Section 2.2Statements, Connectives, and Quantifiers

Objectives1. 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.

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 .

Key Terms•Dominance of Connectives – symbolic

connectives are categorized from least dominant to most dominant.▫Least dominant – Negation

Conjunction/Disjunction ConditionalMost dominant – Biconditional

Using the Dominance of Connectives

StatementMost Dominant

Connective Highlighted in

Red

Statement’s Meaning Clarified

with Grouping Symbols

Type of Statement

p q ~r p q ~r p (q ~r) Conditionalp q ~r p q ~r (p q) ~r Conditionalp q r p q r p (q r) Biconditionalp q r p q r (p q) r Biconditionalp 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:▫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: 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.