
Well, i need to be looking at p.
04:27
I need to be looking at not q.
04:31
And then whatever is in here and here, i'm going to put it together with an n, and the result is going to be in the column that we're filling in...