Discrete Mathematics Propositional Logic Pdf

1. Basic concept of proposition logic

1.1 Proposition and Connection Word

1.1.1 proposition

Inexperienceddeclarative sentence

1.1.2 true value

Proposition judgment

If a proposition is or correct, it is called a true life, the true value is "true", often used T or 1; a proposition is wrong or incorrect, called a fake life, Truth is "false", often used F or 0
The true value of any proposition is unique

example:

1.1.3 Propositional Volume (proposition variation)

Usually use lowercase English letters P, Q, R, ... or lowercase letters P₁, P₂, P₃, ..., called propositions, called propositions or proposition variables

1. The difference between proposition variations and propositions:

Proposition has specific meaning and determined true value
The proposition variation is only clear that there is a specific meaning and determined true value when a proposition is

2. Propositional variations generally only represent an abstract proposition, which may be t or f, but usually also referred to as proposition variations for propositions

3. The proposition that can no longer decompose is called simple proposition or atomic proposition, general form "... is ...". Proposition made from atomic propositions called composite propositions

example:

Π and e are unreasonable
There is at least one of 6 and 8
Say that Teacher Liu is not good to lectone.
I will buy a book without a rain.

1.1.4 Propositional Position (Propony Operator)

Way the proposition is called a life-question clutch or proposition operator

1. Set P as a proposition,Negative Words"¬" is a one-dollar word

¬P reads "non-P" or "negation"
If the true value of P is true, the true value of the ¬P is false; it is, if P is false, then ¬P is true
Neither the meaning of the joint word is quite a "no" in the natural language, "there is no", no "," negative "," well is not "" refuel ", etc.

2. Set P, Q is the proposition,Uniform word"∧" is a binary link

P∧Q read as "P and Q" or "P, Q"
When and only when P, Q is true, the true value of P∧Q is true.
The integration of the combined group is equivalent to "P and Q", "P and Q," P and Q, "P, while Q," P and Q "P and Q," P and Q "," Both P, again Q, "Not only P, and Q", "despite P, still Q", "though P, but Q", etc.

3. Set P, Q is the proposition, Precise words "∨" is a binary link

PVQ reads "P or Q" or "P, Q]
When and only when the true value of P, Q is false, the true value of PVQ is a false
The meaning of the unified clutting words is equivalent to "p or Q" in the natural language, "either p, or q", "not P is Q", etc.

4. Set P, Q is the proposition,Vary"p⊕Q" is a binary linkage

P⊕Q reading "P Dior or Q
When and only when the true value of P, Q is the same, the true value of P⊕Q is a fake.
"Different or" also known "" cannot be determined or "

5. Set P.Q as a proposition,Incubation"→" is a binary linkage

P → Q read as "if P, Q"
When and only when P is true, the true value of Q is false, the true value of P → Q is false
P called the premise, Q is called conclusion
The meaning of the conclusion is equivalent to "if p," because P, so Q, "as long as P, just q", "only Q, only P," only when q, then p "," P is the sufficient condition of Q "" Q is the necessary conditions of P "" Since P So Q ", etc.

6. Set P, Q is proposition,Equivalent"<=>" is a binary linkage

P <=> Q read as "P and only q" or "p, q equivalent
When and only when the true value of P.Q is the same, the true value of P <=> Q is true.
"Equivalent" is also called two conditions.
The meaning of equivalent coupling words is equivalent to "p, and only when q" "P is Q", "P, Q is the same", "" "" ""

1.1.5 proposition formula and its classification

The compound proposition is a symbol string composed of symbols such as propositions, logical links, and braces; but in turn, the symbol strings consisting of these symbols are not necessarily propositions.

Set A as a proposition formula, b is a consecutive symbol string in A, and B is a proposition formula, and then B is a subsite as a

The proposition formula is not a proposition, only when each proposition variant in the formula is assigned, the true value of the formula is determined, thus becoming a proposition

The reconciliation must be satisfied, but it is not true.
Any two reply-based parallel or unpacking is still a reply
Any two contradictory drawn or allocation is still a contradiction

1.1.6 true value table

A proposition formula can be calculated and represented by a true value table in each true value assignment (combination of each proposition error).

