Chapter 3 Section 1 - Slide 1 Copyright © 2009 Pearson Education, Inc. AND

Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 1 - Slide 2 Chapter 3 Logic

Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 1 - Slide 3 WHAT YOU WILL LEARN Statements, quantifiers, and compound statements Statements involving the words not, and, or, if… then…, and if and only if Truth tables for negations, conjunctions, disjunctions, conditional statements, and biconditional statements

Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 1 - Slide 4 Section 1 Statements and Logical Connectives

Chapter 3 Section 1 - Slide 5 Copyright © 2009 Pearson Education, Inc. HISTORY—The Greeks: Aristotelian logic: The ancient Greeks were the first people to look at the way humans think and draw conclusions. Aristotle ( B.C.) is called the father of logic. This logic has been taught and studied for more than 2000 years.

Chapter 3 Section 1 - Slide 6 Copyright © 2009 Pearson Education, Inc. Mathematicians Gottfried Wilhelm Leibniz ( ) believed that all mathematical and scientific concepts could be derived from logic. He was the first to seriously study symbolic logic. In this type of logic, written statements use symbols and letters. George Boole (1815 – 1864) is said to be the founder of symbolic logic because he had such impressive work in this area.

Chapter 3 Section 1 - Slide 7 Copyright © 2009 Pearson Education, Inc. Logic and the English Language Connectives - words such as and, or, if, then Exclusive or - one or the other of the given events can happen, but not both. Inclusive or - one or the other or both of the given events can happen.

Chapter 3 Section 1 - Slide 8 Copyright © 2009 Pearson Education, Inc. Statements and Logical Connectives Statement - A sentence that can be judged either true or false.  Labeling a statement true or false is called assigning a truth value to the statement. Simple Statements - A sentence that conveys only one idea and can be assigned a truth value. Compound Statements - Sentences that combine two or more simple statements.

Chapter 3 Section 1 - Slide 9 Copyright © 2009 Pearson Education, Inc. Negation of a Statement Negation of a statement – change a statement to its opposite meaning.  The negation of a false statement is always a true statement.  The negation of a true statement is always a false statement.

Chapter 3 Section 1 - Slide 10 Copyright © 2009 Pearson Education, Inc. Quantifiers Quantifiers - words such as all, none, no, some, etc… Be careful when negating statements that contain quantifiers.

Chapter 3 Section 1 - Slide 11 Copyright © 2009 Pearson Education, Inc. Negation of Quantified Statements Form of statement All are. None are. Some are. Some are not. Form of negation Some are not. Some are. None are. All are. None are. Some are not. All are. Some are.

Chapter 3 Section 1 - Slide 12 Copyright © 2009 Pearson Education, Inc. Example: Write Negations Write the negation of the statement. Some candy bars contain nuts. Solution:Since some means “at least one” this statement is true. The negation is “No candy bars contain nuts,” which is a false statement.

Chapter 3 Section 1 - Slide 13 Copyright © 2009 Pearson Education, Inc. Example: Write Negations continued Write the negation of the statement. All tables are oval. Solution:This is a false statement since some tables are round, rectangular, or other shapes. The negation would be “Some tables are not oval.”

Chapter 3 Section 1 - Slide 14 Copyright © 2009 Pearson Education, Inc. Compound Statements Statements consisting of two or more simple statements are called compound statements. The connectives often used to join two simple statements are and, or, if…then…, and if and only if.

Chapter 3 Section 1 - Slide 15 Copyright © 2009 Pearson Education, Inc. Not Statements (Negation) The symbol used in logic to show the negation of a statement is ~. It is read “not”.  Example: The negation of p is: ~ p.

Chapter 3 Section 1 - Slide 16 Copyright © 2009 Pearson Education, Inc. And Statements (Conjunction) is the symbol for a conjunction and is read “and.” The other words that may be used to express a conjunction are: but, however, and nevertheless.  Example: The conjunction of p and q is: p ^ q.

