PREDICATE AND QUANTIFIERS

 

PREDICATES AND QUANTIFIERS

PREDICATES

A Predicate is a sentence depending on variables which becomes a statement upon substituting values in the domain.

Predicate usually represented by the letter P, the notation P(x) is used to represent some unspecified property or predicate that x may have.

If P(x) is a predicate and x has domain D, the truth set of P(x)is the set of all elements of D that make P(x) true when they are substituted for x.

The truth set of P(x) is denoted by: {x∈|P(x)}

Examples

P(x) is “x>5” and x ranges over Z

P(8) is True

P(-1) is False

Note:

Combining the quantifier and the predicate, we get a complete statement of the form ∀xP(x) or ∃xP(x)


QUANTIFIERS

  • Quantifiers are words that refer to quantities such as “some” or “all”.
  • It tells for how many elements a given predicate is True.
  • There are two types of quantifier in predicate logic

    1. Universal Quantifier
    2. Existential Quantifier

 

Universal Quantifier

  • Many mathematical statements state that a property is true for all values of a variable in a particular domain, called the universe of discourse.
  • Such a statement is expressed using a universal quantification.
  • The universal quantification of P(x) is the statement.
  • “P(x) is true for all values of x in the universe of discourse” and is denoted by the notation (x)P(x) or ∀xP(x).
  • The proposition (x)P(x) or ∀xP(x) is read as “for all x,P(x)” or “for every x,P(x)”.
  • The symbol ∀ is called the universal quantifier.

Examples

1.If P(x)={(-x)2=x2}, where the universe consists of all integers, then the truth value of  ∀x(-x)2=x2) is True

2. If Q(x)={2x>x}, where the universe consists of all real numbers, then the truth value of  ∀x(Q(x)) is False

3. If P(x)={ x2<10}, where the universe consists of all positive integers 1,2,3,4, then ∀x(P(x)= P(1) ∧P(2) ∧ P(3) ∧P(4) and so the truth value of ∀x(P(x)=T∧T∧T∧F= False

 

Existential Quantifier

  • The existential quantification of P(x) is the proposition.
  • “There exists at least one x such that P(x) is true”.
  • It is denoted by the notation ∃xP(x).
  • The symbol ∃ is called the existential quatifier.
  • The proposition ∃xP(x) is read as “For some x,P(x).

Examples

1.If P(x)={(-x)2=x2}, where the universe of discourse consists of  all real numbers, then the truth value of  ∃xP(x) is True

2. If Q(x)={2x>x}, where the universe consists of all real numbers, then the truth value of  ∃x(Q(x)) is True

3. If P(x)={ x2<10}, where the universe of discourse consists of the positive integers 1,2,3,4, then ∃x(P(x)= P(1) ∨ P(2) ∨ P(3) ∨ P(4) and so the truth value of ∃x(P(x)=T∨T∨T∨F= True

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