In order to facilitate the constructive truth table, the Special is as follows:

(1) Proposition variations are arranged in words

(2) Assignment of each true value, listed in small to large or from large to small order

(3) If the formula is more complicated, you can list the true value of each sub-formula, and finally list the true value of the formula.

p	q	¬p	p∧q	pVq	p⊕q	p→q	p＜＝＞q
T	T	F	T	T	F	T	T
T	F	F	F	T	T	F	F
F	T	T	F	T	T	T	F
F	F	T	F	F	F	T	T

1.1.7 Symbolization of the proposition

The proposition of the natural language is expressed in the form of the proposition formula, called the symbolization of propositions, which is the primary step of reasoning calculations. The symbolicization of the proposition generally has passed the following three steps:

Find all atomic propositions and symbolize atomic propositions
Find out all the words in the proposition, will join the word symbolization
Composition of atomic propositions and linkages into a compound proposition

2. The proposition logic is equivalent calculation

2.1 equivalent

1. Set A, B is two proposition formulas, if a <=> b is a relief, referred to as A and B equivalend or logic equivalent, remembering a CBB, said AUB is equivalent

2. Judging whether the two proposition formulas are equal to two ways

Truth Table: List the true value table of two formulas, determine whether the output column is the same
Equivalent Relief Algorithm: Pushing a new equivalent by known isometric

3. Isometric calculations cannot directly prove that two formulas are not equal. Proof basic idea of two formulas is to find a true value assignment makes a true, another fake

4. Example P∨ (P∧Q) P

p	q	p∧q
T	T	T
T	F	F
F	T	F
F	F	F

When the proposition variation is more, the task of truth is very good.
The equivalent algorithm is based on the basic equivalent type, applying the rules, replacement rules, and gradually.

5. Theorem (in the rules)

Suppose A is a relief, and all the same proposition varies are converted with the same proposition formula, the result is still a reply.
The value of the call does not depend on the change of the proposition variation. Therefore, after the proposition variation is replaced by any formula, it is still a reply.

2.2 Logistics Treatment and Main Circulation

1. A formula such as A1VA2V ... VAN is referred to as an analysis paradigm, wherein AI (i = 1, ... n) is a combined type. The given proposition formula A, the parallel parallel of the equivalent value is referred to as a fractional paradigm of A.

2. The formula of the A1∧a2∧ ... ∧an is referred to as the uniform paradigm, wherein AI (i = 1, ..., n) is a parallel. The given proposition formula A, the equivalence paradigm of the A is referred to as a dispensing paradigm for A.

Possible connections are only: ∨, ∧, ~
A analyzer is a contradiction. When it is only a contradictory, each of its all-in-one is included, at least one complementary pair is included.
A unparalleled paradigm is a relief, and it is only a complementary pair, that is, when it is a relief, that is, each fractionary pair contains at least one complementary pair.
Any analytical parallel panel is a dispersion paradigm; any partial parallel pair
Set B is a decimal paradigm of A *, then b * is a paradigm for A

3. Theorem (existing theorem): Any proposition formula exists with an equivalence paradigm and an equivalent paradigm of the equivalent.

4. Self-style steps:

Transform other linkages into ∨, ∧, ~

♤ p→q≡¬p∨q

♤ p↔q≡(¬p∨q)∧(¬q∨p)

≡ (p∧q)∨(¬p∧¬q)

Simplify the double negative, and write all ~ write into the text, make it only for proposition variations

♤¬(¬p)≡p

♤¬(p∨q) ≡¬p∧¬q

♤¬(p∧q) ≡¬p∨¬q

Using the assignment law, it will eventually become a parallel or analytical paradigm

♤p∨(q∧r)≡(p∨q)∧(p∨r)

♤p∧(q∨r)≡(p∧q)∨(p∧r)

5. Theorem (existing theorem): Any proposition formula exists with an equivalence paradigm and an assembled paradigm. However, the parallel parallel and the uniform paradigm may not be unique.

2.3 minimal items and great items

2.3.1 minimal

Concept:

If the N a list of variables P1P2, ..., PN composed of PN composed of Q1 ∧Q2∧ ... ∧Qn satisfies Qi = Pi or ¬Pi (1 ≤ i ≤ N):

(1) Each proposition variation is different from its negative, but one of the two must appear and only once;

(2) The first proposition variable or its negation appears on the top of the left,

Then, the combined Q1 ∧Q2∧ ... Qn is a very small item

2. Despect the proposition variation as 1, the negation of the order is 0, so each minimal term corresponds to a binary number, the binary number is

This very small real value is a true assignment. Convert it into a decimal number

I, as the next standard, the minimal term can be represented as MI.