Chapter 3 Section 1 - Slide 17 Copyright © 2009 Pearson Education, Inc. Example: Write a Conjunction Write the following conjunction in symbolic form: The dog is gray, but the dog is not old. Solution: Let p and q represent the simple statements. p: The dog is gray. q: The dog is old. In symbolic form, the compound statement is p Λ ~ q

Chapter 3 Section 1 - Slide 18 Copyright © 2009 Pearson Education, Inc. Or Statements (Disjunction)

Chapter 3 Section 1 - Slide 19 Copyright © 2009 Pearson Education, Inc. Example: Write a Disjunction Write the statement in symbolic form. Carl will not go to the movies or Carl will not go to the baseball game. Solution: Let p and q represent the simple statements. p: Carl will go to the movies. q: Carl will go to the baseball game. In symbolic form, the compound statement is

Chapter 3 Section 1 - Slide 20 Copyright © 2009 Pearson Education, Inc. If-Then Statements (continued) The conditional is symbolized by and is read “if-then.”  The antecedent is the part of the statement that comes before the arrow.  The consequent is the part that follows the arrow.  Example: If p, then q is symbolized as: p  q.

Chapter 3 Section 1 - Slide 21 Copyright © 2009 Pearson Education, Inc. Example: Write a Conditional Statement Let p: Nathan goes to the park. q: Nathan will swing. Write the following statements symbolically. a.If Nathan goes to the park, then he will swing. b.If Nathan does not go to the park, then he will not swing. Solutions a) b)

Chapter 3 Section 1 - Slide 22 Copyright © 2009 Pearson Education, Inc. If and Only If Statements (Biconditional) The biconditional is symbolized by and is read “if and only if.” If and only if is sometimes abbreviated as “iff.” The biconditional p q is the conjunction of the two conditionals p  q and q  p: p q = (p  q) (q  p)

Chapter 3 Section 1 - Slide 23 Copyright © 2009 Pearson Education, Inc. Example: Write a Statement Using the Biconditional Let p: The dryer is running. q: There are clothes in the dryer. Write the following symbolic statements in words. a) b) Solutions: a.There are clothes in the dryer if and only if the dryer is running. b.It is false that the dryer is running if and only if there are no clothes in the dryer.

Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 1 - Slide 24 Section 2 Truth Tables for Negation, Conjunction, and Disjunction

Chapter 3 Section 1 - Slide 25 Copyright © 2009 Pearson Education, Inc. Truth Table A truth table is used to determine when a compound statement is true or false.

Chapter 3 Section 1 - Slide 26 Copyright © 2009 Pearson Education, Inc. Negation Truth Table TCase 2 F FCase 1 T ~p p

Chapter 3 Section 1 - Slide 27 Copyright © 2009 Pearson Education, Inc. Conjunction Truth Table The conjunction is true only when both p and q are true. FFFCase 4 FTFCase 3 FFTCase 2 TTTCase 1 qp

Chapter 3 Section 1 - Slide 28 Copyright © 2009 Pearson Education, Inc. Disjunction The disjunction is true when either p is true, q is true, or both p and q are true. FFFCase 4 TTFCase 3 TFTCase 2 TTTCase 1 qp

Chapter 3 Section 1 - Slide 29 Copyright © 2009 Pearson Education, Inc. General Procedure for Constructing Truth Tables 1.Determine if the statement is a negation, conjunction, disjunction, conditional, or biconditional. The answer to the truth table appears under: ~ if it is a negation B if it is a conjunction O if it is a disjunction  if it is conditional  if it is biconditional

Chapter 3 Section 1 - Slide 30 Copyright © 2009 Pearson Education, Inc. General Procedure for Constructing Truth Tables (continued) 2.Complete the columns under the simple statements, p, q, r, and their negations ~p, ~q, ~r, within parentheses, if present. If there are nested parentheses work with the innermost first. 3.Complete the column under the connective within parentheses, if present. You will use the truth values of the connective in determining the final answer in step 5. 4.Complete the column under any remaining statements and their negation.

Chapter 3 Section 1 - Slide 31 Copyright © 2009 Pearson Education, Inc. General Procedure for Constructing Truth Tables (continued) 5.Complete the column under any remaining connectives. Answer will appear under the column determined in step 1. For a conjunction, disjunction, conditional or biconditional, obtain the value using the last column completed on the left side and on the right side of the connective. For a negation, negate the values of the last column completed within the grouping symbols on the right of the negation. Number the columns in the order they were completed.

Chapter 3 Section 1 - Slide 32 Copyright © 2009 Pearson Education, Inc. Example: Truth Table with a Negation and Conjunction Construct a truth table for ~(~q p). Solution: Construct a standard four case truth table. pq~(~qBp)p) TTFFTTFF TFTFTFTF T F T T F T F T F T F F T T F F Then fill-in the table in order, as follows: False only when p is true and q is false