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.