Notes on Discrete Mathematics So why do I need to learn all this nasty mathematics? 1. But isn't . Functions on recursive structures. single gigantic PDF file at ruthenpress.info Basic Structures: Sets, Functions, Sequences, and Sums .. .. Discrete Structures: A course in discrete mathematics should teach students how to work. 2 Peano Axioms and Countability. Peano Axioms and the set of Natural Numbers Addition.
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Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Page 1 DISCRETE MATHEMATICAL STRUCTURES Theory ond Applications Page 2. Discrete mathematics deals with objects that come in discrete bundles, e.g.,. 1 or 2 babies. Why study discrete mathematics in computer science? It does not. Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. This tutorial has an ample amount of both theory and mathematics. The readers are PART 6: DISCRETE STRUCTURES.
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You have been selected to serve on jury duty for a criminal case. The attorney for the defense argues as follows: If my client is guilty, then the knife was in the drawer.
Either the knife was not in the drawer or Jason Pritchard saw the knife.
If the knife was not there on October 10, it follows that Jason Pritchard did not see the knife. Furthermore, if the knife was there on October 10, then the knife was in the drawer and also the hammer was in the barn.
But we all know that the hammer was not in the bam. Therefore, ladies and gentlemen of the jury, my client is innocent.
Question:Is the attorney's argument sound? How should you vote? Formal logic strips away confusing verbiage and allows us to concentrate on the underlying reasoning being applied.
In fact, formal logic-the subject of this chapter-provides the foundation for the organized, careful method of thinking that characterizes any reasoned activity-a criminal investigation, a scientific experiment, a sociological study. In addition, formal logic has direct applications in computer science. The last two sections of this chapter explore a programming language based on logic and the use of formal logic to verify the correctness of computer programs.
Also, circuit logic the logic governing computer circuitry is a direct analog of the statement logic of this chapter. We shall study circuit logic in Chapter 7. Section 1.
A statement or proposition is a sentence that is either true or false. Consider the following: Ten is less than seven. How are you?
She is very talented. There are life forms on other planets in the universe. Sentence a is a statement because it is false. Because item b is a question, it cannot be considered either true or false. It has no truth value and thus is not a statement. Sentence c is neither true nor false because "she" is not specified; therefore c is not a statement.
Sentence d U is a statement because it is either true or false; we do not have to be able to decide which. Connectives and Truth Values In English, simple statements are combined with connecting words like and to make more interesting compound statements.
The truth value of a compound statement depends on the truth values of its components and which connecting words are used. If we combine the two true statements "Elephants are big" and "Baseballs are round," we would consider the resulting statement, "Elephants are big and baseballs are round," to be true. In this book, as in many logic books, capital letters near the beginning of the alphabet, such as A, B, and C, are used to represent statements and are called statement letters; the symbol A is a logical connective representing and.
Table 1. Each row of the table represents a particular truth value assignment to the statement letters, and the resulting truth value for the compound expression is shown. TABLE 1.
The logical connective here is implication, and it conveys the meaning that the truth of A implies or leads to the truth of B. In the implication A -4 B, A stands for the antecedent statement and B stands for the consequent statement.
The truth table for implication is less obvious than that for conjunction or disjunction. To understand its definition, let's suppose your friend remarks, "If I pass my economics test, then I'll go to the movie Friday. If your friend passes the test but doesn't go to the movie, the remark was false.
If your friend doesn't pass the test, then-whether he or she goes to the movie or not-you could not claim that the remark was false. You would probably want to give the benefit of the doubt and say that the statement was true.