there are some examples and theorems about logical quantifiers in the Word Document below . you can download it:

http://www.freewebs.com/hkkass

or

http://www.hkkass.blogspot.com/

I include extracts of this Document below:

Definition: a property is something like x >0 or x=0 in which x is a variable in some set. Such a formula is shown by p(x), q(x) ,etc. if x is fixed then p(x) is a proposition, i.e. it is a true or a false sentence.

Example 1: let p(x) be the property 0 < x where x is a real number. p(1) is true and p(0) is false.

Example 2: a property can have two or more variables. Let p(x,y) be x=y. in this case p(1,1) is true but p(0,1) is false because 0 is not equal to 1.

Definition: let p(x) be a property on the set X, i.e. p(x) is a property and x varies in the set X. a) The symbol (forall x in X)(p(x)) means for every x in the set X the proposition p(x) is true. b) The symbol (exists x in X)(p(x)) means there is some x in the set X for which the proposition p(x) is true. If X=empty set , i.e. if the set X is empty, (forall x in X)(p(x)) is defined to be true and (exists x in X)(p(x)) is defined to be false.

Example 1: (forall x in |R)(x=0 or x > 0 or x < 0) is a true proposition.

Example 2: (exists x in |R)(x (power) 2+1=0) is false, because no real number satisfies x(power)2+10=0.

Example 3: (forall x in |R)(x < y) is a property. y varies in |R. As a result (forall x in |R)(forall y in|R) (x < y) is a proposition, i.e. it is a true or a false sentence. In fact (forall x in |R)(forall y in|R)(x < y) is false but (forall x in |R)(forall y in (x,infinity)(x < y) is true; here (x,infinity) is the interval containing real numbers greater than x.

some theorems:

for proofs of the following theorems see the address above

Theorem 1: if (forall x in A)(p(x)) and (forall x in A)(p(x) implies q(x)) then (forall x in A)(q(x)) .

Theorem 2: suppose a is a singleton, i.e. a set with only one element. We have "(forall x in a)(p(x))" is equivalent to p(a)

Theorem 22: if (exists y in B)(forall x in A)(r(x,y)) then (forall x in A)(exists y in B)(r(x,y)).

here r(x,y) is a property on A * B.