The Logic of Quantified Statements

1
The Logic of Quantified Statements
2
Definition of Predicate

• Predicate is a sentence that
• ? contains finite number of variables
• ? becomes a statement when
• specific values are substituted
• for the variables.
• Ex ? let predicate P(x,y) be x2 and xy8
• ? when x5 and y3,
• P(5,3) is 52 and 538
• Domain of a predicate variable is
• the set of all possible values of the variable.
• Ex (cont.) D(x)? D(y)R

3
Truth Set of a Predicate

• If P(x) is a predicate and
• x has domain D,
• then the truth set of P(x) is
• all x?D such that P(x) is true.
• (denoted x?D P(x) )
• Ex P(x) is 5
• Then x?D P(x) 6, 7, 8

4
Universal Statement and Quantifier

• Let P(x) be x should take CS300
• DCS majors be the domain of x.
• Then all CS majors take CS300
• is denoted ?x?D, P(x)
• and is called universal statement.
• ? is called universal quantifier
• expressions for ? for all, for arbitrary,
• for any, for each.

5
Truth and Falsity of Universal Statements

• Universal statement ?x?D, P(x)
• ? is true iff P(x) is true for every x in D
• ? is false iff P(x) is false for at least one
  x.
• (that x is called counterexample)
• Ex 1) Let D be the set of even integers.
• ?x?D ?y?D, xy is even is true.
• 2) Let D be the set of all NBA players.
• ?x?D, x has a college degree is false.
• Counterexample Kobe Bryant.

6
Existential Statement and Quantifier

• Let P(x) be x(x2)24
• D Z be the domain of x.
• Then
• there is an integer x such that x(x2)24
  
• is denoted ?x?D, P(x)
• and is called existential statement.
• ? is called existential quantifier
• expressions for ? there exists, there is
  a,
• there is at least one,
• we can find a.

7
Truth and Falsity of Existential Statements

• Existential statement ? x?D, P(x)
• ? is true iff P(x) is true for at least one
  x in D
• ? is false iff P(x) is false for all x in D.
• Ex 1) Let D be the set of rational numbers.
• ? x?D, x2-2x10 is true.
• 2) Let D Z.
• ? x?D, x(x-1)(x-2)(x-3)
• Why? Hint Use proof by division into
  cases.

8
Negations of Quantified Statements

• The negation of universal statement ?x?D,
  P(x) is
• the existential statement ?x?D, P(x)
• Example The negation of
• All NBA players have college degree
• is There is a NBA player
• who doesnt have college degree.

9
Negations of Quantified Statements

• The negation of existential statement
  ?x?D, P(x) is
• the universal statement ? x?D, P(x)
• Example The negation of
• ? x? Z such that x(x1)
• is ? x? Z, x(x1) 0.