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## Chapter 1 - Section 1.1 - Propositional Logic - Exercises - Page 12: 1

Log In Sign Up. Download Free PDF. Ian Seepersad. Download PDF. A short summary of this paper. Section 1. Note that we were able to incorporate the parentheses by using the words either and else.

This has been slightly reworded so that the tenses make more sense. Contrapositive: If I do not stay at home, then it will not snow tonight. Inverse: If it does not snow tonight, then I will not stay home. Contrapositive: Whenever I do not go to the beach, it is not a sunny summer day. Inverse: Whenever it is not a sunny day, I do not go to the beach. Contrapositive: If I do not sleep until noon, then I did not stay up late. A truth table will need 2n rows if there are n variables.

To construct the truth table for a compound proposition, we work from the inside out. In each case, we will show the intermediate steps.

For parts a and b we have the following table column three for part a , column four for part b. For parts a and b we have the following table column two for part a , column four for part b.

This time we have omitted the column explicitly showing the negation of q. It is irrelevant that the condition is now false. This cannot be a proposition, because it cannot have a truth value. Indeed, if it were true, then it would be truly asserting that it is false, a contradiction; on the other hand if it were false, then its assertion that it is false must be false, so that it would be true—again a contradiction.

Thus this string of letters, while appearing to be a proposition, is in fact meaningless. This is a classical paradox. We will use the male pronoun in what follows, assuming that we are talking about males shaving their beards here, and assuming that all men have facial hair. If we restrict ourselves to beards and allow female barbers, then the barber could be female with no contradiction. If such a barber existed, who would shave the barber?

If the barber shaved himself, then he would be violating the rule that he shaves only those people who do not shave themselves. On the other hand, if he does not shave himself, then the rule says that he must shave himself.

Neither is possible, so there can be no such barber. Note that we can make all the conclusion true by making a false, s true, and u false. Thus the system is consistent. This system is consistent. This requires that both L and Q be true, by the two conditional statements that have B as their consequence.

Note that there is just this one satisfying truth assignment. This is similar to Example 17, about universities in New Mexico. If A is a knight, then his statement that both of them are knights is true, and both will be telling the truth.

But that is impossible, because B is asserting otherwise that A is a knave. Thus we conclude that A is a knave and B is a knight. We can draw no conclusions. A knight will declare himself to be a knight, telling the truth. A knave will lie and assert that he is a knight. If Smith and Jones are innocent and therefore telling the truth , then we get an immediate contradiction, since Smith said that Jones was a friend of Cooper, but Jones said that he did not even know Cooper.

If Jones and Williams are the innocent truth-tellers, then we again get a contradiction, since Jones says that he did not know Cooper and was out of town, but Williams says he saw Jones with Cooper presumably in town, and presumably if we was with him, then he knew him. Therefore it must be the case that Smith and Williams are telling the truth. Their statements do not contradict each other. Therefore Jones is the murderer.

Can none of them be guilty? If so, then they are all telling the truth, but this is impossible, because as we just saw, some of the statements are contradictory. Can more than one of them be guilty?

If, for example, they are all guilty, then their statements give us no information. So that is certainly possible. This information is enough to determine the entire system. Let each letter stand for the statement that the person whose name begins with that letter is chatting.

Note that we were able to convert all of these statements into conditional statements. In what follows we will sometimes make use of the contrapositives of these conditional statements as well.

First suppose that H is true. Then it follows that A and K are true, whence it follows that R and V are true. But R implies that V is false, so we get a contradiction. Therefore H must be false. From this it follows that K is true; whence V is true, and therefore R is false, as is A. We can now check that this assignment leads to a true value for each conditional statement. There are four cases to consider.

If Alice is the sole truth-teller, then Carlos did it; but this means that John is telling the truth, a contradiction. If John is the sole truth-teller, then Diana must be lying, so she did it, but then Carlos is telling the truth, a contradiction. If Carlos is the sole truth-teller, then Diana did it, but that makes John truthful, again a contradiction. So the only possibility is that Diana is the sole truth-teller.

This means that John is lying when he denied it, so he did it. Note that in this case both Alice and Carlos are indeed lying. Since Carlos and Diana are making contradictory statements, the liar must be one of them we could have used this approach in part a as well.

Therefore Alice is telling the truth, so Carlos did it. Note that John and Diana are telling the truth as well here, and it is Carlos who is lying.

There are two cases. Therefore the two propositions are logically equivalent. We see that the fourth and seventh columns are identical. For part a we have the following table. We argue directly by showing that if the hypothesis is true, then so is the conclusion.

An alternative approach, which we show only for part a , is to use the equivalences listed in the section and work symbolically. Then p is false. To do this, we need only show that if p is true, then r is true. Suppose p is true. It now follows from the second part of the hypothesis that r is true, as desired.

Then p is true, and since the second part of the hypothesis is true, we conclude that q is also true, as desired. If p is true, then the second part of the hypothesis tells us that r is true; similarly, if q is true, then the third part of the hypothesis tells us that r is true. Thus in either case we conclude that r is true. This is not a tautology. It is saying that knowing that the hypothesis of an conditional statement is false allows us to conclude that the conclusion is also false, and we know that this is not valid reasoning.

Since this is possible only if the conclusion if false, we want to let q be true; and since we want the hypothesis to be true, we must also let p be false. It is easy to check that if, indeed, p is false and q is true, then the conditional statement is false.

Therefore it is not a tautology. The second is true if and only if either p and q are both true, or p and q are both false. Clearly these two conditions are saying the same thing. We determine exactly which rows of the truth table will have T as their entries. The conditional statement will be true if p is false, or if q in one case or r in the other case is true, i. Since the two propositions are true in exactly the same situations, they are logically equivalent.

## Solution Manual of Discrete Mathematics and its Application by Kenneth H Rosen

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## Solutions Manual (even) For Discrete Mathematics And Its Applications (7th Edition)

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Конечно. Но я думаю, что одно с другим может быть связано самым непосредственным образом. Сьюзан отказывалась его понимать. - Это долгая история. Она повернулась к монитору и показала на работающего Следопыта.

### Chapter 1 - Section 1.1 - Propositional Logic - Exercises - Page 12: 1

Предмет, который она держала, был гораздо меньшего размера. Стратмор опустил глаза и тут же все понял. Время для него остановилось. Он услышал, как стучит его сердце. Человек, в течение многих лет одерживавший победу над опаснейшими противниками, в одно мгновение потерпел поражение. Причиной этого стала любовь, но не .

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#### Sample Solutions for this Textbook

Но в следующее мгновение послышался оглушающий визг шин, резко затормозивших на цементном полу, и шум снова накатил на Сьюзан, теперь уже сзади. Секунду спустя машина остановилась рядом с. - Мисс Флетчер! - раздался изумленный возглас, и Сьюзан увидела на водительском сиденье электрокара, похожего на те, что разъезжают по полям для гольфа, смутно знакомую фигуру. - Господи Иисусе! - воскликнул водитель.  - С вами все в порядке.

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За конторкой с надписью КОНСЬЕРЖ сидел вежливый подтянутый мужчина, улыбающийся так приветливо, словно всю жизнь ждал минуты, когда сможет оказать любезность посетителю.

Он подумал, дома ли Сьюзан. Куда она могла уйти. Неужели уехала без меня в Стоун-Мэнор. - Эй! - услышал он за спиной сердитый женский голос и чуть не подпрыгнул от неожиданности. - Я… я… прошу прощения, - заикаясь, сказал Беккер и застегнул молнию на брюках.

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