00:01
So we want to show that each of these compound propositions is a tautology.
00:06
That is that they're always true regardless of the truth values of their variables.
00:14
And we want to do this using a truth table.
00:16
So let's begin to fill in the truth table for a, which is, if not p and p or q, then q.
00:23
So i already did all the possible truth combinations for p and q, and we will just do the rest of the truth table.
00:29
So p is true so not p is false, false, true, p or q is true if either p or q or both are true.
00:40
So this is true, true, true, and false.
00:47
For this, we need both not p and p or q to be true.
00:51
Well, not p is false here.
00:54
So this is false and not p is false here too.
00:58
Here, not p is true, and p or q is true, and p or q is false here.
01:01
True so this is true and here p or q is false so this is false now if conditional statements are always true except for when the antecedent is true and the conclusion is false so here the antecedent is false and the conclusion is true therefore this is true the antecedent is false and the conclusion is true so this is true the antecedent is true and q is true as well so this is true and then here we have the antecedent it's false and the conclusion is also false therefore this is also true so a tautology now for b we have if if p then q and if q then r then if p then r then so i have begun by writing all the possible combinations of true values for pq and r, and we will just fill in the rest together.
02:08
So if p then q, this is true, this is true.
02:13
Here the antecedent is true and the conclusion is false.
02:16
This q is false, therefore this is false.
02:19
This is true.
02:20
This is true.
02:22
Here the antecedon is true and the conclusion is false.
02:25
So this is false, true and true.
02:29
So if q then r, it's true.
02:33
Here the antecedent is true, but the conclusion is false.
02:37
So this is false.
02:39
This is true.
02:42
This is true.
02:44
This is true.
02:46
This is false.
02:47
This q is true, but r is false.
02:49
And this is true since both of them are false.
02:55
Both q and r are false.
02:58
Therefore, this is actually true.
03:00
Now, we need both p.
03:02
If p then q and if q then r to be true for this conjunction to be true so here it's true here it's false false true true false and true then if p then r will be true if p will be true always except for when p is true and r is so this is true p is true and r is false this is false p is true and this is true this is true this is true and now we have that p is false and r is true so this is still true and we have p is true but r is false therefore this is false we have p is false they're both false so this is true and here they're also both false so this is true now we have that this conditional statement will be true is will be true always fix that for when the antecedent is true and the conclusion is false.
04:24
So here, this is true, this is true.
04:29
Here the antecedent is false, the conclusion is true, so it's true, true.
04:37
True.
04:38
Here, they're both still true, true, true, and true.
04:52
So, b is also a topology.
04:57
Now we're going to do the conditional statement if p and if p then q, then q, i have written all possible truth combinations for p and q and we will do the rest together.
05:08
So if p then q is true always except for when p is true and q is false.
05:13
So here it's true, true, all, and true.
05:19
This conjunction is true if both p and if p then q are true.
05:25
So this is true, this is false, if p is false.
05:31
This is false, since the given q is false, and this is false, since p is false...