# Discrrete mathematics for Computer Science 08QLogic

Gratis

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### Logic with quantifiers

aka

First-Order Logic

Predicate Logic

Quantificational Logic is a proposition with variables

Predicates

• A

predicate

• For example: P(x,y) := “x+y=0”
• (For today, universe is Z = all integers)
• P(-4,3) is

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 is a proposition with variables

Predicates

• A

predicate

• 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
is a proposition with variables

Predicates

• A

predicate

• 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 ⋀

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)”

That is, Q(x) holds for at least one value of x

• – Let Q(x) := “x-7=0”

• – ∀ x Q(x) is false but x Q(x) is true
• – ∀ ∃ ⋁ Then y x (R(x,y) R(y,x)) is …?
• ∀ ∃ ⋀ ⋁ ⋀ y x ((x≥0 x+y=0) (y≥0 y+x=0)): True!

⋀ Let R(x,y) := “x≥0 x+y=0”

∃

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 …

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)) ? ⇒

⤳ ∃ x ￢ (Glitters(x) Gold(x)) ⇒ ⤳ ∃ x ￢ ( ￢ Glitters(x) Gold(x)) rewriting “ ”

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)) by DeMorgan and double negation “There is something that glitters and is not gold”

Gratis