example of quantifier

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 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 $X=\mathrm{\varnothing }$ , 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: is a true proposition.

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

Example 3: is a property. $y$ varies in $ℝ$. As a result is a proposition, i.e. it is a true or a false sentence. In fact is false but is true; here $(x,\mathrm{\infty })$ 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 $(\forall x\in A)(p(x)\to q(x))$ 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 $A\times B$.