离散数学lecture2
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Discrete Mathematics Thomas Honold Predicates and Quantifiers Nested Quantifiers
Discrete Mathematics
Thomas Honold
Institute of Information and Communication Engineering Zhejiang University
·y
z"
The first statement “x “x” “is greater than 3”
3” has two parts:
a variable (subject of the statement) a predicate (property of the subject x which x may or may not have)
Solution
A possible solution is x x 2 0 or, for better readability, x ´x 2 0µ. Here we assume that the domain of the unary predicate x 2 0 is the set of real numbers. The proposition x ´x 2 0µ is false, since substituting x into x 2 0 yields the false proposition 02 0. We call the real number 0 a counterexample to the statement x ´x 2 0µ. 0
Today’s Lecture: Predicate Calculus
Discrete Mathematics Thomas Honold Predicates and Quantifiers Nested Quantifiers
Introduction
Consider the statements "x 3" "x y" "x
Discrete Mathematics Thomas Honold Predicates and Quantifiers Nested Quantifiers
Example
Express the statement “The square of every real number is positive” as a formula of calculus and determine its truth value.
Examples
Since “x y” contains 2 variables, it defines a binary predicate Q ´x y µ whose domain is the set of ordered pairs of integers, say. Then Q ´2 1µ is true (since 2 1) while Q ´1 2µ is false (since 1 ¥ 2). Similarly “x · y z” defines a ternary predicate R ´x y z µ with domain the set of ordered triples of integers, say, and R ´0 0 0µ, R ´ 1 1 0µ, R ´1 1 2µ are all true, while R ´2 2 3µ is false.
Discrete Mathematics Thomas Honold Predicates and Quantifiers Nested Quantifiers
Existential Quantification
The quantifier (“there exists”)
Definition
Suppose again that P ´x µ is a unary predicate. The existential quantification of P ´x µ is the proposition “There exists an x in the domain of P such that P ´x µ is true.” The existential quantification of P ´x µ is denoted by x P ´x µ (read “there exists x such that P ´x µ holds”). The symbol is called the existential quantifier.
Quantification
Another way to make propositions from predicates
Introductory Examples
For each of the following sentences decide whether it is a proposition, unary predicate, or binary predicate.
பைடு நூலகம்
Example
Denote the statement “x 3” by P ´x µ and let P have domain (the integers). Then P ´ 2µ and P ´3µ are false (since propositions 2 3 and 3 3 are false) while P ´5µ is true (since 5 3). The set of integers having property P . The proposition P ´ 3 µ is not defined, since is 4 5 6 7 4 3 is not an integer. 4
Discrete Mathematics Thomas Honold Predicates and Quantifiers Nested Quantifiers
Further Examples
Example
Consider the statement “d is a divisor of n” for positive integers d n 6. Describe the corresponding predicate explicitly.
Discrete Mathematics Thomas Honold Predicates and Quantifiers Nested Quantifiers
Informal Definition
An unary predicate is a function (“propositional function”) of one variable which assigns to every object of some domain a proposition and hence a unique truth value (either T or F). Predicates are denoted by capital letters P Q R If the proposition P ´x µ is true we think of x as “having property P”.
1
For all real numbers x the inequality x 2 True proposition There exists an integer k such that 15 True proposition
0 is true. 2k
· 1.
2
3
There exists an integer k such that n 2k Unary predicate (definition of “n is odd”)
5
Discrete Mathematics Thomas Honold Predicates and Quantifiers Nested Quantifiers
Universal Quantification
The quantifier (“for all”)
Definition
Suppose P ´x µ is a unary predicate. The universal quantification of P ´x µ is the proposition “P ´x µ is true for every x in the domain of P”. The universal quantification of P ´x µ is denoted by x P ´x µ (read “for all x, P ´x µ holds”). The symbol is called the universal quantifier.
1 T F F F F F
2 T T F F F F
3 T F T F F F
4 T T F T F F
5 T F F F T F
6 T T T F F T
Discrete Mathematics Thomas Honold Predicates and Quantifiers Nested Quantifiers
Discrete Mathematics Thomas Honold Predicates and Quantifiers Nested Quantifiers
n-ary Predicates
An n-ary predicate is a propositional function xn µ of n variables. It assigns to every n-tuple of P ´x1 x2 objects of some domain a unique truth value.
Solution
The statement defines a binary predicate D ´d nµ with d n taken from 1 2 3 4 5 6 . The values of D are shown in the table below.
n d
1 2 3 4 5 6
· 1.
4
There exists an integer k such that n kd . Binary predicate (definition of “d divides n”) For all real numbers a b it is true that ´a · b µ2 a2 · 2ab · b 2 . True proposition (Binomial Formula)
Fall 2011
Discrete Mathematics Thomas Honold Predicates and Quantifiers Nested Quantifiers
Outline
1 Predicates and Quantifiers
2 Nested Quantifiers
Discrete Mathematics Thomas Honold Predicates and Quantifiers Nested Quantifiers
If x is assigned a value, the statement “x 3” becomes a proposition (for example, if we put x 1 then it becomes the proposition 1 3 which is false) Similar reasoning applies to the second statement (variables x y and the predicate “is greater than”) and the third (variables x y z and the predicate “the sum of the first two equals the third”.
