Ch 1.3: Quantifiers

Open sentences, or predicates, are sentences that contain one or more variables. They do not have a truth value until variables are instantiated (replaced with particular value).Example: 2x + y = 7Let P(x,y) = “2x + y = 7”What values of x, y make P(x,y) true?P(2,3) is true; P(1,5) is true; P(-2,11) is true; P(3,5) is falseTruth set: Set of values which make open sentence true.Universe: Set of values that can be considered. Note: truth set may change when the universe changes.

Truth set, universe

Example: Q(x) = “x^2 = 9”

When universe is all reals R, the truth set is {-3, 3}.

When universe is natural numbers N, truth set is{3}.

Defn: Two open sentences P(x) and Q(x) are equivalent iff they have the same truth set, given a particular universe.

Example: Let P(x) be “2x + 5 = 7” and Q(x) be “x = 1”, universe = R. Then P(x) and Q(x) are equivalent.

Universal & existential quantifiers

Definitions: Given an open sentence P(x),

universe. entire theis P(x)for set

truth when the trueis and ,P(x)" x,allfor " read, isP(x)

sentence theand ,by denoted is The

xquantifier universal

nonempty. is P(x)for set truth when the

trueis and ,P(x)"such that an x exists there" read, isP(x)

sentence theand ,by denoted is The

xquantifier lexistentia

false. is )( and true,is )( Also,

true.is )( and true,is )(Then

) (universe "0 " Q(x) and "0" P(x)Let : 22

xQxxPx

xQxxPx

xx

RExample

Universal & existential quantifiers

Example: Translate “All apples have spots” into a symbolic sentence with quantifiers. Use A(x) = “x is an apple” and S(x) = “x has spots,” universe = all fruits.

?)()(or )()( xSxAxxSxAx

spots" with applean is x fruit x, allFor " )()( xSxAx

spots" hasthen x

applean is x if fruit x, allFor " )()( xSxAx

.)()( symbolized be should

Q(x)" are P(x) All" form theof sentence a ,

xQxPx general In

Universal & existential quantifiers

Example: Translate “Some apples have spots” into a symbolic sentence with quantifiers. Use A(x) = “x is an apple” and S(x) = “x has spots,” universe = all fruits.

?)()(or )()( xSxAxxSxAx

spots" with applean isx

such that fruit x a exists There" )()( xSxAx

spots" has then x applean is x if

such that fruit x a exists There" )()( xSxAx

.)()( symbolized be should

Q(x)" are P(x) Some" form theof sentence a ,

xQxPx general In

Examples

Example: Translate “Chickens with jobs ride the bus” into a symbolic sentence with quantifiers.

Universe = all animals

bus theridesjob a has x chicken a xxx

Examples

Example: Translate “Some chickens with jobs have a car” into a symbolic sentence with quantifiers.

Universe = all animals

car a hasjob a has x chicken a xxx

Examples

Example: Translate “A function f has an inverse if different inputs give different outputs” into a symbolic sentence with quantifiers.

Universe = R

inversean has f f(y)f(x) yx yx

Examples

Example: Translate “For every natural number there is a real number greater than the natural number” into a symbolic sentence with quantifiers.

Universe = R

number real a number natural a yxyyxx

Equivalence

Definition: Two quantified sentences are equivalent for a particular universe if they have the same truth value in that universe.

Definition: Two quantified sentences are equivalent iff they are equivalent in every universe.

Example: The following quantified sentences are equivalent in N but not equivalent in R, hence they are not equivalent:

0,0 22 xxxx

Equivalence

Example: The following quantified sentences are equivalent.

)()()()(:)1( xPxQxxQxPx

)(~)(~)()(:)2( xPxQxxQxPx

Negation of Quantifiers

Theorem: For the open sentence A(x),

)(~)(~:)1( xAxxAx

)(~)(~:)2( xAxxAx

false )( true)(~)1( xAxxAx :Proof

trueis)(~ xAx

.equivalent are they hence

true,is )(~iff trueis)(~Thus xAxxAx

(1).part useely,alternativor (1),part for proof theSimilar to (2)

Examples

Example: Negate “Chickens with jobs ride the bus” (Universe = all animals)

bus theridesjob a has x chicken a ~ xxx

bus theridesjob a has x chicken a ~~ xxx

bus therides~job a has x chicken a ~~ xxx

bus theridet doesn'job a has x chicken a xxx

Examples

Example: Negate “Some chickens with jobs have a car”

(Universe = all animals)

car a hasjob a has x chicken a ~ xxx

car a has~job a has x chicken a ~ xxx

car a havet doesn' jobless x chicken anot xxx

Examples

Example: Negate “A function f has an inverse if different inputs give different outputs.” (Universe = R)

inversean has ~ f f(y)f(x) yx yx

inversean has ~~ f f(y)f(x) yx yx

inversean has ~~~ f f(y)f(x) yx yx

inversean havet doesn' f f(y)f(x) yx yx

Examples

Example: Negate “For every natural number there is a real number greater than the natural number”

(Universe = R)

number real a number natural a ~ yxyyxx

number real a number natural a ~~ yxyyxx yxyyxx number real a ~number natural a

yxyyxx ~ number real a ~number natural a

yxyyxx number real anot number natural a

Unique existence quantifier

Definition:

)."(such that x unique a exists There" read is

)( !n propositio the,)( sentenceopen an For

xP

xPxxP

universe. thefromelement oneexactly contains)(

for set truth when the trueis)( ! sentence The

xP

xPx

Examples

Nin trueis prime andeven is ! xx

RN in false isbut ,in trueis 2 ! xx

Uniqueness equivalence & negation

yxyPyxPxxPx )()()( !

:)( ! of xPxNegation

yxyPyxPxxPx )()(~)( !~

:)( ! of xPxeEquivalenc

yxyPyxPx )(~~)(~

yxyPyxPx ~)(~~)(~

yxyPyxPx )()(~

Homework

Read Ch 1.3

Do 24(1a-j,2a-j,4a-c,f,g,5a-c,f,6a-d,g,10)