Predicate Logic Equivalences


What's a Tautology?

Can you have Tautologies in Predicate Logic expressions?

	must be always true regardless of predicate or domain definitions

Which of the expressions are tautologies?

	(all x)cat(x) -> (exists x)cat(x)

	(all x)cat(x) -> cat(Tom)

	(all x)(cat(x) and blue(x)) -> (all x)cat(x) and (all x)blue(x)


















Classwork
You may work with a partner.

Which of the expressions are tautologies?

	cat(Tom) -> (exists x)cat(x)

	(exists x)cat(x) -> (all x)cat(x)

	(all x)(cat(x) or blue(x)) -> (all x)cat(x) or (all x)blue(x)


















What equivalence laws are valid in Predicate Logic?


Quantifiers (of the same type) are Commutative and Associative.

(all x)(all y)A	   	<=>  (all y)(all x)A
(all x)(A and B)	<=>  (all x)A and (all x)B

(exists x)(exists y)A  	<=>  (exists y)(exists x)A
(exists x)(A or B)	<=>  (exists x)A or (exists x)B


What's DeMorgan's law for Quantifiers?

not ((all x)A)		<=>  (exists x)(not A)
not ((exists x)A)	<=>  (all x)(not A)


How does the Distributive law work for Quantifiers?
(x can't be a free variable in B)

(all x)A or B		<=>  (all x)(A or B)
(exists x)A and B	<=>  (exists x)(A and B)


















Use algebra to show the equivalence is valid.

	(exists x)P(x) -> Q <=> (all x)(P(x) -> Q)


















Classwork
You may work with a partner.

Use algebra to show the equivalence is valid.

	not (exists x) (A(x) and (B(x) or C(x))) <=>
	(all x) (A(x) -> (not B(x) and not C(x)))