[mathematical logic] propositions and connectives (propositions | propositional symbolization | truth connectives | no | conjunction | disjunction | non truth connectives | implication | equivalence)

One . proposition Concept

1. proposition Concept

( 1 ) The main content of propositional logic ( Logic Reasoning proposition | Smallest unit | The simplest and most basic part )

The main content of propositional logic :

• 1. Logic , Reasoning And proposition Relationship : Logic Main research reasoning process , reasoning process must rely on proposition To express ;
• 2. Smallest unit : In propositional logic , proposition yes Smallest unit ;
• 3. The simplest part : proposition yes In mathematical logic The most basic , The simplest part ;

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( 2 ) What is a proposition ( Declarative sentence | True and false Must live And Only live firstly )

What is a proposition :

• 1. Propositional concept : proposition yes The statement is objective Statement of what happened outside ;
• 2. True or false : Proposition is Or is it true Or false Of Declarative sentence ;
• 3. Propositional features :① Declarative sentence ; ② True and false must rank first , Only one of them ;
• 4. Description of proposition determination : The following two cases are propositions ;
  • ① For what will happen in the future : As long as it is True and false are only one of them , And it's a declarative sentence , Then this is the proposition , Although now I don't know whether it's true or false , But it must be Not true or false ;
  • ② Unproved theorem : Such as Goldbach conjectures , We I don't know whether it's true or false , But its If proved It must be true or false Of Declarative sentence , Therefore, it is also a proposition ;

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2. proposition give an example

( 1 ) Examples of propositions ( False or untrue | I will know in the future It must be true or false | It will prove in the future Must be true or false )

The following sentences are propositions :

• 1.( 8 Less than 10 ; ) : State 8 and 10 The relationship between , yes True proposition ; This has happened ;
• 2.( 8 Greater than 10 ; ) : State 8 and 10 The relationship between , The statement is wrong , It's a false proposition ; This is impossible ; But it is Declarative sentence also Not true or false ;
• 3.( At the end of the 21st century , Humans will live in space ; ) : It's a declarative sentence , It hasn't happened yet , But it must be true or false , Whether it will happen in the future Not sure , however We don't know Doesn't mean it doesn't exist , You will know at some time , Such as At the end of the 21st century 1 second ;
• 4.( Any one > 5 The even number of can be expressed as the sum of two primes - Goldbach conjectures ) : crown jewels , It's a proposition , It's a declarative sentence , But now I don't know whether it's true or false ; But it will eventually prove this conjecture ;
• 5.(

  2

  \sqrt{2}

  2
  ​ In the decimal expansion of 12345 Appear even many times ; ) : It's true or false , But true or false, I don't know when to know ;

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( 2 ) Not propositional examples ( Not a statement | No judgment | True or false is uncertain | paradox )

It's not a proposition :

• 1.( 8 Greater than 10 Do you ? ) : Not a statement , yes questions ;
• 2.( No Smoking ! ) : Not a statement , yes An imperative sentence , No judgment , True or false is uncertain ;
• 3.( X Greater than Y . ) : It's a declarative sentence , however True and false Not sure ;
• 4.( I'm lying . - paradox ) : It's a declarative sentence , But it belongs to paradox ;
  • ① Outer meaning : If I'm lying , This proposition is false ; If I didn't lie , This proposition is true ;
  • ② If the proposition is true : It means I'm lying , The meaning is This proposition is false , There is a contradiction ;
  • ③ If the proposition is false : That means I didn't lie , The meaning is This proposition is true , There is a contradiction ;

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Two . Compound proposition And Proposition symbolization

1. Conjunctions and Compound proposition

( 1 ) Complex proposition introduce ( Whether a compound proposition is true or false is judged by the truth of its small proposition )

Complex proposition : from Simple proposition can structure more Complex propositions ;

• 1. midsemester , Zhang San No, Pass the exam ;
• 2. Among them, the exam , Zhang San and Li Si all Passed the exam ;
• 3. Among them, the exam , Zhang San and Li Si Zhong someone Take an examination of the 90 branch ;
• 4. If Zhang San can take the exam 90 branch , Then Li Si also Ability to test 90 branch ;
• 5. Zhang San can take the exam 90 branch If and only if Li Si You can also take the exam 90 branch ;

