Modern computers, based on "ancient" electronic computers, rely on certain postulates as basic principles of work. They are called the laws of algebra of logic. For the first time such a discipline was described (of course, not as detailed as in its modern form) by the ancient Greek scholar Aristotle.
Representing a separate section of mathematics, within the framework of which the propositional calculus is studied, the algebra of logic has a number of clearly constructed conclusions and conclusions.
In order to better understand the topic, we will analyze concepts that will help in the future to learn the laws of the algebra of logic.
Perhaps the main term in the studied discipline is the statement. This is a statement that cannot be both false and true. Only one of these characteristics is always inherent in him. In this case, it is conventionally accepted that truth be given the value 1, falsehood - 0, and the statement itself should be called a certain Latin letter: A, B, C. In other words, the formula A = 1 means that the statement A is true. Statements can be dealt with in a variety of ways. Briefly consider those actions that can be performed with them. We also note that the laws of the algebra of logic cannot be learned without knowing these rules.
1. The disjunction of two statements - the result of the operation "or". It can be either false or true. The symbol "v" is used.
2. The conjunction. The result of such an action, performed with two utterances, will be a new utterance, true only when both initial utterances are true. The operation "and", the symbol "^" is used.
3. Implication. Operation "if A, then B". The result is a statement that is false only if A is true and B. is false. The symbol “->” is used.
4. Equivalence. Operation "A if and only if B, when." This statement is true in cases where both variables have the same estimates. The character "<->" is used.
There are also a number of operations close to implication, but they will not be considered in this article.
Now, we will consider in detail the basic laws of the algebra of logic:
1. Commutative or relocational states that the change of places of logical terms in operations of conjunction or disjunction does not affect the result.
2. Combinative or associative. According to this law, variables in conjunction or disjunction operations can be grouped.
3. Distribution or distribution. The essence of the law is that the same variables in equations can be put out of brackets without changing the logic.
4. The law of de Morgan (inversion or negation). The negation of the conjunction operation is equivalent to the disjunction of the negation of the original variables. The negation of the disjunction, in turn, is equal to the conjunction of the negation of the same variables.
5. Double negation. Denial of a statement twice gives the result of the original statement, three times - its denial.
6. The idempotency law is as follows for logical addition: xvxvxvx = x; for multiplication: x ^ x ^ x ^ = x.
7. The law of non-contradiction states: two statements, if they are contradictory, cannot be true at the same time.
8. The law of exclusion of the third. Among two contradictory statements, one is always true, the other is false, the third is not given.
9. The law of absorption can be written in this way for logical addition: xv (x ^ y) = x, for multiplication: x ^ (xvy) = x.
10. The law of bonding. Two adjacent conjunctions are able to stick together, forming a conjunction of a lower rank. At the same time, the variable by which the initial conjunctions stuck together disappears. Example for logical addition:
(x ^ y) v (-x ^ y) = y.
We examined only the most used laws of the algebra of logic, which in fact can be many more, since often logical equations take a long and ornate form, which can be reduced by applying a number of similar laws.
As a rule, for the convenience of calculating and identifying results, special tables are used. All existing laws of the algebra of logic, the table for which has the general structure of a grid rectangle, are painted by distributing each variable in a separate cell. The larger the equation, the easier it is to deal with it using tables.