Rosen and published by Unknown which was released on 09 July with total pages We cannot guarantee that Discrete Mathematics and Its Applications book is available in the library, click Get Book button to download or read online books. Join over Rosen's Discrete Mathematics and its Applications presents a precise, relevant, comprehensive approach to mathematical concepts. This world-renowned best-selling text was written to accommodate the needs across a variety of majors and departments, including mathematics, computer science, and engineering.
As the market leader, the book is highly flexible, comprehensive and a proven pedagogical teaching tool for instructors. 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. But these are equivalent by the commutative and associative laws.
An conditional statement in which the conclusion is true or the hypothesis is false is true, and that completes the argument. We can let p be true and the other two variables be false. We apply the rules stated in the preamble. The table is in fact displayed so as to exhibit the duality.
The two identity laws are duals of each other, the two domination laws are duals of each other, etc. The statement of the problem is really the solution. Each line of the truth table corresponds to exactly one combination of truth values for the n atomic propositions involved. We can write down a conjunction that is true precisely in this case, namely the conjunction of all the atomic propositions that are true and the negations of all the atomic propositions that are false.
If we do this for each line of the truth table for which the value of the compound proposition is to be true, and take the disjunction of the resulting propositions, then we have the desired proposition in its disjunctive normal form. This exercise is similar to Exercise One such assignment is T for p and F for q and r.
To say that p and q are logically equivalent is to say that the truth tables for p and q are identical; similarly, to say that q and r are logically equivalent is to say that the truth tables for q and r are identical. Clearly if the truth tables for p and q are identical, and the truth tables for q and r are identical, then the truth tables for p and r are identical this is a fundamental axiom of the notion of equality. Therefore p and r are logically equivalent. We are assuming—and there is no loss of generality in doing so—that the same atomic variables appear in all three propositions.
If q is true, then the third and fourth expressions will be true, and if r is false, the last expression will be true. In each case we hunt for truth assignments that make all the disjunctions true. This text is designed for students preparing for future coursework in areas such as math, computer science, and engineering. Discrete Mathematics and Its Applications has become a best-seller largely due to how effectively it addresses the main portion of the discrete market, which is typically characterized as the mid to upper level in rigor.
The strength of Rosen's approach has been the effective balance of theory with relevant applications, as well as the overall comprehensive nature of the topic coverage. This concise, undergraduate-level text focuses on combinatorics, graph theory with applications to some standard network optimization problems, and algorithms. More than exercises, many with complete solutions. Rosen has become a bestseller largely due to how effectively it addresses the main portion of the discrete market, which is typically characterized as the mid to upper level in rigor.
Judith Gersting's Mathematical Structures for Computer Science has long been acclaimed for its clear presentation of essential concepts and its exceptional range of applications relevant to computer science majors. Discrete Mathematics and its Applications, Seventh Edition, is intended for one- or two-term introductory discrete mathematics courses taken by students from a wide variety of majors, including computer science, mathematics, and engineering.
This renowned best-selling text, which has been used at over institutions around the world, gives a focused introduction to the primary themes in a discrete mathematics course and demonstrates the relevance and practicality of discrete mathematics to a wide a wide variety of real-world applications Digital is becoming increasingly important and gaining popularity, crowning Connect as the digital leader for this discipline.
McGraw-Hill Education's Connect, available as an optional, add on item. Problems are randomized to prevent sharing of answers and may also have a "multi-step solution" which helps move the students' learning along if they experience difficulty.
Coverage emphasizes the major themes of discrete mathematics as well as the reasoning that underlies mathematical thought. Students learn to think abstractly as they study the ideas of logic and proof.
Answers to ODD numbered problems are in the back of the book. Rosen's Discrete Mathematics and its Applications presents a precise, relevant, comprehensive approach to mathematical concepts.
This world-renowned best-selling text was written to accommodate the needs across a variety of majors and departments, including mathematics, computer science, and engineering. As the market leader, the book is highly flexible, comprehensive and a proven pedagogical teaching tool for instructors.
This text is designed for students preparing for future coursework in areas such as math, computer science, and engineering. Discrete Mathematics and Its Applications has become a best-seller largely due to how effectively it addresses the main portion of the discrete market, which is typically characterized as the mid to upper level in rigor.
The strength of Rosen's approach has been the effective balance of theory with relevant applications, as well as the overall comprehensive nature of the topic coverage. This concise, undergraduate-level text focuses on combinatorics, graph theory with applications to some standard network optimization problems, and algorithms. More than exercises, many with complete solutions.
Rosen has become a bestseller largely due to how effectively it addresses the main portion of the discrete market, which is typically characterized as the mid to upper level in rigor. Judith Gersting's Mathematical Structures for Computer Science has long been acclaimed for its clear presentation of essential concepts and its exceptional range of applications relevant to computer science majors.
Discrete Mathematics and its Applications, Seventh Edition, is intended for one- or two-term introductory discrete mathematics courses taken by students from a wide variety of majors, including computer science, mathematics, and engineering. This renowned best-selling text, which has been used at over institutions around the world, gives a focused introduction to the primary themes in a discrete mathematics course and demonstrates the relevance and practicality of discrete mathematics to a wide a wide variety of real-world applications Digital is becoming increasingly important and gaining popularity, crowning Connect as the digital leader for this discipline.
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