3. Nature:

1 End of the formula containing N life variables, all the number of possible minimal items and the number of explanations of the formula are 2 ⁿ

2 Each minimal item is only true in one explanation.

3 extremely small items two or two, and mi∧mj≡f (i j) because at least one pair of complementary pairs

4 The formula of the precaution composition of 2 ⁿ ministries must be a reply

2.3.2 great item

Concept:

If n-quotes variables P1, P2, ..., PN composed of Q1 ∨Q2 ∨ ... ∨Qn satisfy Qi = Pi or ¬Pi (1 ≤ i ≤ N) is:

(1) Each proposition variation is different from its negative, but one of the two must appear and only once;

(2) Item i or its negation appears on the i-bit from left.

Then, the drawback Q1 ∨Q2∨ ... ∨QN is a great term

2. Describe the proposition variation as 0, the denying of the order as 1, so each great item corresponds to a binary number, the binary number is

This great top truth is a fake assignment. Converting it into a decimal number I, as the foot label, the extreme major item can be represented as mi.

3. Nature:

1 End of the formula containing N-kind variants, all possible numbers of great items and the number of explanations of the formula are 2 ⁿ

2 Each great item is only in an explanation

3 extremely large items two or two, and mi∨mj≡t (i j), because at least one pair of complementary pairs

4 Formulas that are justified by 2 ⁿ

3. Theorem

Set MI and MI are extremely small items and great items formed by propositions P1, P2, ..., PN, then Mi¬¬mi.

2.3.3 Logistics

1. Concept: If all the all-in-parallel formations composed of n-list variations are extremely small, it is called the primary analysis to represent σ representation.

2. Given the proposition formula A, the primary analyzer of the equivalent value is called the primary analysis of A.

3. That is, the parallel formation of limited minimal items is called a primary analysis.

4. Spectational steps for the primary analysis of the proposition formula A:

1). Aspiration of A, an analytical formula A ';

2). If the A 'is not included in the A', the proposition variation Pi is not included or its negation ¬Pi, the B is developed into the following formula: B≡B∧ (PI∨¬PI) ≡ (PI∧B) ∨ (B∧¬pi)

3). Subject to the proposition variations, contradictions and repeated minimal items "eliminated", such as P. P-replacement, p∧¬P with F replacement, MIVMI, with MI replacement;

4. Such as M1VM2VM3 is represented by σ (1.2, 5)

2.3.4 Maincoming

1. Concept: If all the parallel formations in the incoming paradigm composed of N life, it is called the main combination of π.

2. Given the proposition formula A, the main combine of the equivalence of the A, is called the main reference formula of A.

3. That is, the acquisition paradigm consisted only by the limited number of extremely large items is called the main partial reference.

4. Subject to the main combined step of the proposition formula A:

1). Ask a uniform paradigm A ';

2). If the A 'is not included in the A', the proportion variation PI is not included or its negation ¬Pi, the B is developed as follows: B≡B∨ (PI∧¬PI) ≡ (PI∨B) ∧ (B∨¬pi)

3). Subject to proportional variations, reluctance and repeated great items "elimination"

4). Put the great item in order of small to large, and use π. Such as M1∧m2∧m5 is represented by π (1.2, 5)

5. Theorem (the primary analysis

Any formula containing n-quotes varies, there is a unique and equivalent and especially the primary analysis of this N-life variable.

6. Theorem (mainly in combination)

Any formula containing n-quotes varies, there are unique equivalents and the main combine of this N-scenritable variation.

7. The purpose of the primary analysis

Judging whether the two propact formula is equivalent

Since the primary analysis of any proposition formula is unique, if AUB, the same analyzer of A and B have the same primary analysis, if A, B has the same primary analysis, there must be AUB .

Judgment the type of proposition formula

Set A is a proposition formula containing N life variables

☝A is a reply, and when and only the primary analysis of A is included in all 2 ⁿ

☝A is contradictory, and when and only when A's primary analysis is not included in the empty formula

☝ If the primary segment of the primary analysis is included in the primary item, then A is satisfying

The true and fake assignment of the formula

In the above example (2) (p → Q) ∧ ∧ ≡ (1.3) "ft", "TT" is true assignment, "ff", "tf" is false assignment

2.4 Complete set of proposed joint words

1. Set C is a collection of linkages, if any propact formula composed of N life variations, there is only a formula containing only C-z-circular clutches, and that c is a complete linkage collection, or C It is the complete set of linkages

2. Set C is a collection of coupling words, which can be referred to as a redundant coupling word for other coupling words defined by other C.

