Predicates


















Is the argument valid?
Can you prove it using propositional logic?

	All students love pizza.
	Bob is a student.
	Therefore, Bob loves pizza.


You need predicates and quantifiers.


Write the statements using predicates.

	Bob is a student.

	Mary and Paul are siblings.


The following facts define when the son(X,Y) predicate is true.
The predicate is false for all other cases.

	son(Bob, Jim)
	son(Jim, Joe)

What's the truth value for each expression?

	son(Jim, Bob)
	son(Bob, Jim)
	son(Bob, Joe) and son(Jim, Joe)
	son(Bob, Joe) or son(Jim, Joe)


















Universal Quantifier (for all)

What do the expressions mean?

	(all x) student(x)

	(all y) (student(y) -> smart(y))

What does the upside down A symbol mean?

	(all x) P(x)

	1. P(x) is true for every x in the Domain

	2. P(Bob) and P(Jim) and P(Joe)
	   (if the domain is {Bob, Jim, Joe})


The following facts define when the son(X,Y) predicate is true.
The predicate is false for all other cases.

The domain is {Bob, Jim, Joe}.

	son(Bob, Jim)
	son(Jim, Joe)

What's the truth value for each expression?

	(all X) son(Jim, X)
	(all X) not son(Jim, X)
	(all X) not son(X, Bob)


















Existential Quantifier (there exists)

What do the expressions mean?

	(exists x) student(x)

	(exists y) (student(y) and smart(y))

What does the backwards E symbol mean?

	(exists x) P(x)

	1. P(x) is true for at least one value in the Domain

	2. P(Bob) or P(Jim) or P(Joe)
	   (if the domain is {Bob, Jim, Joe})


The following facts define when the son(X,Y) predicate is true.
The predicate is false for all other cases.

The domain is {Bob, Jim, Joe}.

	son(Bob, Jim)
	son(Jim, Joe)

What's the truth value for each expression?

	(exists X) son(Jim, X)
	(exists X) not son(Jim, X)
	(exists X) son(X, Bob)


















Nested Quantifiers

Does order matter for Quantifiers?

Is (all x)(all y)P(x,y) different from (all y)(all x)P(x,y)?
Is (all x)(exists y)P(x,y) different from (exists y)(all x)P(x,y)?

You can think about quantifiers like loops in a program.

Rewrite each expression without quantifiers.
Use the domain {Bob, Jim}.

	(exists x)(all y) son(x,y)
	((all y) son(Bob,y)) or ((all y) son(Jim,y))
	(son(Bob,Bob) and son(Bob,Jim)) or (son(Jim,Bob) and son(Jim,Jim))

	(all y)(exists x) son(x,y)
	((exists x) son(x,Bob)) and ((exists x) son(x,Jim))
	(son(Bob,Bob) or son(Jim,Bob)) and (son(Bob,Jim) or son(Jim,Jim))

	(assume son(x,y) means x is the son of y)
	there is someone who is the son of everyone
	everyone has a son


What do the expressions mean?
(assume loves(x,y) means x loves y)

	(all x)(exists y) loves(x,y)

	(exists y)(all x) loves(x,y)

	(all y)(exists x) loves(x,y)

	(exists x)(all y) loves(x,y)


















What's the truth value for each expression?

The domain is people in the real world.
The predicate mother(X,Y) is X is the mother of Y.
The predicate male(X) is X is male.
The predicate female(X) is X is female.

	(all x)(exists y) mother(x,y)

	(exists x)(all y) mother(x,y)

	(exists x)(exists y) (mother(x,y) and male(y))


















Classwork
You may work with a partner.

What's the truth value for each expression?

	(all y)(exists x) mother(x,y)

	(all x)(all y) (mother(x,y) -> male(y))