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( 2 ) Conjunctions and Compound proposition

Conjunctions and Compound proposition :

• 1. Conjunctions : Above No, , If that , Equal conjunctions Become Conjunctions ;
• 2. Compound proposition : By the conjunction and proposition Connected by More complex propositions Become Compound proposition ;
• 3. Simple proposition : relatively , Can't be broken down into It's simpler The proposition of Become a simple proposition ;
• 4. True or false compound proposition : Compound proposition Of True and false Completely from Make it Simple proposition Of True or false decision ;
• 5. Simple proposition and Compound proposition Division yes Relative ;

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2. Proposition symbolization

( 1 ) Proposition symbolization

Proposition symbolization :

• 1. Proposition symbolization : take proposition Symbolization , Write it down as

  p

  ,

  q

  ,

  r

  ,

  ⋯

  p , q , r , \cdots

  p,q,r,⋯ , Be similar to Algebra. in Use

  a

  a

  a representative 1 The numbers are the same ;

• 2. Symbols are variables :

  • ① On behalf of the digital : In algebra , Using letters

    a

    a

    a Instead of Numbers , Which number does it represent Not sure , Just know that this is a number ;

  • ② Stands for proposition : Empathy , Propositional symbols

    p

    ,

    q

    ,

    r

    p, q, r

    p,q,r Instead of proposition , What propositions does it represent I'm not sure , Just know that this is a proposition ;

• 3. Changyuan and Argument :

  • ① Changyuan : In algebra Letter a When it definitely represents a number , be called Changyuan ;
  • ② Argument : In algebra Letter a When representing uncertain numbers , be called Argument ;

• 4. Propositional constant and Propositional argument :

  • ① Propositional constant : proposition

    p

    p

    p representative determine When the proposition of , be called Propositional constant ;

  • ② Propositional argument : proposition

    p

    p

    p representative Not sure When the proposition of , be called Propositional constant ;

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( 2 ) Propositional symbol Value Numbered

proposition True or false Symbolization :

• 1. really ( True ) : Write it down as $1$

  T ;

• 2. false ( False ) : Write it down as $0$

  F ;

• 3. Proposition value : Propositional argument p Take value $0$

  1 , Value

  p It's a false proposition ;

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3、 ... and . Conjunctions

( 1 ) Negative connectives

Negative connectives :

• 1. Definition : set up p by A proposition , Compound proposition Not p be called p The negative of , Write it down as

  ¬

  p

  \lnot p

  ¬

  \lnot

  ¬p ;

  ¬ Become a negative conjunction ;

• 2. Truth table :

  ¬

  p

  \lnot p

  ¬p It's true here p For false ;

p	$\neg p$
0	1
1	0

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( 3 ) Conjunctions

Conjunctions :

• 1. Definition : set up p , q by Two propositions , Compound proposition " p and q " be called p , q Conjunctive of , Write it down as

  p

  ∧

  q

  p \land q

  ∧

  \land

  p∧q ,

  ∧ be called Conjunctions ;

• 2. Truth table :

  p

  ∧

  q

  p \land q

  p∧q really If and only if p And q At the same time, it's really ;

p	q	$p\wedge q$
0	0	0
0	1	0
1	0	0
1	1	1

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( 3 ) Disjunctive connectives

Disjunctive connectives :

• 1. Definition : set up p , q by Two propositions , Compound proposition " p perhaps q " be called p , q Disjunctive of , Write it down as

  p

  ∨

  q

  p \lor q

  ∨

  \lor

  p∨q ;

  ∨ Referred to as Disjunctive connectives ;

• 2. Truth table :

  p

  ∨

  q

  p \lor q

  p∨q It's true , If and only if p And q At least one is true ;

p	q	$p\vee q$
0	0	0
0	1	1
1	0	1
1	1	1

• 3. give an example : Among them, the exam , Someone from Zhang San and Li Si middle school took the exam 90 branch ;
  • ① p representative Zhang San Take an examination of the 90 branch ;
  • ② q representative The Li Sikao 90 branch ;
  • ③ $p\vee q$

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( 4 ) Contains connectives

Contains connectives :

