Section 3.1 Statements and Logical Connectives
What You Will Learn Statements, quantifiers, and compound statements Statements involving the words not, and, or, if then, and if and only if 3.1-2
HISORY he Greeks: Aristotelian logic: he ancient Greeks were the first people to look at the way humans think and draw conclusions. Aristotle (384-322 B.C.) is called the father of logic. his logic has been taught and studied for more than 2000 years. 3.1-3
Mathematicians Gottfried Wilhelm Leibniz (1646-1716) 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. 3.1-4
Mathematicians George Boole (1815 1864) is said to be the founder of symbolic logic because he had such impressive work in this area. Charles Dodgson, better known as Lewis Carroll, incorporated many interesting ideas from logic into his books. 3.1-5
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 3.1-6
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. 3.1-7
Statements and Logical Connectives 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 and can be assigned a truth value. 3.1-8
Negation of a Statement Negation of a statement change a statement to its opposite meaning. he negation of a false statement is always a true statement. he negation of a true statement is always a false statement. 3.1-9
Quantifiers Quantifiers - words such as all, none, no, some, etc Be careful when negating statements that contain quantifiers. 3.1-10
Negation of Quantified Statements orm of statement All are. None are. Some are. Some are not. orm of negation Some are not. Some are. None are. All are. 3.1-11
Negation of Quantified Statements All are. Some are. None are. Some are not. 3.1-12
Example 1: Write Negations Write the negation of the statement. Some telephones can take photographs. Solution Since some means at least one this statement is true. he negation is No telephones can take photographs, which is a false statement. 3.1-13
3.1-14 Example 1: Write Negations Write the negation of the statement. All houses have two stories. Solution his is a false statement, since some houses have one story, some three or more. he negation Some houses do not have two stories or Not all houses have two stories or At least one house does not have two stories are all true statements.
Compound Statements Statements consisting of two or more simple statements are called compound statements. he connectives often used to join two simple statements are and, or, if then, and if and only if. 3.1-15
Not Statements (Negation) he symbol used in logic to show the negation of a statement is ~. It is read not. he negation of p is: ~ p. 3.1-16
And Statements (Conjunction) is the symbol for a conjunction and is read and. he conjunction of p and q is: p q. he other words that may be used to express a conjunction are: but, however, and nevertheless. 3.1-17
Example 2: Write a Conjunction Write the following conjunction in symbolic form: Green Day is not on tour, but Green Day is recording a new CD. 3.1-18
Example 2: Write a Conjunction Solution Let t and r represent the simple statements. t: Green Day is on tour. r: Green Day is recording a new CD. In symbolic form, the compound statement is ~t r. 3.1-19
Or Statements (Disjunction) he disjunction is symbolized by and read or. In this book the or will be the inclusive or (except where indicated in the exercise set). he disjunction of p and q is: p q. 3.1-20
Example 3: Write a Disjunction Let p: Maria will go to the circus. q: Maria will go to the zoo. Write the statement in symbolic form. Maria will go to the circus or Maria will go the zoo. Solution p q 3.1-21
Example 3: Write a Disjunction Let p: Maria will go to the circus. q: Maria will go to the zoo. Write the statement in symbolic form. Maria will go to the zoo or Maria will not go the circus. Solution q ~p 3.1-22
Compound Statements When a compound statement contains more than one connective, a comma can be used to indicate which simple statements are to be grouped together. When we write the compound statement symbolically, the simple statements on the same side of the comma are to be grouped together within parentheses. 3.1-23
Example 4: Understand How Commas Are Used to Group Statements Let p: Dinner includes soup. q: Dinner includes salad. r: Dinner includes the vegetable of the day Write the statement in symbolic form. Dinner includes soup, and salad or vegetable of the day. Solution p (q r) 3.1-24
Example 4: Understand How Commas Are Used to Group Statements Let p: Dinner includes soup. q: Dinner includes salad. r: Dinner includes the vegetable of the day Write the statement in symbolic form. Dinner includes soup and salad, or vegetable of the day. Solution (p q) r 3.1-25
Example 5: Change Symbolic Statements into Words Let p: he house is for sale. q: We can afford to buy the house. Write the symbolic statement in words. Solution p ~q he house is for sale and we cannot afford to buy the house. 3.1-26
Example 5: Change Symbolic Statements into Words Let p: he house is for sale. q: We can afford to buy the house. Write the symbolic statement in words. Solution ~p ~q he house is not for sale or we cannot afford to buy the house. 3.1-27