3. Complete sets of combined words without redundant joint words are called extremely small complete sets

4. Theorem: If all the consecutive words in a set of gynees can be defined by a set of S₂, S₂ is also a combination of linkages.

5. Theorem: The collection of joints will be completely set:

(a)S₁={~(¬),∧,∨}
(b)S₂={~,∧}
(c)S₃={~,∨}
(d)S4={~,→}

6. Proof s₁ = {~ (¬), ∧, ∨} is complete

Any proposition formula composed of N life variations is modified with the only primary analyzer, and only links in the primary analysis (¬), ∧, ∨. So {~ (¬), ∧, ∨} is the complete set of linkages

3. The reasoning theory of proposition logic

3.1 Propositional Logic

1. Reasoning is the thinking process of introducing conclusions from premise

2. Prerequisites Originally, it is a known proposition formula A1, A2, ... AN

3. Conclusion is the proposition formula for the introduction of application reasoning rules

4. The correct reasoning or effective reasoning is referring to the A1 ∧A2 ∧ ... ∧an → b is a reply

5. At this time, B is logical inference or effective conclusions of A1, A2, ... AN.

6. The basic method of explaining the problem is:

Symbolize the proposition
Writing premise, conclusions and reasoning forms
Judging the correctness of the reasoning form

It is considered here that the effectiveness of the inference form structure is considered, not the correctness of the conclusion.

7. Judging whether a reasoning form is correct, from the definition, it is to judge whether a means of repeat,

8. Decideting the formula type

Truth table
Equivalent algorithm
Logistics

9. Use the method of determining the type of formula to determine the correctness of the push-form, has the advantages of generality, extensive, but not enough:

Cannot visually see the prior festivals A to the conclusion B's deduction process, and it is also difficult to promote to predicate logic

10. Certification method for establishing reasoning calculations (also known as interpretation

From the premise A1, A2, ... AN start
Based on the use of basic reasoning formulas and several reasoning rules
Gradually push the performance conclusion B

11. Main reasoning rules

a) premise introduction

On any steps of proven, the premise

b) Conclusion Introduction Rule

On any steps of the certificate. The conclusions proven can be used as a follow-up proof

The same rules in the calculation of proposition logic

d) Replacement rules

On any steps proven: Any subordistance formula in the proposition formula can be replaced with proposition formula with equivalence

4. First Order Logic Basic Concept

4.1 predicate and quantifier

1. Individual Word: Yes, the word representing the object of thinking, indicating the specific or abstract object of independent existence

2. Specifically, the identified individual word is called individual common items, generally used in A, B, and C

3. Abstract, uncertain individual words are called individual variations, generally use x, y, and z

4. The range of value of individual changes is called individual domains or arguments

5. Individual domains of all things in the universe are called the full total body domain

6. The word representing the nature of individual words or the relationship between each other is called predicate

7. If there is only one individual word in the proposition, the word that represents the nature or attribute of the individual words is called predicate. This is a dollar (mesh) predicate, with P (x), q (x), indicated.

8. Example:

Human(Socrates)
Mortal(Socrates)

9. If the individual words in the proposition are more than one, then the words indicating the relationship between these individual words are called predicates. This is a multi-diversity predicate, and there are predicates P (X1 ..., XN) of n individuals, called N yuan (me) predicates, with P (X, Y), Q (X, Y), R (X, Y, Z), ...

10. Example:

Greater (x,y)

Equal (x,y)
Friend (Yu Bobao, hour)

11. Accurately, predicates P (x), q (x, y), ... is the form of proposition rather than proposition

Because neither specify the meaning of predicate symbols P, Q, and individual words x, y is also individual variations without representing a particular thing, thereby impossible to determine the true value of p (x) .q (x, y)
The proposition form is talented only when the predicate determines the meaning of the predicate, and the individual word is set to individual common items, the proposition form is talented as proposition.

12. The word used to represent the number of individuals is the quantifier, and the word plus measurement is called the quantization of the predicate.

13. It can be seen as a restriction, constraint of the individual words, but not a specific description of one, two, three ..., but discuss two most common quantity limit words:

Full name quantifier

Quantifier

4.2 predicate formulas and classification

1. Pre-predicate formula (WFF) in predicate logic is defined as:

(1) Propositional, proposition variations and atomic predicate formulas (excluding predicates without joint words) are predicate formulas:

(2) If A is a predicate formula, then ~ a is also a predicate formula

(3) If A and B are predicate formulas, the symbol strings of the logical linkage words links A and B are also predicate formulas.

