Universal Quantifier - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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CSE115/ENGR160 Discrete Mathematics

Logical equivalences

• S≡T: Two statements S and T involving

predicates and quantifiers are logically

equivalent

• If and only if they have the same truth value no matter which predicates are substituted into these statements and which domain is used for the variables.
• Example:

i.e., we can distribute a universal quantifier

over a conjunction

2

∀ x ( p ( x )∧ q ( x )) ≡∀ xp ( x )∧∀ xq ( x )

• (→) If a is in the domain, then p(a)˄q(a) is

true. Hence, p(a) is true and q(a) is true.

Because p(a) is true and q(a) is true for every

element in the domain, so

is true

• (←) It follows that are

true. Hence, for a in the domain, p(a) is true

and q(a) is true, hence p(a)˄q(a) is true. If

follows is true 4

∀ x ( p ( x ) ∧ q ( x )) → ∀ xp ( x )∧∀ xq ( x )

∀ xp ( x )∧∀ xq ( x )

∀ xp ( x ) and ∀ xq ( x )

∀ x ( p ( x ) ∧ q ( x ))

∀ x ( p ( x ) ∧ q ( x )) ≡ ∀ xp ( x ) ∧∀ xq ( x )

Negating quantified expressions

5

Negations of the following statements “There is an honest politician” “Every politician is dishonest” (Note “All politicians are not honest” is ambiguous) “All Americans eat cheeseburgers”

Translating English into logical

expressions

• “Every student in this class has studied

calculus”

Let c(x) be the statement that “x has studied

calculus”. Let s(x) be the statement “x is in

this class”

7

( ) ( )?if thedomainconsistsof all people

( ) ( )if thedomainconsists of allpeople

( )if thedomain consistsof studentsof thisclass

x s x c x

x s x c x

x c x

∀ ∧

∀ →

∀

Using quantifiers in system

specifications

• “Every mail message larger than one

megabyte will be compressed”

Let s(m,y) be “mail message m is larger than y

megabytes” where m has the domain of all

mail messages and y is a positive real number.

Let c(m) denote “message m will be

compressed”

8

∀ m ( s ( m , 1 ) → c ( m ))

1.5 Nested quantifiers

10

where ( )is ( , )and ( , )is 0

sameas ( )

( ) 0

∃ + =

∀

∀ ∃ + =

q x yp x y p x y x y

xq x

x y x y

(( 0 ) ( 0 ) ( 0 ))

( ( ) ( ) )

( )

∀ ∀ > ∧ < → <

∀ ∀ ∀ + + = + +

∀ ∀ + = +

x y x y xy

x y z x y z x y z

x y x y y x

Let the variable domain be real numbers

where the domain for these variables consists of real numbers

Quantification as loop

• For every x, for every y
  • Loop through x and for each x loop through y
  • If we find p(x,y) is true for all x and y, then the statement is true
  • If we ever hit a value x for which we hit a value for which p(x,y) is false, the whole statement is false
• For every x, there exists y
  • Loop through x until we find a y that p(x,y) is true
  • If for every x, we find such a y, then the statement is true

11

∀ x ∀ yp ( x , y )

∀ x ∃ yp ( x , y )

Order of quantification

13

and thedomainis realnumber

Let ( , ) be thestatement ,

x yp x y y xp x y

x yp x y

p x y x y y x

( , ) ( , )?

( , ):Foreveryrealnumber thereisarealnumber s.t.

( , ):Thereisarealnumber s.t.foreveryrealnumber ,

andthedomainisrealnumber

Let ( , )bethestatement 0 ,

y xp x y x yp x y

x yq x y x y q(x,y)

y xq x y y x q(x,y)

q x y x y

∃ ∀ ≡∀ ∃

∀ ∃

∃ ∀

• =

Quantification of two variables

Translating mathematical

statements

• “The sum of two positive integers is always

positive”

16

all integers

wherethedomain for both variablesconsists of

∀ x ∀ y (( x > 0 )∧( y > 0 ) →( x + y > 0 ))

allpositive integers

wherethedomain for both variablesconsists of

∀ x ∀ y ( x + y > 0 )

Example

• “Every real number except zero has a

multiplicative inverse”

17

wherethedomain for both variablesconsists of real numbers

∀ x (( x ≠ 0 ) → ∃ y ( xy = 1 ))

Translating statements into English

• where c(x) is “x has

a computer”, f(x,y) is “x and y are friends”, and

the domain for both x and y consists of all

students in our school

where f(x,y) means x and y are friends, and

the domain consists of all students in our

school

19

∀ x ( c ( x )∨∃ y ( c ( y )∧ f ( x , y )))

∃ x ∀ y ∀ z (( f ( x , y )∧ f ( x , z )∧( y ≠ z )) →¬ f ( y , z ))

Negating nested quantifiers

• There does not exist a woman who has taken

a flight on every airline in the world

where p(w,f) is “w has taken f”, and q(f,a) is “f

is a flight on a 20

x y xy

x y xy

x y xy

x y xy

w a f p w f q f a

w a f p w f q f a ≡ ∀ ∃ ∀ ¬ ∨ ¬