So we want to show that each of these compound propositions is a pathology. Um, that is that there they're always true regardless of the truth values of their variable. And we want to do this using a truth table. So let's begin to fill in the truth table for a which is it's not pee on peor que than Q. So I already did all the possible truth combinations for pink. You know, we'll just do the rest of the truth table. So ps truth or not pee this whole Oh true true peor que is true if either peor que or both are true So this is true, true, true and full for this when you both not p and P or cute to be true well not being sold here So this is full on not you too Here not be a fault. Luppi is true and peor que is true So this is true and here he or curious fall fall Now if conditional statements are always true except for when the at dissident is true and the conclusion is sold. So here we are interested in the fall and the conclusion is true. Therefore, this is true, the incident is full and the conclusion is true. So this is true, The interest Eden is true and Q is true as well. So this is true. And then here we have the decedent is fall and the conclusion is also false. Therefore, this is also true. So a is a pathology now for B. We have if if p s and Q And if Cuban are, then if people are so, I have begun by writing all the possible combinations of true value of her p q and R and we'll just fill in the rest together. So if even Q. This is true. This is true here the entities in this sure of the conclusion of falls the killer falls for this hole. This is true. This is true here the other day and assurance conclusion is full of this home true and true If Cuban are it's true here the entities in the troupe of the conclusion that whole fall This is true. This is true. This is true. This is true. This is false musculature but our fault and true since both of them are full both q and r. Our phone therefore this is actually true Now we need both. Pete is even Q and a Cuban are to be true for this consumption to be true. But here it's true Here it's a whole ball. True, True. Oh huh. And trip Then if people are will be true If Pete will be true always Except for when p is true and are cold So this is true history and our souls is this hole He is true This is true. This is true And now we have bad piece falls in our truth is still true And we have he is true But our souls for this hole we have peace Well, they're both fault So this is true And here they're all so beautiful. So this is true Now we have that this conditional statement will be true We'll be true Always fix up for when the antis CNN is true on the conclusion that holds So here's this is true This is true here The entity that is holding the conclusion is sure it's true True. Here's their boat Still true, true, true and true So be is also a apology. Now we're gonna do the conditional statement If p and if P and Q on Q. I have written all possible truth combinations for P and Q and Rebels is there up together. So if you think you it's true. Always except for when he is true and you it's cool. So here is true. True, All true, This convention is true if both p and F P thank you are term. So this is true. This is, 0 60 years old. This is Hawkes is disputing your sauce and this is folk. Oh, now this conditional statement is true. If it's true, always except for when the EMP decedent. It's true, but the conclusion that's full So here the anticipated this true and the conclusion is sure that this is true. It's about fault. So it's true this scene. That's true. But the conclusion is full. Ana, the antecedent is false and the conclusion is also follows. So this is actually true, and then the antecedent is home and the conclusion is also false. So this is true and we have that this is a pathology here. It's similar. I haven't all the possible truth combinations for peak human arm rules. A troll under arrest, citrus table together So peor que is true If P or two or both are true is true True, true True Oh, both you and he are full True, True Both They're both full if peace and are so is true Except for when the p is true and are full So Peter troops are true He is true And our fold this whole He is true and are true He is healthy are true Oh, it is Oh, and our truth is still true He is true and ours phone with full pay And are both folks assurance being our foothold So it's true similarly here it's always true except for when cute is true and are full Does it since true foe True, true true true falls since cues True But our souls in truth now this conjunction peor que And if p then our and Cuban are true If all three of thieves true So this is true This is Hall. This is true. This is true. This is a whole Oh oh and fall now for this conditional statement we have that if the antecedent is true But the conclusion is false and it solved otherwise it's true so here is true, since both of them are true because it's true. Since Moses on our fault, we have its truth. It wasn't more true. Both of them are true. True, uh, on to see them this false But the conclusion is true here the antecedent is true. This whole on the conclusion is true. I can't sit in this hole with conclusion is fall fantasy isn't his whole illusion holes in my here 1234 way have that the truth table is ultra regardless of the combination of truth Alistair pick you and are therefore he is also a tautology.