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Discrete mathematics: propositional symbolization of predicate logic
2022-07-05 00:58:00 【Xiaobai Xiaozheng】
Individual words
- Individual constant or individual constant : Use x, y, z Express
- Individual variable or individual variable : Use a, b, c Express
- Individual domain or discourse : Value of individual argument
- Total individual domain : Everything in the universe
The predicate
- Concept : A word that indicates the nature of an individual or the relationship between them
- Illustrate with examples :A(x) Can mean x It's the students ,B(x,y) Can mean x Greater than y
quantifiers
- Full name quantifier ∀:∀x Represents all x
- 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)))
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