Example 5: Change Symbolic Statements into Words Let p: he house is for sale. q: We can afford to buy the house. Write the symbolic statement in words. Solution ~(p q) It is false that the house is for sale and we can afford to buy the house. 3.1-28
If-hen Statements he conditional is symbolized by and is read if-then. he antecedent is the part of the statement that comes before the arrow. he consequent is the part that follows the arrow. If p, then q is symbolized as: p q. 3.1-29
Example 6: Write Conditional Statements Let p: he portrait is a pastel. q: he portrait is by Beth Anderson. Write the statement symbolically. If the portrait is a pastel, then the portrait is by Beth Anderson. Solution p q 3.1-30
Example 6: Write Conditional Statements Let p: he portrait is pastel. q: he portrait is by Beth Anderson. Write the statement symbolically. If the portrait is by Beth Anderson, then the portrait is not a pastel. Solution q ~p 3.1-31
Example 6: Write Conditional Statements Let p: he portrait is pastel. q: he portrait is by Beth Anderson. Write the statement symbolically. It is false that if the portrait is by Beth Anderson, then the portrait is a pastel. Solution 3.1-32 ~(q p)
If and Only If Statements he biconditional is symbolized by and is read if and only if. If and only if is sometimes abbreviated as iff. he statement p q is read p if and only if q. 3.1-33
Example 8: Write Statements Using the Biconditional Let p: Alex plays goalie on the lacrosse team. q: he itans win the Champion s Cup. Write the symbolical statement in words. p q Solution Alex plays goalie on the lacrosse team if and only if the itans win the Champion s Cup. 3.1-34
Example 8: Write Statements Using the Biconditional Let p: Alex plays goalie on the lacrosse team. q: he itans win the Champion s Cup. Write the symbolical statement in words. q ~p Solution he itans win the Champion s cup if and only if Alex does not play goalie on the lacrosse team. 3.1-35
Example 8: Write Statements Using the Biconditional Let p: Alex plays goalie on the lacrosse team. q: he itans win the Champion s Cup. Write the symbolical statement in words. ~(p ~q) Solution It is false that Alex plays goalie on the lacrosse team if and only if the itans do not win the Champion s Cup. 3.1-36
Logical Connectives 3.1-37
Section 3.2 ruth ables for Negation, Conjunction, and Disjunction
What You Will Learn ruth tables for negations, conjunctions, and disjunctions 3.2-39
ruth able A truth table is used to determine when a compound statement is true or false. 3.2-40
Negation ruth able p ~p Case 1 Case 2 3.2-41
Compound Statement ruth able p q Case 1 Case 2 Case 3 Case 4 3.2-42
Conjunction ruth able p q p q Case 1 Case 2 Case 3 Case 4 he conjunction is true only when both p and q are true. 3.2-43
Disjunction ruth able p q p q Case 1 Case 2 Case 3 Case 4 he disjunction is true when either p is true, q is true, or both p and q are true. 3.2-44
Negation Negation ~p is read not p. If p is true, then ~p is false; if p is false, then ~p is true. In other words, ~p will always have the opposite truth value of p. 3.2-45
Conjunction Conjunction p q is read p and q. p q is true only when both p and q are true 3.2-46
Disjunction Disjunction p q is read p or q. p q is true when either p is true or q is true, or both p and q are true. In other words, p q is false only when both p and q are false. 3.2-47
Constructing ruth ables 1. Determine if the statement is a negation, conjunction, disjunction, conditional, or biconditional. he answer to the truth table appears under: ~ if it is a negation if it is a conjunction if it is a disjunction if it is conditional if it is biconditional 3.2-48
Constructing ruth ables 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 pair first. 3.2-49
Constructing ruth ables 3. Complete the column under the connective within the parentheses, if present. You will use the truth values of the connective in determining the final answer in step 5. 3.2-50
Constructing ruth ables 4. Complete the column under any remaining statements and their negation. 3.2-51
Constructing ruth ables 5. Complete the column under any remaining connectives. he answer will appear under the column determined in step 1. or 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. 3.2-52
Constructing ruth ables 5. (continued) or a negation, negate the values of the last column completed within the grouping symbols on the right of the negation. Circle or highlight the answer column and number the columns in the order they were completed. 3.2-53
Example 3: ruth able with a Negation Construct a truth table for ~(~q p). 3.2-54
Example 3: ruth able with a Negation Construct a truth table for ~(~q p). Solution p q ~ (~q p) 4 alse only when p is false and q is true. 1 3 2 3.2-55
Example 7: Use the Alternative Method to Construct a ruth able Construct a truth table for ~p ~q. 3.2-56
Example 7: Use the Alternative Method to Construct a ruth able Solution Construct a truth table with four cases. p q 3.2-57
Example 7: Use the Alternative Method to Construct a ruth able Solution Add a column for ~p ~q. Use columns ~p and ~q to find ~p ~q. p q ~p ~q ~p ~q It is true only when ~p and~q are true. 3.2-58
Example 9: Determine the ruth Value of a Compound Statement Determine the truth value for each simple statement. hen, using these truth values, determine the truth value of the compound statement. 15 is less than or equal to 9. 3.2-59
