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1.. Find all Lhe truth values for p, 4 and that Inakc: the LHS of this logical imnplication GTI: [(p 8 4) €r] ^ (p ~ > 4) = p V q. Is Ghis vallid implicati...

Question

1.. Find all Lhe truth values for p, 4 and that Inakc: the LHS of this logical imnplication GTI: [(p 8 4) €r] ^ (p ~ > 4) = p V q. Is Ghis vallid implication? Why o why not?Prove by verhal argument (Io truth table) thatP ^ (q (r) ==7 (p ^q) V (p Ar)-Can this implication be reversed Lo conclude Ghat thesc arc logically equivalent? If give prOof; if not explain why nol

1.. Find all Lhe truth values for p, 4 and that Inakc: the LHS of this logical imnplication GTI: [(p 8 4) €r] ^ (p ~ > 4) = p V q. Is Ghis vallid implication? Why o why not? Prove by verhal argument (Io truth table) that P ^ (q (r) ==7 (p ^q) V (p Ar)- Can this implication be reversed Lo conclude Ghat thesc arc logically equivalent? If give prOof; if not explain why nol



Answers

Use truth tables to verify these equivalences.
$\begin{array}{ll}{\text { a) } p \wedge \mathbf{T} \equiv p} & {\text { b) } p \vee \mathbf{F} \equiv p} \\ {\text { c) } p \wedge \mathbf{F} \equiv \mathbf{F}} & {\text { d) } p \vee \mathbf{F} \equiv \mathbf{T}} \\ {\text { e) } p \vee p \equiv p} & {\text { f) } p \wedge p \equiv p}\end{array}$

In this problem we will be looking at logical equivalence is and recall that two statements are logically equivalent if in all possible cases, um they have the same truth values. So the first logical equivalence that we're looking at is this logical equivalence right here where P. And true is logically equivalent to just people. And so the only variable in this logical equivalence. In fact in all of the logical equivalence is we will be looking at for this problem is P. So in the truth table we first write our variables and P is our variable which has two possibilities true and false. So then let's look at what happens when we have P. And true. So P when it's true the statement P and true becomes true and true. And recall that if you have an and statement, if both operators are true it's true, otherwise false. So if it's true and true, then it would be true. Now when peace false, the statement will become false and true. Which again if you have an end statement, both operators need to be true for the whole statement to be true. In that case it would be false. And it's pretty easy to see at the truth table right now that in fact in all possible cases P. And true has the same the truth value as trustee. And therefore it is logically equivalent to P. Um The next uh logical equivalence we're looking at is if P or false is logically equivalent to P. So let's look at what happens when we have P. Or false. So when peace true the statement becomes true or false. And if you recall in an or statement or a disjunction only one opera and needs to be true or the whole statement to be true. So when it's true or false then um the whole statement is true. Now on the other hand, if it's if P is false and we have false or false, both operations are false and so in an or with an or operator um this evaluates to false and it's clear to see that um in all of the possible cases P or phones has the same truth value as trustee. So this uh is logical equivalent as we can see through the truth table. The next logical equivalence we're looking at is if P and false is logically equivalent of false. And that just means that in all possible cases P and false evaluates to false. So let's look at P and false. So P and false when P is true, the whole statement becomes true and false. Uh And again and and for an end operator, both operators need to be true for the whole statement to be true. So true and false only has one opera and which is true. Therefore this is false. Uh and likewise, if P is false, we have false and false and neither of the operations are true, so it's false. And as we can see that in all possible cases P and false is false. Therefore it's logically equivalent to false the next state. The next logical equivalence, we're looking at S. P or true is logically equivalent to true. And this just means that if in all possible cases P or true is evaluates true. So when P is true, the statement becomes true or true and in an oar operator, but if one operation is true, then the whole statement is true. Therefore, if we have true or true, both operators are true. And so this evaluates to true. Now when peace false, we have false or true. And since the or operator only needs one operator to be true for the whole statement to be true then false or true is true. And as we can see that P or true always evaluates the true and it's therefore logically equivalent to true. Um The next logical equivalent. So we're looking at is P. Or P is logically equivalent to P. And so when peace true, this statement evaluates to true or true since both our parents that your operator is true, it's R E. S P. And because it's both true, then the uh this statement evaluates to true Now when peace false again both operations of the operators piece so it will be false or false. And since the or operator needs at least one operation to be true for the statement to be true, this evaluates to false. And finally we are looking at if P and P is logically equivalent to be So in the case where peace true again, both operations here is P. So that would be true and true. And since true about since the and operator evaluates the true when both operates are true, then this would be true and um when P is false, the statement would be false and false. And since neither um of the two operations is true and an and operator needs both operators to be true for the whole statement to be true, then PNP would evaluate to false. And it's clear to see that PNP evaluated the same truth values as just P in all possible cases. And so we have shown that P and P is logically equivalent to P and that's the end of this problem. Thanks for watching.

In this problem, we are given that T. Is a tautology and for any proposition P. So he could be true or false. We'd like to determine the truth value of this. Maybe it's always true, maybe it's always false. And one way to do this is to make a truth table. You can see if it's always true or false. So P could be either show or false T. Because he's a tautology, regardless of P is sure false, he is always true. So let's be careful there. He is not arbitrary proposition that can be true or false. Like P. Okay, next up the C. P and T. That's only true when both T and T. Or true. Otherwise if at least one of these two is false, then the end is also false. And then finally the proposition that we want which is P and T implies peak. Now here, if you look at a proposition of this form A implies B. This is true if is false or if both A and B are true. So here, if if the A. Is false, then I automatically can say that this is true regardless of being. So if I come over here, this is my A. That's my B. I see that my A. Is false here, so that makes this true over here. I see that my A. Is true, but my baby is also true. So I do follow these cases here and we can see at the end that our proposition is always true. So we'll conclude, and this is regardless of whether or not P. Itself is sure balls.

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.

In this problem, we are given that he is a tautology, which means that he is always true. And we want to determine the truth value of the proposition. P implies P and T. So here's one way to do this. You can use the truth table for P. P. S. And arbitrary proposition so it can be true or false. However, since T is a tautology, it always has to be true. Okay, next up we have P E n T. That's only true. If both P and T. Or true, otherwise it's false. And then finally, we have our implication here and our final column. Okay, now, no. Yeah, if you have an implication of the form A implies B. This is true in the two cases below the hypothesis is false or if both A and B are true, otherwise your implications falls. So let's use that over here with A. And B. So here, if I look at P if I see my A is false, I'm in this case which is over here. So automatically, the implications through in the first row, my A. Is true, but my baby is also true. So I'm in the second case. And as a result, we can see that are final proposition is always true, so we conclude that this proposition it's true, and that does not depend on whether P is sure falls.


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