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Predicates and Quantifiers

Section 1.

Predicates

• (^) A predicate is a statement that

contains variables.

• (^) Example: P ( x ) : x > 3 Q ( x ,y) : x = y + 3 R ( x , y , z ) : x + y = z

Quantifiers

• (^) Two types of quantifiers
  • (^) Universal
  • (^) Existential
• (^) Universe of discourse - the

particular domain of the variable in

a propositional function

Universal

Quantification

• (^) P ( x ) is true for all values of x in the

universe of discourse.

 x P ( x )

• (^) “for all x , P ( x )”
• (^) “for every x , P ( x )”
• (^) The variable x is bound by the universal quantifier, producing a proposition

For universal quantification P ( x )  P ( x 1 )  P ( x 2 )  …  P ( x n )

• (^) If the elements in the universe of discourse can be listed, U = { x 1 , x 2 , …, xn }  x P ( x )  P ( x 1 )  P ( x 2 )  …  P ( xn )
• (^) Example U = {positive integers not exceeding 3} and P ( x ): x^2 < 10 - (^) What is the truth value of  x P ( x ) P(1) ^ P(2) ^ P (3) T ^ T ^ T T

Existential

Quantification

• (^) P ( x ) is true for some x in the

universe of discourse

  x P ( x )

• (^) “for some x , P ( x )”
• (^) “There exists an x such that P ( x )”
• (^) “There is at least one x such that P ( x )”
• (^) The variable x is bound by the existential quantifier, producing a proposition

For existential quantification P ( x )  P ( x 1 )  P ( x 2 )  …  P ( x n )

• (^) If the elements in the universe of discourse

can be listed, U = { x 1 , x 2 , …, xn }

 x P ( x )  P ( x 1 )  P ( x 2 )  …  P ( xn )

• (^) Example U = {positive integers not exceeding 4} and P ( x ): x^2 > 10
  • (^) What is the truth value of  x P ( x ) P(1) v P(2) v P(3) v P(4)

Binding Variables

• (^) Bound variable: if a variable is quantified
• (^) Free variable: Neither bound nor

assigned a specific value

• (^) Example:  x P ( x )  x Q ( x , y )
• (^) Scope of Quantifiers: Part of a logical

expression to which a quantifier is

applied

• (^) Example:  x ( P ( x )  Q ( x ))   x R ( x )

Translating from

English

• (^) Many ways to translate a given sentence
• (^) Goal is to produce a logical expression

that is simple and can be easily used in

subsequent reasoning

• (^) Steps:
  • (^) Clearly identify the appropriate quantifier(s)
  • (^) Introduce variable(s) and predicate(s)
  • (^) Translate using quantifiers, predicates, and logical operators

Example

• (^) Every student in this class has

studied calculus

• (^) Solution 1
  • (^) Assume, U = {all students in CSE 2813}
• (^) Solution 2
  • (^) Assume, U = {all people}

More Example

• (^) C ( x ): x is a CSE student
• (^) E ( x ): x is an ECE student
• (^) S ( x ): x is a smart student
• (^) U = {all students in CSE 2813}

More Example (Cont..)

• (^) Everyone is a CSE student.  x C ( x )
• (^) Nobody is an ECE student.  x ~E ( x ) or ~  x E ( x )
• (^) All CSE students are smart students.  x [ C ( x )  S ( x )]
• (^) Some CSE students are smart students.  x [ C ( x )  S ( x )]

Use implication or

conjunction?

• (^) Existential quantifiers usually take

conjunctions

• (^) Some CSE students are smart

students.

 x [ C ( x )  S ( x )] Correct  x [ C ( x )  S ( x )] Incorrect

More Example

• (^) No CSE student is an ECE student.
  • (^) If x is a CSE student, then that student is not an ECE student.  x [ C ( x )  ~E ( x )]
  • (^) There does not exist a CSE student who is also an ECE student. ~ x [ C ( x )  E ( x )]
• (^) If any ECE student is a smart student then he is also a CSE student.  x [( E ( x )  S ( x ))  C ( x )]