Discrete mathematics: propositional symbolization of predicate logic

Individual words

1. Individual constant or individual constant ： Use x, y, z Express
2. Individual variable or individual variable ： Use a, b, c Express
3. Individual domain or discourse ： Value of individual argument
4. Total individual domain ： Everything in the universe

The predicate

1. Concept ： A word that indicates the nature of an individual or the relationship between them
2. Illustrate with examples ：A(x) Can mean x It's the students ,B(x,y) Can mean x Greater than y

quantifiers

1. Full name quantifier ∀：∀x Represents all x
2. There are quantifiers ∃：∃x Indicates that x

exercises

The individual domain is limited to (a) and (b) When the conditions , Symbolize the following proposition ：
proposition ：
(1) For any x, There are x²-5x+6=(x-2)(x-3).
(2) There is x, bring x+1=0.
Conditions ：
(a) Individual domain D₁ Is a set of natural numbers .
(b) Individual domain D₂ Is a set of real numbers .

Explain ： Make F(x)：x²-5x+6=(x-2)(x-3),G(x)：x+1=0
For conditions (a), Individual domain D₁ After symbolization
(1) ∀xF(x), True proposition .
(2) ∀xG(x), False proposition , Natural number is greater than or equal to 0.
For conditions (b), Individual domain D₂ After symbolization
(1) ∀xF(x), True proposition .
(2) ∀xG(x), True proposition .

Proposition symbolization

The basic formula

F(x)：x Have the quality of F
G(x)：x Have the quality of G

（1） There are properties in the individual domain F All individuals have properties G
Proposition symbolization ：∀x(F(x)→G(x))

（2） There are properties in the individual domain F And nature G The individual of
Proposition symbolization ：∃x(F(x)∧G(x))

Propositional symbolization steps

（1） Determine the scope of individual domain （ Human beings gather 、 Total individual domain …）
（2） Definite predicate （G(x)：x Prime number 、F(x)：x Born myopia …）
（3） Get the symbolic result of proposition

exercises

Not all rabbits run faster than turtles .
Explain ：
（1） The individual domain is the total individual domain
（2）F(x)：x It's a rabbit ,G(y)：y It's a turtle
--------H(x,y)：x Than y Run fast
（3）¬∀x(F(x)∧∀y(G(y)→H(x,y)))