Discrete Mathematics Thomas Honold Predicates and Quantifiers Nested Quantifiers
Discrete Mathematics
Thomas Honold
Institute of Information and Communication Engineering Zhejiang University
·y
z"
The first statement “x “x” “is greater than 3”
3” has two parts:
a variable (subject of the statement) a predicate (property of the subject x which x may or may not have)
Solution
A possible solution is x x 2 0 or, for better readability, x ´x 2 0µ. Here we assume that the domain of the unary predicate x 2 0 is the set of real numbers. The proposition x ´x 2 0µ is false, since substituting x into x 2 0 yields the false proposition 02 0. We call the real number 0 a counterexample to the statement x ´x 2 0µ. 0
Today’s Lecture: Predicate Calculus
Discrete Mathematics Thomas Honold Predicates and Quantifiers Nested Quantifiers
Introduction
Consider the statements "x 3" "x y" "x
Discrete Mathematics Thomas Honold Predicates and Quantifiers Nested Quantifiers
Example
Express the statement “The square of every real number is positive” as a formula of calculus and determine its truth value.
Examples
Since “x y” contains 2 variables, it defines a binary predicate Q ´x y µ whose domain is the set of ordered pairs of integers, say. Then Q ´2 1µ is true (since 2 1) while Q ´1 2µ is false (since 1 ¥ 2). Similarly “x · y z” defines a ternary predicate R ´x y z µ with domain the set of ordered triples of integers, say, and R ´0 0 0µ, R ´ 1 1 0µ, R ´1 1 2µ are all true, while R ´2 2 3µ is false.
Discrete Mathematics Thomas Honold Predicates and Quantifiers Nested Quantifiers
Existential Quantification
The quantifier (“there exists”)
Definition
Suppose again that P ´x µ is a unary predicate. The existential quantification of P ´x µ is the proposition “There exists an x in the domain of P such that P ´x µ is true.” The existential quantification of P ´x µ is denoted by x P ´x µ (read “there exists x such that P ´x µ holds”). The symbol is called the existential quantifier.
Quantification
Another way to make propositions from predicates
Introductory Examples
For each of the following sentences decide whether it is a proposition, unary predicate, or binary predicate.
பைடு நூலகம்
Example
Denote the statement “x 3” by P ´x µ and let P have domain (the integers). Then P ´ 2µ and P ´3µ are false (since propositions 2 3 and 3 3 are false) while P ´5µ is true (since 5 3). The set of integers having property P . The proposition P ´ 3 µ is not defined, since is 4 5 6 7 4 3 is not an integer. 4
Discrete Mathematics Thomas Honold Predicates and Quantifiers Nested Quantifiers
Further Examples
Example
Consider the statement “d is a divisor of n” for positive integers d n 6. Describe the corresponding predicate explicitly.
Discrete Mathematics Thomas Honold Predicates and Quantifiers Nested Quantifiers
Informal Definition
An unary predicate is a function (“propositional function”) of one variable which assigns to every object of some domain a proposition and hence a unique truth value (either T or F). Predicates are denoted by capital letters P Q R If the proposition P ´x µ is true we think of x as “having property P”.
1
For all real numbers x the inequality x 2 True proposition There exists an integer k such that 15 True proposition
0 is true. 2k
· 1.
2
3
There exists an integer k such that n 2k Unary predicate (definition of “n is odd”)
5
Discrete Mathematics Thomas Honold Predicates and Quantifiers Nested Quantifiers
Universal Quantification
The quantifier (“for all”)
Definition
Suppose P ´x µ is a unary predicate. The universal quantification of P ´x µ is the proposition “P ´x µ is true for every x in the domain of P”. The universal quantification of P ´x µ is denoted by x P ´x µ (read “for all x, P ´x µ holds”). The symbol is called the universal quantifier.
1 T F F F F F
2 T T F F F F
3 T F T F F F
4 T T F T F F
5 T F F F T F
6 T T T F F T
Discrete Mathematics Thomas Honold Predicates and Quantifiers Nested Quantifiers
Discrete Mathematics Thomas Honold Predicates and Quantifiers Nested Quantifiers
n-ary Predicates
An n-ary predicate is a propositional function xn µ of n variables. It assigns to every n-tuple of P ´x1 x2 objects of some domain a unique truth value.
Solution
The statement defines a binary predicate D ´d nµ with d n taken from 1 2 3 4 5 6 . The values of D are shown in the table below.
n d
1 2 3 4 5 6
· 1.
4
There exists an integer k such that n kd . Binary predicate (definition of “d divides n”) For all real numbers a b it is true that ´a · b µ2 a2 · 2ab · b 2 . True proposition (Binomial Formula)
Fall 2011
Discrete Mathematics Thomas Honold Predicates and Quantifiers Nested Quantifiers
Outline
1 Predicates and Quantifiers
2 Nested Quantifiers
Discrete Mathematics Thomas Honold Predicates and Quantifiers Nested Quantifiers
If x is assigned a value, the statement “x 3” becomes a proposition (for example, if we put x 1 then it becomes the proposition 1 3 which is false) Similar reasoning applies to the second statement (variables x y and the predicate “is greater than”) and the third (variables x y z and the predicate “the sum of the first two equals the third”.