1. Statements, Connectives, and Quantifiers
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 .
notion

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

23. Using the Dominance of Connectives
Statement
Most Dominant Connective Highlighted in Red
Statement’s Meaning Clarified with Grouping Symbols
Type of Statement
p q ~r
p q ~r
p (q ~r)
Conditional
p q  ~r
(p q)  ~r
p  q  r
p  (q  r)
Biconditional
p q  r
(p q)  r
p q r **
and have the same level of dominance
The meaning is ambiguous ?
**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: t (~s)
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 w represent the following statements: ~(s t)
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 w represent 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.