Proposition and logic

What is proposition?

1. can distinguish between true and false statements, valid statements or invalid statements

2. either true or false must be true or false.

3. A proposition without a combination of words is an atomic proposition.

Examples:

Beijing is the capital of China (t)

1 + 1 = 2 (t)

The capital of Japan is Osaka (f)

X = B (not a proposition)

Please listen to me (not a proposition or a imperative sentence)

What's today? (No proposition, no question)

 

What is a composite proposition?

A new proposition formed by combining existing atomic propositions (which cannot be broken down into simpler declarative statements) with logical join Operators

 

No (that is, no):

P is a proposition, so the opposite (opposite) of P is also a proposition, called P's negative proposition,

 

Conjunction (that is, And, intersection): A proposition composed of PQ. It is connected by and is called P to combine Q. True is true, false is false.

Disjunction (that is, or, and): A proposition composed of the proposition PQ. It is connected by or, and called P to extract Q. True is true, false is false.

Note: the abbreviation "v" does not correspond to the natural language "or" 100%! A natural language may occur simultaneously (or simultaneously) or differently (exclusive or), while an analysis can occur simultaneously (or simultaneously)

At eight o'clock A.M. this afternoon, I went to a history lesson or a math lesson. This is a rejection or event that cannot happen at the same time.

I may have rice or noodles at noon. This is optional or can happen at the same time. Only the union or can be used for analysis. Therefore, be sure to pay attention to the expression!

 

Exclusive or:

Proposition PQ: The proposition P is different or Q. The two proposition conclusions are the same as false. The difference is true. The difference or refers to the exclusion or!

 

Conditional connection:

Hypothesis (premise)-> conclusion (result): A New Proposition composed of proposition p and q. if and only if true Q is false, proposition P-> q is false, and the rest, the conclusion is true, and true or false is false.

Similar to the natural language, if ...... So ......, But not all. The conditional proposition is a kind of suggestion. For example, if the first part is F, no matter what the latter part is, the true value of the conditional proposition is T.

 

Inverse Proposition: the inverse proposition of proposition P-> q (if p, then Q or if p is Q, etc.) is Q-> P

Inverse negative proposition: the original proposition is Q if p, and the inverse negative proposition is P if not Q. The original proposition is equivalent to the true or false of its inverse negative proposition.

Negation of a proposition: It only denies the conclusion of this proposition. the negation of a proposition is opposite to that of the original one.

Negative proposition: if the conditions and conclusions of one proposition are the negation of the conditions of the other and the negation of the conclusion, the two propositions are mutually negative. The inverse proposition is equivalent to the negative proposition. If the inverse proposition is true, the negative proposition is true. If the inverse proposition is false, the negative proposition is false.

Original proposition: If P, then q; negative proposition: If not P, then not Q. Note: A negative proposition denies both the conditions and conclusions, while a negative proposition only denies the conclusions.

Examples:

Original proposition implication: If it rains, the team wins.

Negative proposition contrapositive of this implication: if the team does not win, it will not rain. The original proposition is equivalent to its inverse negative proposition.

Reverse proposition converse of this implication: if the team wins, it will rain

Negative proposition inverse of this implication: If it doesn't rain, the team won't win. The inverse proposition is equivalent to the negative one.

 

Common English description of conditional connectors:

If P, then q

If p, q

P is sufficient for Q (commonly known as P is a sufficient condition for q)

Q if p

Q when P

P implies Q

P only if Q

A necessary condition for P is Q (P is a sufficient condition for Q, and Q is a necessary condition for P)

A sufficient condition for Q is P

Q whenever P

Q is necessary for P

Q follows from P

 

Two conditions, if and only if (cannot be considered as equivalent symbols in mathematics)

Proposition PQ forms a new proposition P <-> q. When p and q are true and false, the proposition conclusion is true. Otherwise, all are false, different or opposite, and different or different are true, the two conditions are the same and true.

Similar to the conditional proposition, dual conditions can be considered as "IFF", and can only be defined as true or false based on the combination word.

 

 

Priority of logical operators

Whether or not the first priority is (not), followed by the Union, extraction, condition, and finally the two conditions

 

Translation of proposition Formulas

Example:

If you can access the Internet on campus, you are a computer professional, or you are not a freshman.

A: You can access the Internet on campus.

C: You are a computer professional.

F: You are a new student.

A-› (C ν ¬ F)

A, C, and F are all proposition variables. A-› (C ν ¬ F) is called a proposition formula. ACF is the component of a proposition formula, and there is no true or false proposition formula! Not all proposition variables + logical concatenation + parentheses + strings are proposition formulas! Follow the following rules:

Basis: the variable element of a single proposition is a combination formula.

Induction: If a is a combination formula, Not A is also a combination formula.

If A and B are the combination formulas, (a), (a v B), (a-> B), (a v B)↔B) they are all combination formulas.

Boundary: When and only when there is a limited number of applications, the formula is the sum of the obtained results.

 

The automatic reply cannot be sent when the file system is full

P: The file system is full

Q: The automatic reply can be sent

P → baiq

 

Boolean search)

Logical concatenation words are widely used in the search of a large number of information sets, such as web search. in a strict sense, Boolean logical search uses Boolean logical operators to connect various search terms, then, the computer performs logical operations to find the method of the required information. It is widely used and has the highest usage frequency. A boolean logical operator is used to connect a query term to form a logical search type.

 

Logical confusion Reasoning

There are two types of people in a village. The first is gentlemen who always tell the truth. The gentleman's opposition is Rogue. The rogue always tells lies. One day, you met two people, a and B, judge who a is and who B is.

A said: B is a gentleman.

B said: the two of us are not the same.

Analysis; A is a gentleman, B must be a gentleman, B's words are true, then a is a rogue, which is in conflict with B's words and is excluded. If a is a rogue, B must be a rogue. B is false, and AB is a type of person. B is also a rogue and correct.

Conclusion: AB is a rogue.

 

Bitwise AND logical operators

 

Equality of propositions

Permanent truth type, double argument type, tautology: A proposition formula is always true, and the conclusion of a proposition is true no matter what the component is assigned.

Contradiction: the conclusions of a proposition formula are always false, regardless of how the components are assigned.

 

If P↔Q is always true, so p and q are logically constant. Use the symbol P ≡ Q

Proposition Law

 

Proposition and logic