00:01
So in this question, we're asked to construct truth tables for six different compound propositions that are given to us.
00:08
We could start off with the first compound proposition a, which is p, conditional statement, negation of q.
00:19
So we could start off with the individual propositions.
00:24
We have p, we have q, and we have the negation of q.
00:27
So we need proposition q in order to find the negation of q.
00:34
So we'll write down the different truth values.
00:38
True, true, false, false, true, false.
00:43
True, false.
00:47
Now, we know that the negation of q is the opposite of q.
00:52
So if it's true for q, it'll be false for the negation.
00:56
And then if it's false, it'll be true.
01:01
Now we're going to write down the compound proposition that's given to us.
01:06
So p, conditional statement, negation of q.
01:11
So we know with the conditional statement that if this proposition is true and this proposition is false, then it'll be false.
01:21
Anything else, the truth value is true.
01:24
So we'll look at the first and the third columns to solve for this.
01:29
So here we have true.
01:30
False we know that is false we have true true that'll be true we have false false that will be false true as well and then we have false true that'll be true so this will be the truth table for the first compound proposition now let's look at b so let's get rid of a let's get rid of this one and if we look at b let's write it down.
02:12
Negation of p by conditional statement, q.
02:22
So from this compound proposition, we know that we need to write down three different propositions.
02:30
So it will be p, q, and it will no longer be the negation of q, but it will be the negation of p.
02:37
So let's get rid of this column and replace it with a new one, which will be the negation of p.
02:47
So the negation of p.
02:48
So the will be opposite of p.
02:50
So this is true true false false this will be false false true, true.
02:58
Now let's write down the compound proposition negation of p biconditional statement and q so we know that with the bi -conditional statement both propositions need to have the same truth values in order to be true.
03:16
If they differ it'll be false.
03:20
Therefore we look at, we're going to look at the third column and the second column.
03:29
So these differ, so this will be false.
03:33
These both are false, so this will be true.
03:36
These both are true, so this will be true as well.
03:40
And these both differ, so this will be false.
03:43
So this will be the truth table for b.
03:49
Now let's get rid of this and solve c.
03:54
Get rid of that one.
03:57
Let's get rid of this.
04:00
Let's write down c.
04:03
So c we're given p, conditional statement, q, inclusive or, negation of p, conditional statement, q.
04:25
So again, we see, looking at the compound position, we have three variables or three different propositions.
04:33
We have p and q, and then we have the negation of p.
04:36
So we'll rewrite it and again this will be the opposite of whatever p is true true now let's solve the first compound proposition so we write that down p conditional statement q again we know that the conditional statement if the first proposition is true and the second proposition is false it'll be false anything else it's true so we're looking at the first two columns for this one so true true it'll be true, true false, it'll be false, and then it's false true, that'll be true, and false false false, that'll be true.
05:17
Now let's look at the second compound proposition.
05:20
Let's write that down, the negation of p, conditional statement, q.
05:25
Again, if the first proposition is true and the second is false, it'll be false.
05:31
Anything else, it's true.
05:33
So now we're looking at the third column, and the second.
05:37
Second column.
05:38
So the third one is false true.
05:41
You know that'll be true.
05:44
False, false.
05:45
You know, that'll be true.
05:47
True, true.
05:49
True.
05:50
True.
05:50
True.
05:52
That'll be false.
05:54
Now we write down the complete compound proposition.
05:58
So it's p, conditional statement, q.
06:03
Inclusive or, negation of p, conditional statement q.
06:11
So we know that for the inclusive or each proposition can be true.
06:21
They both could be true in order to be true.
06:23
If they're both false, then it's false.
06:27
So we're looking at this column and this column.
06:30
So true, true, we have true.
06:33
False true.
06:34
We have true.
06:36
True, true.
06:36
We have true.
06:38
And true false, we have true.
06:40
So this is a true statement.
06:42
For this for c.
06:48
Now let's get rid of this.
06:50
It's a software d.
06:55
Gone now d.
07:01
We have p, conditional statement, q, and negation of p, conditional statement, q.
07:19
So again, if we look at this compound proposition, we have three different propositions.
07:24
We have p, q and the negation of p...