Discrrete mathematics for Computer Science 08QLogic

Teks penuh

(1)

Logic with quantifiers

aka

First-Order Logic

Predicate Logic

Quantificational Logic

(2)

Predicates

A

predicate

is a proposition with

variables

For example: P(x,y) := “x+y=0”

(For today, universe is Z = all integers)

P(-4,3) is

(3)

Predicates

A

predicate

is a proposition with

variables

For example: P(x,y) := “x+y=0”

P(-4,3) is False

P(5,-5) is

(4)

Predicates

A

predicate

is a proposition with

variables

For example: P(x,y) := “x+y=0”

P(-4,3) is False

P(5,-5) is True

P(6,-6) ¬P(1,2) is

(5)

Predicates

A

predicate

is a proposition with

variables

For example: P(x,y) := “x+y=0”

P(-4,3) is False

P(5,-5) is True

P(6,-6) ¬P(1,2) is True

(6)

Quantifiers

x Q(x) := “for all x, Q(x)”

That is, Q(x) holds for each and every value of x

x Q(x) := “for some x, Q(x)”

(7)

Quantifiers

is AND-like and is OR-like

If the universe is {Alice, Bob, Carol}

then

x Q(x) is the same as

Q(Alice) Q(Bob) Q(Carol)

x Q(x) is the same as

Q(Alice) Q(Bob) Q(Carol)

In general the universe is infinite …

(8)

Rhetoric and Quantifiers

Let Loves(x,y) := “x loves y”

“Everybody loves Oprah”: x Loves(x, Oprah)

What does “Everybody loves somebody” mean?

x y Loves(x,y)?

y x Loves(x,y)?

“All that glitters is not gold”

x (Glitters(x)

Gold(x)) ?

¬∀

x (Glitters(x) Gold(x)) ?

(9)

Negation and Quantifiers

¬∀

x P(x) x

≡ ∃

P(x)

¬∃

x P(x) x

≡ ∀

P(x)

So negation signs can be pushed in to the

predicates but the quantifiers flip

¬∀

x (Glitters(x) Gold(x))

⤳ ∃

x

(Glitters(x) Gold(x))

⤳ ∃

x

(

Glitters(x) Gold(x)) rewriting “ ”

⤳ ∃

x (Glitters(x)

Gold(x)) by DeMorgan and

double negation

“There is something that glitters and is not gold”

Figur

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Referensi

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