Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 10 Logic Section 10.1 Statements Section 10.2 Conditional Statements Section 10.3 Equivalent Statements Section 10.4 Valid Arguments

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Statements Open your book to page 93. (a command) Did you enjoy the concert? (a question) What an exciting game! (an exclamation) The Sun rises in the east. (a true declarative statement) Three plus two is nine. (a false declarative statement) That was a good movie. (an ambiguous sentence) If we examine the sentences we use in daily life, we find they can be classified into several categories, which include the following: Section 10.1

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. More on Statements Statements are declarative sentences that are unambiguous and can be classified as true or false but not both. The letters p, q, r, … are used to denote statements. Statements that are constructed by combining two or more simple statements are called compound statements. A conjunction is formed from two statements that are connected using the word “and”. The notation “p  q” is used to denote “p and q” in a conjunction. Two statements connected with the word “or” forms a disjunction. A disjunction is denoted by “p  q”. A statement that means the opposite of a given statement is the negation of the given statement. The negation of statement p is denoted by ~p.

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Example a)When is your next class? b)Andrew Jackson was a great president c)In 2010, February will have 29 days. d)That was a hard test. Classify each of the following sentences as a statement or not a statement. SOLUTION a)This is a question, not a declarative sentence, so it is not a statement. b)This is an ambiguous sentence, therefore, it is not a statement. c)This is a false declarative statement, so it is a statement. d)This is not a statement because it is an exclamation. It is also ambiguous.

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Example CONJUNCTION The statement “1992 was a leap year, and the Fourth of July is a national holiday in the United States” is true because both statements “1992 was a leap year” and “The Fourth of July is a national holiday in the United States” are true. The statement “1992 was a leap year, and February has 30 days” is false because the statement “February has 30 days” is false.

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Example a)The toast is burned, and the eggs are cold. b)The cafeteria opens for lunch at 11:00 A.M. c)Students in English 102 are required to analyze two poems, or they are required to analyze two short stories. Identify each of the following statements as conjunction, disjunction, or neither. SOLUTION a)Conjunction b)Neither c)Disjunction

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Example a)p  q b)~q c)p  (~q) Let p be the statement “Susan's dog is a poodle” and let q be the statement “Jake has a black cat.” Write the following statements: SOLUTION a)Susan’s dog is a poodle, or Jake has a black cat. b)Jake does not have a black cat. c)Susan’s dog is a poodle, and Jake does not have a black cat.

HW 10.1 Pg Odd

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Conditional Statements Section 10.2 A conditional (implication) statement is a statement of the form “If p then q.” We denote it by p  q. In an implication, p  q, we call the p the hypothesis and q the conclusion.

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Example Let p be the statement “I study for hours” and q be the statement “I can make an A on the exam.” a)Write the statement p  q. b)Write the statement q  p. c)Write the statement with p as the hypothesis and q as the conclusion. SOLUTION a)If I study for hours, then I can make an A on the exam. b)If I can make an A on the exam, then I study for hours. c)If I study for hours, then I can make an A on the exam.

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Converse, Inverse, and Contrapositive From a given conditional “If p then q” we can form the following conditionals: Converse: “If q, then p.” ( q  p) Inverse: “If not p, then not q.” (~p  ~q) Contrapositive: “If not q, then not p.” (~ q  ~ p)

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Example Write the converse, inverse, and contrapositive of “If it does not rain, then we want to go on a picnic.” SOLUTION Converse: “If we want to go on a picnic, then it does not rain.” Inverse: “If it does rain, then we do not want to go on a picnic.” Contrapositive: “If we do not want to go on a picnic, then it does rain.”

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Biconditional Another fundamental compound statement is the biconditional statement denoted by p  q, read “p if and only if q.” The truth table for the biconditional statement is: Notice that p  q is true exactly when p and q have the same truth values.

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Example Find the truth table for ~p  (p  q) SOLUTION It is easier to analyze ~p  (p  q) if the truth table includes columns for ~p and p  q.

HW 10.2 Pg

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc Equivalent Statements Tautology – A compound statement that is always true Self Contradictory – A compound statement that is always false Equivalent Statements – Two statements that have exactly the same truth values

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Tautology Section 10.3 A compound statement that is always true EXAMPLE Show that the following statement is a tautology

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Self Contradictory Section 10.3 A compound statement that is always false EXAMPLE Show that the following statement is self contradictory

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Equivalent Statements Section 10.3 Two compound statements are logically equivalent, or equivalent, if they have exactly the same truth values. EXAMPLE Show that p  q and q  p are not logically equivalent. SOLUTION Form the truth table and compare the truth values to show this property. The truth values of p  q and q  p differ in rows 2 and 3, so the statements are not equivalent.

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Equivalent Statements Section 10.3 Two compound statements are logically equivalent, or equivalent, if they have exactly the same truth values. EXAMPLE Show that p  q and ~q  ~p are logically equivalent.

HW 10.3 Pg

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Valid Arguments An argument with premises p 1, p 2, p 3, …, p n and a conclusion r is valid if r is true when all of p 1, p 2, p 3, …, p n are true. Law of Detachment For any pair of statements p and q, the argument Premises: p  q p Conclusion:q is a valid argument.

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Example Premises:If Jenny has a job, she saves money. Jenny has a job. Conclusions:Jenny saves money. Is this a valid argument? Writing this in symbolic form, we have the notation p: Jenny has a job. q: Jenny saves money. Premises:p  q p Conclusion:q Notice that if both p  q and p are both true, q is also true, so the argument is valid. Check the truth table:

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Theorem An argument with premises p 1, p 2, p 3, …, p n and a conclusion r is valid, if p 1  p 2  p 3  …  p n  r Is true for all possible truth values of p 1, p 2, p 3, …, p n and r.

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Example Show that the following argument is valid: Premises:p  q q Conclusion:p SOLUTION Using the Valid Argument Theorem we can prove the argument is valid by showing (p  q)  q  p is true for all p and q. Checking the truth table: The argument is valid because only T appears in the last column.

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Four Valid Arguments Law of Detachment Premises:p  q p Conclusion:q Syllogism Premises:p  q q  r Conclusion:p  r Disjunctive Syllogism Premises:p  q ~p Conclusion:q Indirect Reasoning Premises:p  q ~q Conclusion:~p

10.4 Pg Odd