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Prove that the following sequents cannot be valid:(i) (P = Q) v ~ R F (~ Q^R) = P(ii) P = Q, R=~ SF (PV R) = (Qv S)Let W1, Wz,W be wffs in Propositional Logic.Prove...

Question

Prove that the following sequents cannot be valid:(i) (P = Q) v ~ R F (~ Q^R) = P(ii) P = Q, R=~ SF (PV R) = (Qv S)Let W1, Wz,W be wffs in Propositional Logic.Prove that W,Wz FW < F (W = (Wz = W)):Prove that | (Wi = (Wz = W)) < > W1,Wz F W .

Prove that the following sequents cannot be valid: (i) (P = Q) v ~ R F (~ Q^R) = P (ii) P = Q, R=~ SF (PV R) = (Qv S) Let W1, Wz,W be wffs in Propositional Logic. Prove that W,Wz FW < F (W = (Wz = W)): Prove that | (Wi = (Wz = W)) < > W1,Wz F W .



Answers

If $a^{\frac{1}{x}}=b^{\frac{1}{y}}=c^{\frac{1}{z}}$ and $a, b, c$ be in G.P. then prove that $x, y, z$ are in A.P.

It's there. So when you moved here, So we're going to draw truth tables. Mmm. They look like this. Me for proposition you And it's his team two. Herbie, we have kina room T Oh, you all the go, Uh true and falls hurt. See, it's similar to part B, but it's going to be end instead of war. And this gives us tall faults in party. The truth tier consists p chain you take Mr Song. Port E says p he or I mean balls. True phones. Finally he too holds true. Holds true, sure.

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 question it is given there a to the power X. Is equal to B to the power Y. Is equal to C. To the power jet and abc RN GP. Then we have to prove that Xy Jack are in HP. Mhm slate eight of the power X is equal to we to repower. Why is equal to C to the power jet? Is it cold Turkey? When? Okay. Italy Power X is equal to K. Then is equal to K. To the power one upon. It's consider this as a question one when B to the power why is equal to K. Yeah. Then musical too. Get to the power one upon Why consider this as equation too Public then shoot the power Jed is equal to key. Thank you. Then C. Is equal to care to the power one upon gent. Consider this. Believe it as a question three mr No mm B. See yeah uh N G P. Therefore the square is equal to S. E. Non certitude. The value of a BNC from a christian 12 and three. We are being stop care to the power one upon X. Into. Okay. To the power one of one. Inject get to the power two upon. Why is it clear to k to the power one upon tax plus one upon Jared because A to the power em into a to the power and is equal to eight to the power mm. Plys and therefore to Y two upon why is equal to one upon X plus one A phone yet therefore once upon X one upon white one upon jade. This was not are in A. P. Mhm. Therefore X why Jag R N H. P approved. It's just professional. Okay.

In this question it is even there a minus X upon P X. Is equal to a minus Y upon cuba is equal to a minor jet upon our object. And speak you are in ap then we have to prove that Xy jack are in HP click. Uh huh. A minus X upon P X. Is equal to a minus Y upon cuba. I is equal to a minus joint upon object is equal to K. Well that's different When a minus tax upon PX. Is it cold turkey critics then B is equal to a minus X upon. Yeah. Tex Consider this as a question one. Mhm. When? Yeah. Mhm. A minus white upon Q. Y is equal to K. His mother Just then Q is equal to a minus Y upon Ky consider this as a question two. Mhm. When? Yeah. Yeah A minus jag. A fallen. Our jag is equal to K then or is he going to a minus jet upon gauge it. Consider this as equation three Given B Q. Ah in epi therefore two Q. Is equal to B. Bless our substitute the value of P. Q. And R. From a question one, two and three to into a minus Y Upon Ky is equal to a minus X upon giggs plus a minus jet upon gidget two into a upon ky minus light upon the way is equal to a upon gxe minus ex upon dicks plus a upon okay aged minus just a phone kinetic to a upon Ky minus do upon key is equal to a upon K X -1 upon key plus a upon K. Jade minus once upon key to a upon giving -2 upon key is equal to he upon giggs plus. AARP on Yeah Cajun minus to a con. Okay, Credified comment on from both sides -2 by K. Mhm. To a upon the way is equal to taking common A upon key. Show yourself one upon x plus. Mhm. Well upon injured cancel comment on a upon white from both sides. Two upon wine. Physical, too one upon x plus well upon jake. Therefore, once upon X well upon wine one upon jade adding AP therefore, X. Why Jed R n h P two?


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