• 1. Definition : set up p , q by proposition , Compound proposition " If p , be q " be called p Yes q Of Implication , Remember to do

  p

  →

  q

  p \to q

  →

  \to

  p→q , among also called p by This implication Of The front part , become q by This implication Of Afterpiece ;

  → by Implicative connectives ;

• 2. Truth table :

  p

  →

  q

  p \to q

  p→q false If and only if p really and q false ;

p	q	$p\to q$
0	0	1
0	1	1
1	0	0
1	1	1

• 3. give an example : If Zhang San can take the exam 90 branch , Then Li Si can also take the exam 90 branch ;

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( 5 ) Equivalent connectives

Equivalent connectives :

• 1. Definition : set up p , q by proposition , Compound proposition " p If and only if q " Referred to as p , q The equivalent of , Remember to do

  p

  q

  p \leftrightarrow q

  \leftrightarrow

  pq ,

  Remember to make equivalent connectives ;

• 2. Truth table :

  p

  q

  p \leftrightarrow q

  pq really If and only if p , q It's true at the same time or At the same time, it's false ;

p	q	$pq$
0	0	1
0	1	0
1	0	0
1	1	1

• 3. give an example : Zhang San can take the exam 90 branch If and only if Li Si You can also take the exam 90 branch

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3、 ... and . Examples of propositional symbolization

( 1 ) Proposition symbolization ( Look closely at three examples )

Proposition symbolization :

• 1. iron and oxygen compound , but iron and nitrogen Do not combine ;

  • ① proposition p : Iron and oxide combine ;
  • ② proposition q : Iron and nitrogen combine ;
  • ③ Compound proposition :

    p

    ∧

    (

    ¬

    q

    )

    p \land ( \lnot q )

    p∧(¬q) ;

• 2. If I get off work early , Just go to the store , Unless I'm tired ;

  • ① proposition p : I get off work early ;
  • ② proposition q : Go to the shop ;
  • ③ proposition r : I'm exhausted ;
  • ④ Compound proposition :

    (

    (

    ¬

    r

    )

    ∧

    p

    )

    →

    q

    ( ( \lnot r ) \land p ) \to q

    ((¬r)∧p)→q : The premise of going to the store yes Not tired also Get off work early ;

• 3. Li Si is a computer student , He lives in 312 room or 313 room ;

  • ① proposition p : Li Si is a computer student ;
  • ② proposition q : Li Si lives in 312 room ;
  • ③ proposition r : Li Si lives in 313 room ;
  • ④ Compound proposition : $p\wedge ((q\vee r)\wedge (\neg (q\wedge r)))$

    ¬(q∧r) ;

  • ⑤ Compound proposition :

    p

    ∧

    (

    (

    q

    ∧

    (

    ¬

    r

    )

    )

    ∨

    (

    (

    ¬

    q

    )

    ∧

    r

    )

    )

    p \land ( ( q \land ( \lnot r ) ) \lor ( ( \lnot q ) \land r ) )

    p∧((q∧(¬r))∨((¬q)∧r)) ; here Li Si To live in 312 Don't live 313, Li Si lives in 313 Don't live 312 Only one of them can be taken ;

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( 2 ) Proposition symbolization Be careful ( ① Conjunctions And Daily vocabulary is inconsistent | ② True or false propositions are understood by definition | ③ You can't take your seat according to the number | ④ Some words can also be expressed as five connectives )

Note on the symbolization of propositions :

• 1. Conjunctions are not completely consistent with everyday vocabulary : Above Five connectives Not , Disjunction , Syntaxis , implication , Equivalent , originate For daily use Corresponding vocabulary , But not exactly ;
• 2. True or false propositions are understood by definition : The true and false values of compound propositions composed of connectives According to they Of Definition To understand the , Cannot be understood according to the meaning of everyday language , Such as Rougamo and so on Everyday meaning ;
• 3. You can't take your seat according to the number : Don't see or It means

  ∨

  \lor

  ∨ Disjunction , If the above lives 312 or 313 The situation of , Want to consider Only live in 312 , Only live in 313 , Live in 312 and 313 The situation of ;

• 4. Some words can also be expressed as these five connectives : Such as “ however ” It can be expressed as “

  ∧

  \land

  ∧” ;

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