Discrete Mathematics-Syllabus-01-Propositional Logic

1. Equivalent calculus and reasoning calculus of propositional logic

reference

Summary of discrete mathematics knowledge points (5): implication; reasoning theory of proposition; method of logical deduction; proof of validity of reasoning

1.1 Proposition

命题: The declarative sentence we make to the determined object is called a proposition (propositions and statements proposition or statement). The proposition is true when the judgment is true, otherwise it is false.

It’s raining today is a proposition √
What are you doing non-declarative X
I only shave everyone who doesn’t shave themselves Paradox X

原子命题: Propositions that do not contain logical connectives are usually called atomic propositions or atoms
复合命题: Propositions composed of atomic propositions and logical connectives are called compound propositions (compositive propositions or compound statements).

1.2 Commonly used connectives

否定: symbol $\neg$ is called a negative connective
合取: the symbol$\wedge$ is called the conjunction conjunction
析取: symbol$\vee$ is called a disjunctive connective.
蕴含或条件: symbol$\to$ called entailment or conditional connective.
双向蕴含或等价: symbol$\leftrightarrow$ is called a double implication or equivalent connective.

Connective Priority
$()$ > $\neg$ > $\wedge$ > $\vee$ > $\to$ > $\leftrightarrow$

1.3 Proposition formula

命题常元: Represents a specific simple proposition
命题变元: Represents an arbitrary proposition, a variable whose value is true or false
命题公式: an expression containing propositional variables. That is, $P\vee Q$ is a propositional formula

公式的赋值
Definition: If the proposition formula AAAll propositional variables contained in$A$$p_1,p_2,p_3,p_4\dots p_n$, give $p_1,p_2,p_3,p_4\dots p_n$Specify a set of truth values, called AAAn interpretation or assignment of$A.$Make$The\; assignment\; of\; the\; truth\; value\; of\; A$ to true is called a true assignment, and the assignment that makes the truth value of A false is a false assignment.

指派或赋值:用$a,\beta$ etc. means when$A$ pair value status$When\; \alpha$ is true, it is said to assign$\alpha$ Seishin$A$ , or$\alpha$ isAAThe true assignment of$A.$Denote as$a\left(A\right)=1$
for all possible assignments, formula$The\; value\; of\; A$ can be described by the following table, the truth table

真值表: A list of the values ​​of a propositional formula under all possible assignments, which contains n variants, has 2 to the nth power of assignments.

Classification of Propositional Formulas-Tautologies-Contradiction-Satisfiable Forms

A is said to be a tautology or perpetual truth if the value assigned to A in all its cases is true
If the value assigned to A in all its cases is false, then A is said to be a contradiction If there is at least one assignment
that can make the truth value of A true, then A is said to be satisfiable

Equivalence relation - logically equivalent logically equivalent

逻辑等价: When the propositional formula $A\leftrightarrow When\; B$ is a tautology, it is called$A$ is logically equivalent to$B$ , denoted as$A\iff B$
注意: $A\leftrightarrow B$ 和 $A\iff B$ is differentiated, symbol$A\leftrightarrow B$ is a logical connective and is an operator. And$A\iff B$ is a relation symbol, indicating the logical equivalence relationship between A and B.

1.4 Equivalence Calculus and Reasoning of Propositions

basic equivalence

(1) Double negation law $\neg \neg \iff A$
（2）幂等律 $A\wedge A\iff A,A\vee A\iff A$
（3）交换律 $A\wedge B\iff B\wedge A,A\vee B\iff B\vee A$
（4）结合律
$A\wedge (B\wedge C)\iff (A\wedge B)\wedge C$,
$A\vee (B\vee C)\iff (A\vee B)\vee C$
$A\leftrightarrow (B\leftrightarrow C)\iff (A\leftrightarrow B)\leftrightarrow C$
（5）分配律
$A\wedge (B\vee C)\iff (A\wedge B)\vee (A\wedge C)$
$A\vee (B\wedge C)\iff (A\vee B)\wedge (A\vee C)$
$A\to (B\to C)\iff (A\to B)\to (A\to C)$
（6）德摩根律 $\neg (A\wedge B)\iff \neg A\vee \neg B,\neg (A\vee B)\iff \neg A\wedge \neg B$
（7）吸收律 $A\wedge (A\vee B)\iff A,A\vee (A\wedge B)\iff A$
（8）零律 $A\vee 1\iff 1,A\wedge 0\iff 0$
（9）同一律 $A\wedge 1\iff A,A\vee 0\iff A$
(10)subtract$A\vee \neg A\iff 1$
(11) Law of Contradiction$A\wedge \neg A\iff 0$
(12) implies the equivalence$A\to B\iff \neg A\vee B$
(13) Equivalent Equivalence
$A\leftrightarrow B\iff (A\to B)\wedge (B\to A)$
$A\leftrightarrow B\iff (\neg A\vee B)\wedge (\neg B\vee A)$
$A\leftrightarrow B\iff (A\wedge B)\vee (\neg A\wedge \neg B)$
(14) Hypothetical translocation$A\to B\iff \neg B\to \neg A$
(15) Equivalent Negative Equivalent$A\leftrightarrow B\iff \neg A\leftrightarrow \neg B$
（16）归谬论 $(A\to B)\wedge (A\to \neg B)\iff \neg A$

logically implies tautology

When the propositional formula $A\to B$ is a tautology, called$A$ logically implies$B$ , denoted as$A\Rightarrow B$ , need to pay attention to tautology entailment$\Rightarrow$ with common implication$\to$ relationship.

tautological implication pushes to

$\Rightarrow$ is the propositional formula$A$ and propositional formula$The\; reasoning\; relation\; of\; B$ ,$\to$ is the connection relation of two atomic propositions.

Resolution

Resolution is the method by which computers reason

1.5 Relationship between propositional formula and truth table

For any dependent proposition variable $p_1,p_2,p_3,p_4\dots p_n$The propositional formula AAFor$A$$p_1,p_2,p_3,p_4\dots p_n$The truth value of is according to the propositional formula $The$ dateAA$The\; truth\; value\; of\; A$ , thus establishing from$p_1,p_2,p_3,p_4\dots p_n$to $A$ 's truth table.
Conversely, if given by$p_1,p_2,p_3,p_4\dots p_n$to $The\; truth\; table\; of\; A$ , the propositional formula AAcan be written by the following method$A$对$p_1,p_2,p_3,p_4\dots p_n$logical expression of .

written by column T

written by column F

1.6 The complete set of connectives

Reference:
[Discrete Mathematics] Mathematical Logic Chapter 1 Propositional Logic (4) The Complete Set of Connectives

complete set

Basic concept of duality

1.7 Paradigm

Paradigm definition and generation steps

Principal disjunctive and principal conjunctive normal forms

Master disjunctive normal form:

Assuming that a propositional formula A contains n propositional variables, if the simple conjunctions in the disjunctive normal form of A are all minimal terms, then the disjunctive normal form is called the principal disjunctive normal form of A.

  若干个极小项的析取（并集）。

Primary Conjunctive Normal Form:

Assuming that a propositional formula A contains n propositional variables, if the simple analytic forms in the disjunctive normal form of A are all maximal terms, then the disjunctive normal form is called the principal disjunctive normal form of A.

若干个极大项的合取（交集）。

Maximum term, minimum term: