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DISCRETE MATH: SHOW WORK If needed SIMPLIFY if needed:Given that p = T, q = F, and r = T. Find the following truth valuesa.) ¬(𝑞𝑞∧𝑟...

Question

DISCRETE MATH: SHOW WORK If needed SIMPLIFY if needed:Given that p = T, q = F, and r = T. Find the following truth valuesa.) ¬(𝑞𝑞∧𝑟𝑟)→𝑝𝑝b.) 𝑞𝑞 ↔(𝑝𝑝 ∧ 𝑟𝑟)

DISCRETE MATH: SHOW WORK If needed SIMPLIFY if needed: Given that p = T, q = F, and r = T. Find the following truth values a.) ¬(𝑞𝑞∧𝑟𝑟)→𝑝𝑝 b.) 𝑞𝑞 ↔(𝑝𝑝 ∧ 𝑟𝑟)



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.

Hey, it's Claire. So when you're in and we're gonna see if these are technologies so going of our definition of proposition is a tautology if the proposition is always true. So it's true for any combination of truth values for the variables peak, you are an exception. So we look at a we have. This is all gonna bloom you have. True for the first column row. Excuse me. And if this is true, falls, the Mrs Paul's and True Falls for P true for Q and this hospital falls, then P and Q. If P and Q then P has to be true. And if these air falls, the last one is true. So this one, we could see that it is a tautology because the last column off the fallen tree table contains only one team for a part B. These are true then these both have to be true. Since this is end p N Q. If and then then p This is true falls. It must be true. True, false, true, true, True. And if this is falls, then all of them are false and stuff for the last one. So it is a tautology because the last column of the falling tree table contains only one t for this one. But these are true since falls true true truth balls not Pius falls If p then cues falls this is true falls true. The rest are gonna be true Left pmk your falls then the rest are true. This is also a tautology For the same reason last column of the falling tree table continues only one true on our feet d the first column B They're true and the rest are gonna be true. This is true balls for Q and Mrs Falls Walls been true balls and true P and Q Since Pius falls and swells true, true. And then these air falls then end this falls. The rest are true. This is also a tautology Part e. I'm gonna use this table here. Soldiers air race Mmm Yes. This eso for this table. You finished everything. This is for part E. We have a peek your horse then we have If p thank you we have not If p thank you we have not If p thank you that p and for a part f I'll use the table below here U P Q mrs if P then. Q. This is not P. Not if P then not. Q. Mmm, it's this part. I could probably keep true here, so and we have not. Q. And not if. P then Q. Then knock you doing and blue again. So we have 1st 2 are true. Then if you think you is true, falls true your true and falls balls True. True? Yeah, Falls Trail And we are true balls True. Then we have the last one. They're both falls. We have true falls. True. And this one is a tautology since the last Colin has only one t but the last one part of it these are cheer than this one has to be true, not give p thank you's falls on its balls. True 2nd 1 doesn't ask you falls in true True True If it's false, untrue. That's true. Falls not cues falls true. This is true bones. True, True. Last one is a tautology

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.

Okay, so we wanna verify the associative ball. We will begin by verifying that the left side is equivalent to the right side of the block by using a truth table. If the left side on the right side have the same truth table, then that means that they are logically equivalent. So I have began by writing all the possible combinations of truth values for P Q and R. Now we will begin. Why don't dress so peor Que is true? It's either peor que or both are true right here, troll. True, true, True. Oh, both pink your fall again True, since P is true, True and oh, true now you're accused or are true is either pure Q r r or both True hope You're curious trip for all of these and that automatically makes pure cure are true. Now Pierre or cure falls here But our is true. So this is true. And here Pierre accused health and are also holds So this is Oh, here are true Excuse Drew are true or both killing All true. This is true. This is true here, Kim True, both cumin all true are true. Both cured off falls here and she was true here. And both Q and R Full here to told now pee or cure our truest piotre or cure our true secure are a stripper all the 1st 5 minutes on a medically put the truth and then here cure our is true. But he is true. So true. Hear cure our true again true and here cure are all and also our fault. That is, we can observe that P. R Q or our has the same truth table at p. Your cure are so therefore they are logically equivalent. Now we will do the second part, which is P and Cube, and our is logically Colin to P and Q and R. So once again I have. So then all the possible combinations of truth value for human arm we will do over so P and Q is true on Lee when p and cure both true otherwise home. So this is true here. Q is full. So it's all he is all here. Both pick your true. Both are all Pete is full. He was full and both are full p and Q. And our is true when P and Q is true and when our true. So here p and Q. And are all true European cues all fall. Hear pinchers True but are false is full for all the rest. It's false. P into a hole. Humans are true Q. And are both true. So here it's true here for fall. They're both true here. True here are false. Full here to fall p. I mean our fall. Those are false and both our phones here. So killing our true here and P is also true human. Our fault here and Q and R is true here and but p this fall. So this fall and for all of these Q and R false, This cannot be true. So P and Q and R has the same. Does not have interest table up. Oh, I accidentally wrote, True for all of them when I meant fault because Q and R are all falls here. So no, we can see that P and Q and R have the same truth table for all possible combinations of P Q and are therefore they are logically equivalent. Therefore, we have verified both associate of law


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