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Section 2.2 Statements, Connectives, and Quantifiers

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Objectives 1.Identify English sentences that are statements. 2.Express statements using symbols. 3.Form the negation of a statement. 4.Express negations using symbols. 5.Translate a negation represented by symbols into English. 6.Express quantified statements in two ways. 7.Write negations of quantified statements.

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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”

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

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

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

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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.

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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.

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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.

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Example 7: Form the negation. It is raining.

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

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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.

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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.

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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

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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

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

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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 .

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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.

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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.

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

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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 .

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

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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) Conditional p q ~r (p q) ~r Conditional p q r p (q r) Biconditional p q r (p q) r Biconditional 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.

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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.

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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.

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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.

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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)

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Example 19: 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. ~(s t)

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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)

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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.

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