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Bases of logic, logician of statements, logic sheaves, validity tables.

The algebra of logic (Boolean algebra) is the section of the mathematics which appeared in the XIX century thanks to the efforts of English mathematician D. Boulle. Logic, (from Greek’s logos – a word, concept, a reasoning, reason), or Formal logic – a science about laws and operations of the correct thinking.

First the Boolean algebra had no practical value. However in the XX century it found an application in the description of functioning and development of various electronic schemes. Laws and the device of algebra of logic began to be used at the design of various parts of computers (memory, the processor). Though it not unique scope of this science.

The algebra of logic, first, studies methods of establishment of the validity or falsity of difficult logic statements by means of algebraic methods. Secondly, the Boolean algebra does it in such a manner that the difficult logic statement is described by the function which result of calculation can be either truth or lie (1 or 0). Thus arguments of function (simple statements) also can have only two values: 0 or 1.

The simple logic statement is phrases of type «two more than one», «5.8 is an integer». In the first case we have truth and in the second it is lie. The algebra of logic doesn't concern an essence of these statements. If someone decides that the statement «Earth square» is true, the algebra of logic will accept it as the fact. The matter is that the Boolean algebra is engaged in calculations of result of difficult logic statements on the basis of in advance known values of simple statements.

Logic operations. Disjunction, conjunction and denial

In a natural language we use the various unions and other parts of speech. For example: «and», «or», «or», «not», «if», «then». Example of difficult statements: «it has knowledge and skills», «it will arrive to Tuesday, or on Wednesday», «I will play when I will make lessons», «5 isn't equal to 6». How do we solve it?, What were we told: the truth or not? Somehow logically, even somewhere unconsciously, proceeding from the previous life experience, we understand that the truth at the union «i» comes in case of truthfulness of both simple statements. It is necessary to one to become lie and all difficult statement will be false. And here, at a sheaf «or» there should be the truth only one simple statement, and then all expression becomes true.

The Boolean algebra shifted this life experience on the mathematics device, formalized it, and entered rigid rules of receiving unequivocal result. The unions began to be called here as logic operators.

The algebra of logic provides the set of logic operations. However three of them deserve special attention since with their help it is possible to describe all the others, and, therefore, to use less various devices when designing schemes. Such operations are conjunction (And – logic multiplication), disjunction (OR – logic addition) and denial (NOT). Often conjunction designate &, a disjunction – ||, and denial – line over a variable designating the statement. At conjunction the truth of difficult expression arises only in case of the validity of all simple expressions of which the difficult consists. In all other cases difficult expression will be false.



At a disjunction the truth of difficult expression comes at the validity at least one simple expression entering into it or two at once. It happens that difficult expression consists of more than two simple. In this case it is enough, that one idle time was true and then all statement will be true.

Denial is a monadic operation, because it is carried out in the relation to one simple expression or in the relation to the result of difficult expression. As a result of denial the new statement opposite initial turns out. The rules of seniority of logic operations:

1. Denial (inversion) – logic action of the first step.

2. Conjunction – logic action of the second step.

3. A disjunction – logic action of the third step.

4. Implication – logic sequence.

5. Equivalence – equivalence.

If in the logic expression the actions of various steps are used, the first steps are carried at the beginning, then the second and only after that the third step are held. Any deviation from this order should be designated by brackets.

The validity table for function implication (logic following):

 

À Â À=>Â

 

The validity table for function equivalence (equivalence):

 

À Â À<=>Â

 

Control questions:

1. What is the uniqueness?

2. What is the relation?

3. What is the set?

4. What is the discrete mathematics?

5. Why does the discrete mathematics make the basis of ADP equipment?

6. What is the graph?

7. What does the oriented graph represent?

8. What is the mixed graph?

9. What are the additional characteristics of graphs?

10. What are the methods of submission of graphs?


Date: 2015-12-24; view: 103


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