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Material Taken From: Mathematics for the international student Mathematical Studies SL Mal Coad, Glen Whiffen, John Owen, Robert Haese, Sandra Haese and Mark Bruce Haese and Haese Publications, 2004 Chapter 15A - Propositions Mathematical Logic Converting worded statements into symbols, then applying rules of deduction. Example of deductive reasoning: •All teachers are poor. •I am a teacher. •By using logic, it follows that I am poor. • Logic, unlike natural language, is precise and exact. • Logic is useful in computers and artificial intelligence where there is a need to represent the problems we wish to solve using symbolic language. BrainPop – Binary Video For each of these statements, list the students for which the statement is true: a) I am wearing a green shirt. b) I am not wearing a green shirt c) I am wearing a green shirt and green pants. d) I am wearing a green shirt or green pants. e) I am wearing a green shirt or green pants, but not both. Propositions Statements which may be true or false. • Page 496 in the text. • Questions are not propositions. • Comments or opinions are not propositions. • Example: ‘Green is a nice color’ is subjective; it is not definitely true or false. • Propositions may be indeterminate. • Example: ‘your father is 177 cm tall’ would not have the same answer (true or false) for all people. • The truth value of a proposition is whether it is true or false. Example 1 Which of the following statements are propositions? If they are propositions, are they true, false, or indeterminate? a) b) c) d) 20 4 = 80 25 × 8 = 200 Where is my pen? Your eyes are blue. Notation • We represent propositions by letters such as p, q and r. • For example: – p: It always rains on Tuesdays. – q: 37 + 9 = 46 – r: x is an even number. Negation • The negation of a proposition p is “not p” and is written as ¬p. • The truth value of ¬p is the opposite of the truth value of p. • For example: p: It is Monday. ¬p: It is not Monday. q: Tim has brown hair. ¬q: Tim does not have brown hair. Truth Tables • Using the example: – p: It is Monday. – ¬p: It is not Monday. p T F ¬p F T ¬(¬p) T F Example 2 Find the negation of: a) x is a dog for x {dogs, cats} b) x ≥ 2 for x N c) x ≥ 2 for x Z Section 15B - Compound Propositions Compound propositions Statements which are formed using ‘and’ or ‘or.’ • ‘and’ conjunction – notation: p q • ‘or’ disjunction – notation: p q Conjunction vs. Disjunction Examples Conjunction p: Eli had soup for lunch q: Eli had a pie for lunch. p q: Disjunction p: Frank played tennis today q: Frank played golf today. p q: • p q is true if one or both propositions are true. • p q is only true if both original propositions are true. • p q is false only if both propositions are false. Conjunction/Disjunction and Truth Tables p T T F F q T F T F pq pq T T F T F T F F Conjunction/Disjunction and Venn Diagrams Suppose P is the truth set of p, and Q is the truth set of q. PQ P Q the truth set for pq is PQ the truth set for pq is PQ U PQ Examples 3 and 4 Write p q for the following : p: Kim has brown hair, q: Kim has blue eyes Determine whether p q is true or false: p: A square has four sides, q: A triangle has five sides Examples 5 and 6 Write the disjunction p q for p: x is a multiple of 2, q: x is a multiple of 5. Determine whether p q is true or false p: There are 100 in a right angle, q: There are 180 on a straight line. Exclusive Disjunction Is true when only one of the propositions is true. • notation: p q • p q means “p or q, but not both” • For example, – p: Sally ate cereal for breakfast – q: Sally ate toast for breakfast pq p T T F F q T F T F pq F T T F Exclusive Disjunction pq • In Logic ‘or’ is usually given in the inclusive sense. – “p or q or both” • If the exclusive disjunction is meant, then it’ll be stated. – “p or q, but not both’ or “exactly one of p or q” Example 7 Write the exclusive disjunction pq p: Ann will invite Kate to her party, q: Ann will invite Tracy to her party. for Examples 8 and 9 Consider r: Kelly is a good driver, and s: Kelly has a good car. • Write in symbolic form: a) Kelly is a good driver and has a good car. b) Kelly is not a good driver or has a good car. Consider x: Sergio would like to go swimming tomorrow, and y: Sergio would like to go bowling tomorrow • Write in symbolic form: – Sergio would not like to go both swimming and bowling tomorrow. Example 10 Define appropriate propositions and then write in symbolic form: – Phillip likes ice cream or Phillip does not like Jell-O – Homework (from 2nd edition) • 17A.1 (every other problem) • #1, #2, #4, #5 • 17B.1 (every other problem) • #1, #2 • 17B.2 • #1ac, #2ad, #3a, #6ace, #7aeg, #11