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

Converting worded statements into symbols, 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: I am wearing a green shirt. I am not wearing a green shirt I am wearing a green shirt and green pants. I am wearing a green shirt or green pants. 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.

Which of the following statements are propositions? Example 1 Which of the following statements are propositions? If they are propositions, are they true, false, or indeterminate? 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. F

Find the negation of: x is a dog for x  {dogs, cats} x ≥ 2 for x  N Example 2 Find the negation of: x is a dog for x  {dogs, cats} x ≥ 2 for x  N x ≥ 2 for x  Z Example 2 on page 532 of 2nd edition Skip example 3 (and corresponding practice problems) on page 533 of 2nd edition

Compound propositions Statements which are formed using ‘and’ or ‘or.’ 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 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 false only if both propositions are false. p: Eli had soup for lunch q: Eli had a pie for lunch. p  q: p  q is only true if both original propositions are true.

Conjunction/Disjunction and Truth Tables p q p  q p  q T F

Conjunction/Disjunction and Venn Diagrams Suppose P is the truth set of p, and Q is the truth set of q. P  Q the truth set for pq is PQ P Q the truth set for pq is PQ U P  Q

Write p  q for the following : p: Kim has brown hair, 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 These examples are from Haese & Harris, Math Studies, 2nd edition, Exercise 17B.1, #1d, #2b

Determine whether p  q is true or false Examples 5 and 6 Determine whether p  q is true or false p: There are 100 in a right angle, q: There are 180 on a straight line. Write the disjunction p  q for p: x is a multiple of 2, q: x is a multiple of 5. These examples are from Haese & Harris, Math Studies, 2nd edition, Exercise 17B.2 #1b, #2b

means “p or q, but not both” For example, Exclusive Disjunction Is true when only one of the propositions is true. notation: means “p or q, but not both” For example, p: Sally ate cereal for breakfast q: Sally ate toast for breakfast p q p  q T F

Exclusive Disjunction 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”

Write the exclusive disjunction for Example 7 Write the exclusive disjunction for p: Ann will invite Kate to her party, q: Ann will invite Tracy to her party. These examples are from Haese & Harris, Math Studies, 2nd edition, Exercise 17B.2 #3b

Consider r: Kelly is a good driver, and s: Kelly has a good car. Examples 8 and 9 Consider r: Kelly is a good driver, and s: Kelly has a good car. Write in symbolic form: Kelly is a good driver and has a good car. 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 Sergio would not like to go both swimming and bowling tomorrow. These examples are from Haese & Harris, Math Studies, 2nd edition, Exercise 17B.2 #5b, #6d

Define appropriate propositions and then write in symbolic form: Example 10 Define appropriate propositions and then write in symbolic form: Phillip likes ice cream or Phillip does not like Jell-O – These examples are from Haese & Harris, Math Studies, 2nd edition, Exercise 17B.2 #7b

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