Chapter1

Chapter1 - Chapter 1: The Logic of Compound Statements...

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Unformatted text preview: Chapter 1: The Logic of Compound Statements January 7, 2008 Outline 1 1.1 Logical Form and Logical Equivalence 2 1.2 Conditional Statements 3 1.3 Valid and Invalid Arguments Central notion of deductive logic: argument form Argument: sequence of statements whose goal is to establish the truth of an assertion The assertion at the end of the sequence is called the conclusion while the preceding statements in an argument are called premises The goal of an argument is to show that the truth of the conclusion follows necessarily from the truth of the premises. Example If x is a real number such that x <- 5 or x > 5, then x 2 > 25. Since x 2 6 > 25 then x 6 <- 5 and x 6 > 5. We can introduce the letters p , q , and r to represent statements that occur within our argument: If p or q , then r . Therefore, if not r , then not p and not q . Example Fill in the blanks in the argument (b) so that it has the same form as the argument (a). Then, write the common form of the argument using letters to replace the individual statements. (a) If it rains today or I have a lot of work to do, I wont go for a walk. I have a lot of work to do. Therefore, I wont go for a walk. (b) If MTH314 is easy or then . I will study hard. Therefore, I will get an A in this course. Solution: 1 I will study hard. 2 I will get an A in this course. Common form of the arguments: If p or q , then r . q Therefore, r . Statements Definition A statement (or, proposition ) is a sentence that is true or false but not both. Examples 1 The area of the circle of radius r is r 2 is a statement. So is sin ( / 2 ) = 0. The first is a true statement, while the second one is false. 2 x + 2 y is not a statement; namely, for some values of x and y , e.g. x = 1, y = 2, it is true, while for some other values (e.g. x =- 1, y = 2), it is false. Compound Statements We want to build more complex logical expressions starting with simple statements. We will introduce three new logical symbols ( connectives ): 1 (NOT) 2 (AND) 3 (OR) p means not p or It is not the case that p and is called the negation of p . p q means p and q and is called the conjunction of p and q . p q means p or q and is called the disjunction of p and q . In logical expressions, the symbol is evaluated before or , since it binds statements in a stronger way than conjunction or disjunction. (This is similar to the fact that, in arithmetic, we evaluate multiplication before + or -) For example, we can simplify the expression ( p ) q as p q p but q is often translated as p and q ....
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This note was uploaded on 04/17/2008 for the course MTH 314 taught by Professor Forgot during the Spring '08 term at Ryerson.

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Chapter1 - Chapter 1: The Logic of Compound Statements...

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