(4) If a is a predicate formula, and there is no "X and $ x, (" x) a (x), ($ x), ($ x), is also a predicate formula:

(5) Only a limited number of symbols composed of finishes (1) - (4) are predicate formulas.

2. Predication formula is also known as a combined formula, abbreviation formula

3. Example:

4. Similar to the true value assignment of the proposition formula in proposition logic, you can give different interpretations on predicate logic formulas:

A explanation of the predicate formula consists of below 4:

Non-empty arrest D;
Part D Part of specific elements;
D on some specific functions;
D on some specific predicates.

5. Interpret the specific meaning of the corresponding individuals, individual variations, function symbols, and predicate symbols, and the value range of individual changes

6. If there is at least a portion of the four components of the interpretation, the two explanations are different.

7. A formula can be given a given meaning of different explanations, an explanation can be pairs of different formulas

8. Similar to proposition logic, you can classify predicate formulas:

Set A as a predicate formula, if A is true in any explanation, then A is called a generally effective formula or logic effective formula, example: ("x) p (x) → p (y)
Set A as a predicate formula, if A is fake in any explanation, then A is an unsaturated formula or contradiction, an example: ("x) (p (x) ∧ ~ p (x)); ("x) p (x) ∧ ($ x) ~ p (y)
Set A as a predicate formula, if at least one interpretation makess A true, then A is a satisfactory formula

9. Decision of predicate logic:

Theorem (Church Prayer)

For any predicate formula, there is no feasible method to determine if it is generally effective. The word logic is not determined. However, some subclasses of the predicate formula are determinable

10. Set the schedule formula A0 End Typical variations P1, P2, ..., PN, use N predicate formulas A1, A2, ..., AN, and the resulting formula A is called Substitution of A0.

■ Example: p (y) → q (z), ("x) p (x) → ($ x) q (x) is a restriction example of proposition formula P → Q.

theorem

The replacement example in the proposition formula is a logic effective style, and it is still called a call in the predicate formula; the contradictions in the proposition formula are contradictory.

■ example

Set A as a predicate formula, b is a consecutive symbol string in A, and B is a predicate formula, and then B is a subsite as a

Everything in the jurisdiction is called constraints, and the guidance variables are constrained.
All constraints appear as a constraint variable
In addition to variables, there is a restricted variable, unconstrained

4.3. Natural statement

4.3.1 Basic Method

First, you must break the problem into some atomic propositions and logical linkages.
After decomposing individual words, predicates and quantities of each atomic proposition
Translate natural statements in accordance with the representation rules of the formula

■ All the numbers are odd

■ Generally speaking, "all A is B", "is a b", "all A is B." The form of this type of statement can only use "→" without using "∧"

■ There is no positive number is the number of prime numbers

■ Generally speaking, "Some a is B." The form of such statements can only use "∧" without using "→".

■ All positive intensities are positive or inexpected

■ All positive intensity is positive, or all positive integers is odd

■ This is not "equivalent"!

■ For either positive integers, there is a large integer than it.

There is a positive integer, greater than any positive integer

■ This is not "equivalent"!

5. Equivalence calculation of predicate logic

■ Set A, B is two predicate formulas, if A ↔B is a generally effective formula, then A and B equivalents are called A☰B.

■ Similar to proposition logic, two predicate formulas A, B or other values are only true, and the true value of A and B are identical only if any explanation.

■ The equivalence calculation of predicate logic is still the basic equivalent, and the application equivalent calculation rules will gradually

■ The basic equivalent in the predicate logic is mainly divided into two categories:

One is the substantially equivalent substantive example of the essential, either logic transplanted from the proposition formula.

Another type is the equivalent equivalent of the predicate logic, related to the quantifier

5.1 elimination quantifier isometric

5.2 quantomer negative equivalent / De Motor

5.3 Quantitative Issue Regional Contraction and Expansion Equivalence

A (x) is an formula that contains X free appearance, and the predicate formula B does not contain x, there is:

5.4 quantifier assignment isometric

Set A (x), B (x) is a predicate formula that contains X free appearance, there is:

Set A (X, Y) is a predicate formula that contains X, Y free, is:

This group of equivalents indicate that the same quantifiers are independent of the order of the arrangement, but for different quantities, the order cannot be replaced, ie ("x) ($ y) A (x, y) and ($ y) (" x) A (" X, Y) does not equal

■ Equivalence rules of predicate logic

5.5 Replacement Rules

Instead of rules

5.6 Dispute Rules

■ The same individual variable symbol has both constraints and free appearance, easy to cause conceptual confusion

■ In order to avoid this, the following instead of rules and renummers rules are introduced.

