Technical elements used in the regulation and control of machines as buttons or switches. This is a two-valued element. For a mathematical description of these components was created two valued algebra, which, after its creator, British mathematician Boole, called Boolean algebra. These are the calculations with binary variables. For computations in Boolean algebra defined laws and rules as in other algebras. These are the laws and rules.
Basic laws of commutative, associative and distributive, which are defined for any algebra, thus Boolean (Table 2), are expressed in two forms, disjunction and conjunction. In classical algebra is about addition and subtraction. In Boolean algebra is a logical sum and logical product.
Property |
Disjunction |
Conjunction |
Commutative property |
A ˅ B = B ˅ A |
A ˄ B = B ˄ A |
Associative property |
(A ˅ B) ˅ C = A ˅ (B ˅ C) |
(A ˄ B) C = A ˄ (B ˄ C) |
Distributive property |
(A ˅ B) ˄ C = A ˄ C ˅ B ˄ C |
(A ˄ B) ˅ C = (A ˅ C) (B ˄ C) |
In practice, the logical functors disjunction, conjunction and negation may use other methods of marking. Sample labeling the various ways in Table 3.
A ˅ B |
A + B |
|
A or B |
A ˄ B |
|
|
|
|
¬ A |
not A |
Boolean algebra is complemented by a set of rules that are used to simplify logic functions. List of all the rules in Table 4.
Rules |
Addition |
Multiplication |
Rule of neutrality 0 and 1 |
A + 0 = A |
|
Rule of aggressiveness 0 and 1 |
A + 1 = 1 |
|
Rule of independence elements |
A + A = A |
|
Rule of excluded middle |
|
|
De Morgan’s rule |
|
|
Rule of absorption |
|
|
Rule of absorption negation |
|
|
Rule of double negation |
|
Simplify the function of three variables:
First, it is done before pointing out brackets and braces is a third application of the rule of absorption of negation.
Furthermore, the rule is applied Independence elements is performed multiplication brackets and application rules exclude the third. The result is 0
This means that the function value is always zero, regardless of the combination of values of input variables, a function is called forgery.