3) PREDICATES AND QUANTIFIERS

·         INTRODUCTION TO PREDICATES

- A predicate or propositional function is a description of the property (or properties) a variable or subject may have.

- A proposition may be created from a propositional function by either assigning a value to the variable or by

   quantification.

-In general, the set of all x in the universe of discourse having the attibute P(x) is

  called the truth set of P(x). That is, the truth set of P(x) is :

         

      { x ϵ U |P(x) }

where the truth value can be whether true or false...

Example 1 :

-The propositional function P(x) is given by "x > 0".

-The universe of discourse for x is the set of integers.

-To create a proposition from P, we

  may assign a value for x. For example, 

• setting x = -3, we get P(-3): "-3 > 0", which is false.
• setting x = 2, we get P(2): "2 > 0", which is true.

Example 2 :

-Suppose P(x) is the predicate “ x has fur ".

-The universe of discourse for x is the set of all animals.

- P(x) is a true statement if,

     x is a cat.

 -P(x) is false if,

     x is an alligator.

Example 3 :

There also involve 2 or more variables.consider "x =  y + 3" and we can denote it as Q(x,y).

Q is a predicate and the question is what is the truth value for Q(1,2) and Q(3,0) ?

Answer for Q(1,2),set x = 1 and y = 2 and substitute into "x =  y + 3",which u get false. Same as Q(3,0),which u get true.

QUANTIFIERS

   -A quantifier turns a propositional function into a proposition

    without assigning specifi c values for the variable.

   -There are primarily two quantifi ers,

• universal quantifier
• existential quantifier.

universal quantifier

-The universal quanti cation of P(x) is the proposition

 “P(x) is true for all values x in the universe of discourse.”

Notation : "For all x P(x)" or "For every x P(x)" is written     ∀xP(x).

example 1 :

Let P(x) be the statement “x+1>x.” What is the truth value of the Quantification ∀xP(x), where the universe of discourse consist of all real numbers ?

solution : Since P(x) is true for all real numbers x, the quantification  ∀xP(x)  is true

example 2 :

What is the truth value of ∀ xP(x), where P(x) is the statement "x² < 10" 

and the universe discourse consist of the positive integers not exceed 4 ?

Solution: the statement  ∀ xP(x) is the same as the conjunction 

P(1) ^ P(2) ^ P(3) ^ P(4),

since the universe discourse consist of the integers 1,2,3 n 4. Since P(4) which the statements "4² < 10 " is false, it follows that  ∀ xP(x) is false.

existential quantifier.

-The existential quanti cation of P(x) is the proposition

"There exists an element x in the universe of discourse such that P(x) is true."

Notation: "There exists x such that P(x)" or "There is at least one x such that

P(x)" is written  ƎxP(x).

Example 1 :

Let P(x) denote the statement "x >3" Whats is the truth value of the quantification ƎxP(x), where the universe discourse consist of all real numbers ?

solution : Since "x>3" is true--for instance , when x = 4--the existancial qualification of P(x),

which is  ƎxP(x) is true

Example 2 :

What is the truth value of  ƎxP(x), where P(x) is the statement "x² > 10" 

and the universe discourse consist of the positive integers not exceed 4 ?

Solution:

Since the universe of discourse is {1,2,3,4}, the proposition ƎxP(x) is the same as the disjunction

P(1) v P(2) v P(3) v P(4),

Since P(4) which the statements "4² > 10 " is true, it follows that  ƎxP(x) is true.

TABLE 1  QUANTIFIERS
Statement	When True ?	When False ?
∀xP(x) ƎxP(x)	P(x) is true for every x. There is an x for which P(x) is true.	There is an x for which P(x) is false. P(x) is false for every x.

EXAMPLES OF USING QUANTIFIERS IN REALITY

• quantification in the natural sciences
  
  
     In some instances in the natural sciences a seemingly intangible concept may be quantified by creating ascale ­­­– for example , a pain scale in medical research,or a discomfort scale at the intersection of meteorology and human physiology such as the heat index measuring the combined perceived effect of heat and humadity, or the wind chill factor measuring the combined perceived effects of colds and wind.
  
  
• In the social sciences

          In the social sciences,quantification is an integral part of economics and pshycology