MATHEMATICS IN THE MODERN WORLD Lesson 2: Truth table and Tautologies In previous lesson, truth values for negation of a statement, the conjunction of two statements and the disjunction of two statement. The sum of all interior angles of a triangle is 180 degree. ES PNOTRWN> ACTIVITY: Which of the following sentences are The sum of 15 and 17 is greater than 30. ![]() oe Conjunction is two simple statements are combined with the MATHEMATICS IN THE MODERN WORLD ‘2 f \ = LPNS Tiley Nae) s) statements? Give a short reason for your answer. Disjunction For two statements p andq, p\vgq means either p org is true or both are true. e The negation of any simple statement can be formed by \ putting “not” into the statement. For more knowledge about Logic Statement and Quantifier, please check the link o ‘, provided N y’ - ~~ Sela a ¢ Compound statement is a statement formed by connecting aa two or more statements or by negating a single statement. The word some, several, one of, and part of are quantifiers which illustrate that not all but at least one object or case satisfies the given condition. The word all, any, and every are quantifiers which illustrate that each and every object or case satisfies the given condition. Quantifiers are words that denote the number of objects or cases referred to in a given statement. Example: All of the following statements have the same meaning: p ‘All students are intelligent. The quantifiers “some”, “there exist(s)”, and “at least one” are interchangeable. The quantifiers “all’, “every”, and “each” are interchangeable. English quantifiers include “all”, “none”, “some”, and “not all”. MATHEMATICS IN THE MODERN WORLD Quantifier - A quantifier is a word or phrase telling “how many”. ![]() Negation - The negation of any simple statement can be formed by putting “not” into the statement. A truth table is a table that shows the truth value of a compound statement for all possible cases. The truth value of a statement is either true (denoted by T) or false (denoted by F). A truth table gives the statement's truth value, 7 (true) or F (false), for all possible values of its variable. We can use the truth table to specify each of these compound statements. These statements are read as not p, p or q, p and q, if p then q, respectively. If p and q are statements, then the compound statements ~ P» PVG, Pq, andp>q can be formed using the negation symbol ~ and connectives ~, A and >. Example: “Karel is beautiful and Loren is selfish.” “Karel is beautiful’, “Loren is selfish” joined by the connectives and. The words or phrases (or symbols) used to form compound statements are called connectives. It is formed by joining two propositions. Compound Statements Is a statement formed by connecting two or more statements or by negating a single statement. In mathematical treatment of logic, we avoid an unclear meaning, ambiguous situations, differences of opinions by assuming a statement as either true or false. If p denotes the statement “Zamboanga City is located at the tip of Zamboanga Peninsula”, then instead of saying that the above statement is true or false, we can simply represent value of pas F. A PON> MATHEMATICS IN THE MODERN WORLD To avoid writing the statements in words all the time, we usually label statements with lowercase letters p,q,r. ![]() This is not a statement, we are not told what y is. This is a statement, although few of us can say whether it is true or false. What is the exchange rate from United States dollars to pesos? Solution: This is a statement. The price of a Samsung Galaxy tablet was Php 11, 000 on December 24, 2014. Example: Which of the following are statements? For each statement identified, discuss whether it is true or false. ![]() A statement cannot be a question, an instruction, or an opinion. Statement - ls adeclarative sentence that can be meaningfully classified as either true or false. |In the study of logic or mathematics, we are concerned with only one type of sentence, that is, we assume that we are able to recognize a declarative sentence (or statement) and to form an opinion as to whether it is true or false. Construct and us truth tables to show that statements are equivalent Lesson 1: Logic Statement and Quantifiers English sentences may be classified as declarative, interrogative, exclamatory or imperative. Identify whether a compound proposition is tatulogy or not c. Determine whether or not a sentence is a statement b. Download Mathematics in Modern World Chapter 6 - Logic and more Mathematics Papers in PDF only on Docsity!MATHEMATICS IN THE MODERN WORLD CHAPTER 6: Logic Objectives: a.
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