Bound & Free Variables


Describe the variable bindings in the expression.

	(all x) P(x) and (exists x) Q(x)

	1. (all x) binds x in P(x)
	2. (exists x) binds x in Q(x)
	3. the two x's are different

Compare bound variables to local variables in a program.


Describe the variable bindings in the expression.

	(exists x) Q(x,y)

	Q(Bob,y) or Q(Jim,y) (with domain {Bob, Jim})

	1. (exists x) binds x
	2. variable y is free

(Note that an expression with free variables can't be evaluated.)


Describe the variable bindings in the expression.

	(all y)(exists x)(P(x,y) or (Q(x) and (exists x)R(x,y,z)))

	1. (all y) binds y's in P(x,y) and R(x,y,z)
	2. first (exists x) binds x's in P(x,y) and Q(x)
	3. second (exists x) binds x in R(x,y,z)
	4. variable z is free


















Translating English into Predicate Logic


How do you translate statements of the form all P's have property Q?

	(all x)(P(x) -> Q(x))

	All students are smart.

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

	(all y) (student(y) and smart(y))  (wrong)


How do you translate statements of the form some P's have property Q?

	(exists x)(P(x) and Q(x))

	Some students are smart.

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

	(exists y) (student(y) -> smart(y))  (wrong)


















Translate the statements into predicate logic.


	Some days are not rainy.

	Every fruit is sweeter than all vegetables.

	Every fruit is sweeter than some vegetable.


















Classwork
You may work with a partner.

Translate the statements into predicate logic.


	All days are sunny.

	Every day that is sunny is not rainy.

	Some vegetable is sweeter than all fruits.