example of quantifier

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




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 formulaMathworldPlanetmathPlanetmath is shown by p(x), q(x) ,etc. if x is fixed then p(x) is a propositionPlanetmathPlanetmath, i.e. it is a true or a false sentenceMathworldPlanetmath.

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 (xX)(p(x)) means for every x in the set X the proposition p(x) is true. b) The symbol (xX)(p(x)) means there is some x in the set X for which the proposition p(x) is true. If X= , i.e. if the set X is empty, (xX)(p(x)) is defined to be true and (xX)(p(x)) is defined to be false.

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

Example 2: (x)(x2+1=0) is false, because no real number satisfies x2+10=0.

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

some theorems:

for proofs of the following theorems see the address above

Theorem 1: if (xA)(p(x)) and (xA)(p(x)q(x)) then (xA)(q(x)).

Theorem 2: suppose {a} is a singleton, i.e. a set with only one element. We have ”(x{a})(p(x))” is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to p(a).

Theorem 22: if (yB)(xA)(r(x,y)) then (xA)(yB)(r(x,y)).

here r(x,y) is a property on A×B.

Title example of quantifier
Canonical name ExampleOfQuantifier
Date of creation 2013-05-23 19:14:17
Last modified on 2013-05-23 19:14:17
Owner hkkass (6035)
Last modified by hkkass (6035)
Numerical id 19
Author hkkass (6035)
Entry type Example
Classification msc 03B15
Classification msc 03B10