MATHEMATICAL LOGIC

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The logic studies the form of reasoning, is a discipline that by means of rules and techniques determines if an argument is valid. The logic is widely applied in philosophy, mathematics, computing, physics. In the philosophy to determine if a reasoning is valid or not, since a phrase can have different interpretations, nevertheless the logic allows to know the correct meaning. In mathematics to prove theorems and infer mathematical results that can be applied in research. In the computer for check programs. In general the logic is applied in the daily task, since any work that is carried out has a logical procedure, for example; In order to go shopping at the supermarket, a housewife has to perform a certain logical procedure that allows her to perform this task. If a person wishes to paint a wall, this work has a logical procedure, since he can not paint if he does not prepare the painting beforehand, or he should not paint the lower part of the wall if he did not previously painted the upper part because it would stain what already has painted, also depending on whether he is left or right, he can paint from left to right or from right to left depending on the case, all this is the application of logic.
PROPOSITIONAL LOGIC
Classes of propositions
There are two kinds of propositions:
Simple and compound propositions, also called atomic and molecular respectively.
to. Simple Propositions.- Also called atomic. They are those propositions that can not be divided. Example:
The sky is blue. (true)
Nomenclature: p
b. Compound Propositions- Also called molecular. They are those that are formed by two or more simple propositions joined by the logical operators. Example:
I went to the bank, but the bank was closed.
The readers of this book are young or university students.
If next Wednesday I take out the lottery then I will give you a car.
Logical connective
They are elements that serve as a link between the propositions, to form another, called molecular proposition.
Negation
It is a logical element that acts independently of the proposition.
It reads no p.
RULE.- The denial of a true proposition is false. The denial of a false proposition is true.
Example:
p.- Juan talks
-p.- Juan does not talk

Conjunction
It is the union of two or more propositions by the logical connective "y", "but", "also", "nevertheless", "besides", etc.
Read p and q.
RULE.- The conjunctive proposition is true only when the two propositions are true (p and q), in any other case it is false.
Example:
Q: The house is dirty.
Q: The employee cleans it tomorrow
Q: The house is dirty and the employee cleans it tomorrow

Disjunction
Join propositions through the logical connective "o".
Read p or q.
RULE.- A disjunctive proposition is true when at least one of its components is true. It is false only when all its components are false (p or q).
Q: Pedro plays basketball
Q: Maria plays soccer
PvQ: Pedro plays basketball or María plays soccer.

Negative Conjunction
It is the union of two or more propositions by "ni".
It reads neither p nor q.
RULE.- The result is true only when the two propositions are false (neither p nor q), in any other case it is false
Q: Pedro plays basketball
Q: Maria plays soccer
PvQ: Pedro plays basketball or María plays soccer.

Exclusive Disjunction
It is the union of two or more propositions through the logical connective "o".
Read or p or q, but not both.
RULE.- The proposition is true when the first proposition is true and the second is false or when the first proposition is false and the second is true.
Conditional
It comes to be the combination of two propositions with "if ... then".
Read if p then q.
RULE.- A conditional proposition is false when the first proposition is true and the second is false. It is true in any of the other ways
Example:
Q: If I get the lottery
Q: I will give you a car
Q: If I take out the lottery then I will give you the car.

Biconditional
It is the union of two propositions by "if and only if". Read p if and only if q.
RULE.- A biconditional proposition is true when either its two components are true or its two components are false.