The same individual variants only present a form in a formula, either the constraint, or free
At the same time, different quantities constrained different names, easy to computer processing

■ Instead of rules) All free appearance of a free appearance in the predicate formula A is changed to a certain individual variable symbol that has not appeared in A. The rest is unchanged, and the resulting predicate formula is A ', then A☰A '

■ (Distribution Rules) The guidance variation of the predicate formula A and its change in all constraints in the jurisdiction of the domain have not changed a certain individual variable symbol, the rest is constant, and the resulting words The formula is A ', then A☰A'

■ Dispute rules

6. Reasoning of the front bunch of pattern and predicate logic

■ Set A as a predicate formula, if satisfied

All quantities are located at the left side of the formula;
Neither is not included before all quantities;
The jurisdiction of the quantifier extends to the end of the entire formula.

Then A is the forehead

■ General form of front bungeral:

M is the formula of the content of the content, called the formula or mother formula

■ Basic methods of grafting the avatar:

Step 1. Eliminate the links in the predicate formula →, ↔;
Step 2. Move the negative words of the predicate formula ~ right
Step 3. Transmission of the quantifiers in the predicate formula (using the quantifier allocation isometric, quantitative terrorism contraction and expansion equality), if necessary, change the name

■ Theorem (the forefront existence theorem)

Any predicate formula exists with the forefront of the equivalent, but its forward bid is not unique

■

■ Because there is no real value form method in the predicate logic, there is no general method of discriminating A → B, and the basic reasoning formula and reasoning rules are the basic reasoning method of predicate logic.

■ Basic reasoning formula

■ In addition to the reasoning rules used in the reasoning rules used, the reasoning rules used, including four relevant quantities of eliminated and introduction rules:

Full name promotion rules / full name quantifier introduction rules (UG)
Full name example rules / full name quantifier elimination rules (US)
Presence promotion rules / existence quantifier introduction rules (EG)
There is a quantifier to eliminate the rules (ES)

■ Full title quantifier decommissioned rules:

Where Y is an individual in the field

■ It means that if all the x∈D is nature P, then any one of the di must have properties P

■ The conditions used in this rule are:

In the first form, the substitution x Y should be anything that is not in P (individual variables that occur in X.
In the second form, A is any individual common term
When using Y or A to replace the X (X), it is necessary to replace everything in X free.

■ The full number quantifier introduction rules:

■ where Y is an individual in the field. It means that if any of the individual Y∈D has a nature P, all individuals X in D have properties P

■ The conditions used in this rule are:

The P (Y) should be true regardless of the individual variable item Y, P (Y).
The X of the replaced free Y is unconstrained in P (V).

■ There is a quantifier introduction rule:

Where A is an individual common term in the field. It means that if there is a nature P, ($ x) p (x) must be true.

■ The conditions used in this rule are:

A is a specific individual usual term
The X does not appear in P (a) in P (a)

■ There are quantitatives to remove rules:

■ where A is an individual common item in the field. It means that if there is a certain body in the field D, it has a nature P, then there is a specific individual A having the nature P.

■ The conditions used in this rule are:

A is a true individual common item that makes P true
a does not appear in P (x)
There is no other free individual variable in P (x)
A is not used in the derivation

■ The reasoning calculation process using reasoning rules is:

Step 1. First, the reasoning problem expressed by the natural statement will be converted to predicate formulas
Step 2. If you don't directly use the basic reasoning formula, you can eliminate the quantifier.
Step 3. Infancy rules and formulas under unqualified words
Step 4. Finally, the quantifier is introduced to obtain the corresponding conclusion

Pay special attention to the following two points:

(1) need to eliminate the existence of the quantifier

When you go to the full name quantifier, it is generally necessary to use the existence quantifier to eliminate the rules, and then use the full name quantifier to eliminate the rules.

(2) When using US, UG, ES, EG rules, the jurisdiction of the quantifier must extend to the end of the entire formula.

In other words, in the predicate inference containing multiple quantities, the use of elimination rules should be in order from left to right, and the use of rules should be in order from right to left.