chapter 1- section 1.4 mini notes

The existential quantification of P(x) is the proposition:

"There exists an element x in the domain such that P(x) is true"

UNIVERSAL QUANTIFICATION : Read as

"for all xP(x)" or "for every xP(x)"

EXTENSIAL QUANTIFICATION: READ AS

"there is an x such that P(x) or "there is at least one x such that P(x)".

THE UPSIDE DOWN A IS WHAT (∀)

(∀) is called the universal quantifier

THE UNIVERSE OF DISCOURSE IS ALSO CALLED WHAT

Also called domain of discourse or just the domain

When a quantifier is used on a variable, it binds that variable and the variable is considered WHAT

BOUND

Example: "All Americans eat cheeseburgers." WHATS THE EXPRESSION AND NEGATION

Expression: ∀xP(x) Negation: "There is an American who does not eat cheeseburgers" or ¬∀xP(x)=∃x¬P(x)

Example: "There is an honest politician" WHATS THE EXPRESSION AND NEGATION

Expression: ∃xP(x) Negation: "Every politician is dishonest" or ¬∃xP(x)=∀x¬P(x)

∃y∀xP(x,y), where P(x,y) = "x + y = 0" IS WHAT

FALSE AND MEANS "There is a real number y, such that for every real number x, x + y = 0"

All other variables are WHAT

FREEEEEEEEEE

∀x∃yP(x,y), where P(x,y) = "x + y = 0" IS WHAT

IS TRUE AND MEANS "For every real number x, there is a real number y such that x + y = 0"

QUANTIFICATION EXAMPLES

In English this is done with the words all, some, many, none, and few Example: "a few people like rum raisin ice cream"

WHAT IS A QUANTIFICATION

Instead of assigning values to a proposition to create truth values, we can instead designate the values to which a predicate is true over a range of elements.

Statements involving predicates and quantifiers are logically equivalent if and only if they have the same truth value. NO MATTER WHAT 2 THINGS?

No matter which predicates are substituted into these statements & No matter which domain of discourse is used

The universal quantification of P(x) is the statement:

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

HOW CAN WE NEST QUANTIFIERS

QUNATIFICATIONS OF TWO VARIABLES

Example: Let A(x) denote the statement "Computer x is under attack by an intruder". If CS2 and CS5are under attach by intruders, what are the truth values of A(CS1) and A(CS2)?

SOLUTION A(x)= "Computer x is under attack by an intruder" Obtain the statement A(CS1) by setting x = CS1 in the statement "Computer x is under attack by an intruder", so "Computer CS1 is under attack by an intruder", which is false. Obtain the statement A(CS2) by setting x = CS2 in the statement "Computer x is under attack by an intruder", so "Computer CS2 is under attack by an intruder", which is true.

Example: Let P(x) denote the statement "x > 3". What are the truth values of P(4) and P(2)

SOLUTION P(x)=x>3 Obtain the statement P(4) by setting x = 4 in the statement "x > 3", so 4 > 3, which is true Obtain the statement P(2) by setting x = 2 in the statement "x > 3", so 2 > 3, which is false

Example: Let Q(x) denote the statement "x = y + 3". What are the truth values of Q(1, 2) and Q(3,0)?

SOLUTION Q(x)="x = y + 3" Obtain the statement Q(1, 2) by setting x = 1 and y = 2 in the statement "x = y + 3", so "1 = 2 + 3",which is false. Obtain the statement Q(3, 0) by setting x = 3 and y = 0 in the statement "x = y + 3", so "3 = 0 + 3",which is true.

EXAMPLE Let P(x) be "x = x + 1". What is the truth value of the quantification ∃xP(x), where the domain consists of all real numbers?

SOLUTION Since x = x + 1 is false for all numbers, this is false

EXAMPLE Let P(x) be "x > 3". What is the truth value of the quantification ∃xP(x), where the domain consists of all real numbers?

SOLUTION Since there is at least one number greater than 3, this is true.

Example: Let P(x) be "x < 2". What is the truth value of the quantification ∀xP(x), where the domain consists of all real numbers?

Solution: Since there are many numbers that are not less than 2, this is false. A counterexample can be x = 3.

Example: Let P(x) be "x + 1 > x". What is the truth value of the quantification ∀xP(x), where the domain consists of all real numbers?

Solution: Since there can always be a number 1 more than any real number, this is true.

∀x∀yP(x,y), where P(x,y) = "x + y = y + x" IS WHAT

TRUE AND MEANS "For all real numbers x, for all real numbers y, x + y = y + x"

what would this statement be if the variables are specified x > 3

The first part is the variable, or the subject, such as "x" The second part is the predicate, or the property that the subject of the statement can have, such as "is greater than 3". denote the statement "x is greater than 3" by P(x), where P denotes the predicate "is greater than 3" and x is the variable.

Example: ∃x(x+y=1)

The variable x is bound by ∃x The variable y is free

WHAT ARE THE 2 TYPES OF QUANTIFICATION WE WILL FOCUS ON

UNIVERSAL AND EXTENSIAL

Consider the statement: if x > 0 then x := x + 1

When this statement is encountered in a computer program, the value of the variable x at that point in the execution of the program is inserted into P(x), which is "x > 0". If P(x) is true for this value of x, the assignment statement x := x + 1 is executed, if P(x) is false, the assignment statement is not executed.

WHAT IS THE UNIVERSE OF DISCOURSE

a particular domain in which all values of a variable are true for a given property.

Statement involving a variable such as: x > 3 x = y + 3 Computer x is under attach by an intruder

are neither true or false until the values of the variables are specified

An element for which P(x) is false is called a

counterexample of ∀xP(x).

∃xP(x) is the notation for the

existential quantification of P(x)

∃ is called the

existential quantifier

WHAT IS UNIVERSAL QUANTIFICATION

predicate is true for every element under consideration

P(x) is the value of the

propositional function P at x.

S ≡ T indicates WHAT

that S and T are logically equivalent

Once a value is assigned to the variable x

the statement P(x) becomes a proposition and has a truth value.

The notion ∀xP(x) denotes WHAT

the universal quantification of P(x)

WHAT IS EXISTENTIAL QUANTIFICATION

there is one or more elements under consideration for which the predicate is true

Statements involving predicates and quantifiers are logically equivalent if and only if WHAT

they have the same truth value.