Aim: What is the conjunction and disjunction of the truth?

1. Aim: What is the conjunction and disjunction of the truth? Do Now: True or False Math is a 4-letter word and 4th period starts at 10:25 am. George Washington was the first president of the United States and John Lennon is alive. T F

2. Compound Statement A compound statement is formed by combining simple statements with one or more operators. Conjunction “and” is and operator and is represented symbolically as ˄ Math is a 4-letter word and 4th period starts at 10:25 am. T George Washington was the first president of the United States and John Lennon is alive. F

3. Conjunction - Symbolically Let p represent the simple statement “The earth is a spherical ball.” Let q represent the simple statement “The earth is the third planet in our solar system.” T T The earth is a spherical ball and the earth is the third planet in our solar system. p ˄ q T

4. Symbolic Let p represent the simple statement “Kobe Bryant is a basketball player.” Let q represent the simple statement “Derek Jeter is a basketball player.” T F Kobe Bryant is a basketball player and Derek Jeter is a basketball player. p ˄ q F

5. Evaluating the Truth Let p represent the simple statement “The Mets are a baseball team based in Los Angeles.” Let q represent the simple statement “The Jets are a football team based in Chicago.” F F The Mets are a baseball team based in Los Angeles and The Jets are a football team based in Chicago. p ˄ q F

6. Truth Table - Conjunction If p and q represent simple statements then the truth table for compound statements with the ‘and’ operator (Conjunction) is

7. Evaluating the Truth Let p represent “Staten Island is a borough of NYC.” Let q represent “SI is connected to Brooklyn via the Verrazano Bridge.” Let r represent “SI has a greater population than Brooklyn.” T T F What is the truth value of the following? a) p ˄ r b) p ˄ ~r c) ~p ˄ ~r d) p ˄ q ˄ ~r

8. T T T F F T F F Model Problem Construct the truth table for the following: ~( ~p ˄ q ) ~p ˄ q ~p ~(~p ˄ q) F F T F F T T T F T F T

9. Disjunction A compound statement is formed by combining simple statements with one or more operators. Disjunction “or” is and operator and is represented symbolically as ˅ Home plate is 60’ 6” from the pitching mound or a football field is 100 yards long. T First base is 90 feet from home plate or a football field is 100 feet long. T The Petrides school is located in Brooklyn or October follows November in the calender year. F

10. Truth Table - Disjunction If p and q represent simple statements then the truth table for compound statements with the ‘or’ operator (Disjunction) is

11. Symbols  Words Let m represent “The grass is green.” Let n represent “The hit was a single.” Let s represent “The first baseman made the out.” Rewrite: The grass is green or the hit was a single. a) m ˅ n The hit was not a single or the first baseman made the out. b) ~n ˅ s The hit was not a single or the first baseman did not make the out. c) ~n ˅ ~s

12. Order of Operations ~(n ˄ m)? negation of the statement ‘n and m” Parentheses indicate order of operations. Ex. n is a true statement and c is a false statement. What is the truth value of the compound statement (n ˅ c) ˄ ~(n ˄ c) (T ˅ F) ˄ ~(T ˄ F) T ˄ ~(F) T ˄ T T

13. T T T F F T F F Model Problem Construct a truth table for the following compound statement: ~(p˄ q) ˄ [(p ˅ q) ˄ q] p ˄ q ~(p ˄ q) p ˅ q [(p˅q)˄q] ~(p˄q)˄[(p˅q)˄q] F T F T T F F T T F T F T T T F F T F F

14. Model Problem • Suppose p is T and q if F. Find the truth values for the following statements. • Let p: I eat spinach. q: I am strong. • ~(p ˄ q) • ~p ˄ q