Example 9: Determine the ruth Value of a Compound Statement Solution Let p: 15 is less than 9. q: 15 is equal to 9. Express 15 is less than or equal to 9 as p q. Both p and q are false. p q 3.2-60
Example 9: Determine the ruth Value of a Compound Statement Determine the truth value for each simple statement. hen, using these truth values, determine the truth value of the compound statement. George Washington was the first U.S. president or Abraham Lincoln was the second U.S. president, but there has not been a U.S. president born in Antarctica. 3.2-61
Example 9: Determine the ruth Value of a Compound Statement Solution Let p: George Washington was the first U.S. president. q: Abraham Lincoln was the second U.S. president. r: here has been a U.S. president who was born in Antarctica. he statement can be written in symbolic form as (p q) ~r. 3.2-62
Example 9: Determine the ruth Value of a Compound Statement Solution p: George Washington was the first U.S. president. q: Abraham Lincoln was the second U.S. president. r: here has been a U.S. president who was born in Antarctica. he statement is (p q) ~r. p is true, q is false, r is false. Since r is false, ~r is true. 3.2-63
Example 9: Determine the ruth Value of a Compound Statement Solution he statement is (p q) ~r. p is true, q is false, ~r is true. (p q) ~r ( ) he original compound statement is true. 3.2-64
Section 3.3 ruth ables for the Conditional and Biconditional
What You Will Learn ruth tables for conditional and biconditional Self-contradictions, autologies, and Implications 3.3-66
his image cannot currently be displayed. Conditional p q p q Case 1 Case 2 Case 3 Case 4 he conditional statement p q is true in every case except when p is a true statement and q is a false statement. 3.3-67
Example: ruth able with a Conditional Construct a truth table for the statement ~p ~q. 3.3-68
Example: ruth able with a Conditional Solution Construct a standard four case truth table. p q ~p ~q 1 It s a conditional, the answer lies under. 3 2 3.3-69
Biconditional he biconditional statement, p q means that p q and q p or, symbolically (p q) (q p). case 1 case 2 case 3 case 4 order of steps p q (p 1 3 q) 2 7 (q 4 6 p) 5 3.3-70
Biconditional he biconditional statement, p q is true only when p and q have the same truth value, that is, when both are true or both are false. 3.3-71
Example 4: A ruth able Using a Biconditional Construct a truth table for the statement ~p (~q r). 3.3-72
Example 4: A ruth able Using a Biconditional p q r ~p (~q r) 2 3 1 4 3.3-73 5
Example 7: Using Real Data in Compound Statements he graph on the next slide represents the student population by age group in 2009 for the State College of lorida (SC). Use this graph to determine the truth value of the following compound statements. 3.3-74
Example 7: Using Real Data in Compound Statements 3.3-75
Example 7: Using Real Data in Compound Statements If 37% of the SC population is younger than 21 or 26% of the SC population is age 21 30, then 13% of the SC population is age 31 40. 3.3-76
Example 7: Using Real Data in Compound Statements Solution Let p: 37% of the SC population is younger than 21. q: 26% of the SC population is age 21 30. r: 13% of the SC population is age 31 40. Original statement can be written: (p q) r 3.3-77
Example 7: Using Real Data in Compound Statements Solution Original statement: (p q) r p and r are true, q is false ( ) he original statement is true. 3.3-78
Example 7: Using Real Data in Compound Statements 3% of the SC population is older than 50 and 8% of the SC population is age 41 50, if and only if 19% of the SC population is age 21 30. 3.3-79
Example 7: Using Real Data in Compound Statements Solution Let p: 3% of the SC population is older than 50. q: 8% of the SC population is age 41 50. r: 19% of the SC population is age 21 30. Original statement can be written: (p q) r 3.3-80
Example 7: Using Real Data in Compound Statements Solution Original statement: (p q) r p and q are true, r is false ( ) he original statement is false. 3.3-81
Self-Contradiction A self-contradiction is a compound statement that is always false. When every truth value in the answer column of the truth table is false, then the statement is a self-contradiction. 3.3-82
Example 8: All alses, a Self- Contradiction Construct a truth table for the statement (p q) (p ~q). 3.3-83
Example 8: All alses, a Self- Contradiction Solution he statement is a self-contradiction or a logically false statement. 3.3-84
autology A tautology is a compound statement that is always true. When every truth value in the answer column of the truth table is true, the statement is a tautology. 3.3-85
Example 9: All rues, a autology Construct a truth table for the statement (p q) (p r). 3.3-86
Example 9: All rues, a autology Solution he statement is a tautology or a logically true statement. 3.3-87
Implication An implication is a conditional statement that is a tautology. he consequent will be true whenever the antecedent is true. 3.3-88
Example 10: An Implication? Determine whether the conditional statement [(p q) q] q is an implication. 3.3-89
Example 10: An Implication? Solution he statement is a tautology, so it is an implication. 3.3-90