Section 2.2

1. Section 2.2 Statements, Connectives, and Quantifiers

2. Objectives • Identify English sentences that are statements. • Express statements using symbols. • Form the negation of a statement. • Express negations using symbols. • Translate a negation represented by symbols into English. • Express quantified statements in two ways. • Write negations of quantified statements.

3. Key Terms: • Statement: a sentence that is either true/false, but not both; symbolized by lowercase letters such as: p, q, r, and s. • Simple Statement: contains a single idea. • Compound Statement: contains several ideas combined together. • Connectives: the words used to join the ideas of a compound statement. • Connectives: not, and, or, if…then, if and only if • Negation: a statement that has a meaning that is opposite its original meaning, symbolized by ~p. • ~p: read as “not p”

4. Example 1: • Determine if the sentence is a statement. • As a young and struggling artist, Pablo Picasso kept warm by burning his own paintings.

5. Example 2: • Determine if the sentence is a statement. • Don’t try to study on a Friday night.

6. Example 3: • Determine if the sentence is a statement. • Is the unexamined life worth living?

7. Example 4: • Identify each statement as a simple or compound. If compound, then identify the connective used. • Laura is satisfied with her performance in the musical.

8. Example 5: • Identify each statement as a simple or compound. If compound, then identify the connective used. • If Hillary supports environmental issues, she will succeed in politics.

9. Example 6: • Identify each statement as a simple or compound. If compound, then identify the connective used. • I will sell my old computer and buy a new computer.

10. Example 7: • Form the negation. • It is raining.

11. Example 8: • Form the negation. • The Dallas Cowboys are not the team with the most Super Bowl wins.

12. Example 9: • Let p, q, r, and s represent the following statements: • p: One works hard. • q: One succeeds. • r: The temperature outside is not freezing. • s: It is not true that the heater is working. • Express the following statement symbolically. • One does not work hard.

13. Example 10: • Let p, q, r, and s represent the following statements: • p: One works hard. • q: One succeeds. • r: The temperature outside is not freezing. • s: It is not true that the heater is working. • Express the following statement symbolically. • The temperature outside is freezing.

14. Example 11: • Let p, q, r, and s represent the following statements: • p: Listening to classical music makes infants smarter. • q: Subliminal advertising makes you buy things. • r: Sigmund Freud’s father was not 20 years older than his mother. • Represent each symbolic statement in words. • ~p

15. Example 12: • Let p, q, r, and s represent the following statements: • p: Listening to classical music makes infants smarter. • q: Subliminal advertising makes you buy things. • r: Sigmund Freud’s father was not 20 years older than his mother. • Represent each symbolic statement in words. • ~r

16. Section 2.2 Assignments • TB pg. 85/1 – 20 All • Must write problems and show ALL work to receive credit for the assignment.

17. Key Terms • Quantified Statements – statements containing the words “all”, “some”, and “no (or none)”. • Universal Quantifiers– words such as all and every that state that all objects of a certain type satisfy a given property, symbolized by . • Existential Quantifiers – words such as some, there exists, and there is at least one that state that there are one or more objects that satisfy a given property, symbolized by .

18. Negating Statements w/ Quantifiers • The phrase Not all are has the same meaning as Some are not. • The phrase Not some are has the same meaning as All are not.

19. Example 13: Quantifiers • Rewrite the quantified statement in an alternative way and then negate it. • All citizens over age eighteen have the right to vote.

20. Example 14: Quantifiers • Rewrite the quantified statement in an alternative way and then negate it. • Some computers have a two-year warranty

21. Key Terms • Conjunction – expresses the idea of and, symbolized by . • Disjunction – conveys the notion of or, symbolized by . • Conditional – expresses the notion of if…then, symbolized by . • Biconditional – represents the idea of if and only if, symbolized by .

22. Key Terms • Dominance of Connectives – symbolic connectives are categorized from least dominant to most dominant. • Least dominant – Negation Conjunction/Disjunction ConditionalMost dominant – Biconditional

23. Using the Dominance of Connectives **Grouping symbols must be given with this statement to determine if it is a disjunction or a conjunction.

24. Example 15: • Let r, t, and s represent the following statements: • r: The Republicans will control Congress. • s: Social programs will be increased. • t: Taxes will be cut. • The Republicans will control Congress or social programs will not be increased.

25. Example 16: • Let r, t, and s represent the following statements: • r: The Republicans will control Congress. • s: Social programs will be increased. • t: Taxes will be cut. • If the Republicans do not control Congress and taxes are cut, then social programs will not be increased.

26. Example 17: • Let r, t, and s represent the following statements: • r: The Republicans will control Congress. • s: Social programs will be increased. • t: Taxes will be cut. • Social programs will not be increased if and only if taxes are cut.

27. Example 18: • Let s, t, and w represent the following statements: • s: The sunroof is extra. • t: The radial tires are included. • w: Power windows are optional. • t (~s)

28. Example 19: • Let s, t, and wrepresent the following statements: • s: The sunroof is extra. • t: The radial tires are included. • w: Power windows are optional. • ~(s t)

29. Example 20: • Let s, t, and wrepresent the following statements: • s: The sunroof is extra. • t: The radial tires are included. • w: Power windows are optional. • t  (s ~w)

30. Section 2.2 Assignment II • Classwork: • TB pg. 86/21 – 32 All • Remember you must write the problems and show ALL work to receive credit for this assignment.