Statements, Connectives, and Quantifiers Section 2.2 Statements, Connectives, and Quantifiers

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.

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”

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

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

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

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.

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.

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.

Example 7: Form the negation. It is raining.

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

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.

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.

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

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

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

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 .

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.

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.

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

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

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

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.

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.

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.

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.

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)

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)

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)

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.