Boolean algebra
pages 10-14
1.2 The DITIMI ALGEBRA BOOLE
1.2.1 Definition
The Algebra Boole (Boolean algebra) takes its name from G. Boole (1815-1864), who developed an algebraic system (1854) for the systematic treatment of logic. The axioms of Boole Algebra expressed by E. V. Huntington (1904).
The variables used in Boole Algebra called logical variables because they can get two (2) only values: 0 and 1. That is why the Boole Algebra is the basis for digital electronics.
Boole algebra defined three basic operations:
ð the operation NOT (NOT) with symbol
ð act AND (AND) with a symbol;
ð act OR (H) with the symbol +
The NOT operation
NOT participate in practice only a logical variable and the outcome of the operation is the complement (inverse) of this variable, ie if the variable has the value "0" then the result is "1" and vice versa if the variable has the value " 1 "then the result is" 0 ".
If A is a logical variable, then the NOT operation is expressed by the relation:
The truth table of the NOT operation is presented in Table 1.2.1.
Drank 1.2.1
Truth table of NOT operation
A
0
1
1
0
The AND operation
In practice, logical AND between two variables and the outcome of the operation is "1" if both variables are "1".
If A and B are two logical variables, then the AND operation is expressed by the relation:
Y = A · B
Note: the symbol of the act AND (·) can be omitted in the expressions of algebra Boole (A · B = AB).
The truth table of the AND operation is presented in Table 1.2.2.
Drank 1.2.2
Truth table of AND operation
A
B
Y = A · B
0
0
0
0
1
0
1
0
0
1
1
1
The OR operation
In practice, logical OR between two variables and the outcome of the operation is "1" if at least one of the two variables are "1".
If A and B are two logical variables, the OR operation is expressed by the relation:
Y = A + B
The truth table of OR operation is presented in Table 1.2.3.
Drank 1.2.3
Truth Table of the Act OR
A
B
Y = A + B
0
0
0
0
1
1
1
0
1
1
1
1
1.2.2 Axioms Huntington
1. Neutral element of the AND and OR
The neutral element of the AND operation is 1 and the neutral element of the OR operation is 0.
a x · 1 = 1 · x = x
b x +0 = 0 + x = x
This axiom can be verified from the truth tables of operations AND and OR, by showing that:
0.1 = 1.0 = 1.1 = 0 and 1
and
0 +0 = 0 and 1 +0 = 0 +1 = 1
2. Commutative property of AND and OR operations
The operations AND and OR are commutative property.
a x · y = y · x
b x + y = y + x
3. Distributive property of AND and OR operations
The AND operation is distributive property on the act and the act OR OR is the distributive property to the operation AND.
a x · (y + z) = (x · y) + (x · z)
b x + (y · z) = (x + y) · (x + z)
4. Unique Supplement (NOT)
Each logical variable x has a unique supplement with the following properties:
a x · = 0
b x + = 1
This axiom can be verified from the truth tables of the Act NOT, by showing that:
· 0 = 0 and = 0.1 1 = 1.0; = 0
and
0 + = 0 +1 = 1 and 1 + = 1 +0 = 1
1.2.3 Principle dualism
The validity of expressions of Boole Algebra continues, if one changes the AND and OR operations and data between the neutral (· '+ and 0 "1).
For example, if the expression is x +1 = 1, then, the expression x · 0 = 0 and an expression called the dual of the other.
Boole Algebra Theorems 1.2.4
Theorem 1.
a x · x = x
b x + x = x
Theorem 2.
a x · 0 = 0
b x +1 = 1
Theorem 3.
x =
Theorem 4. Prosetairistiki property
a x · y · z = x · (y · z) = (x · y) · z
b x + y + z = x + (y + z) = (x + y) + z
Theorem 5. Theorem absorption
A. x + x · y = x
b x · (x + y) = x
Theorem 6. De Morgan's theorem
a
b
Comment:
The De Morgan's theorem applies to more than two variables:
a
b
1.2.5. Priority actions
To perform operations on expressions of Boole Algebra is necessary to determine the priority of their execution, as is known in mathematics algebra.
Table of precedence of operations is presented in Table 1.2.4.
Table 1.2.4.
Priority actions
Priority
Practice
1
()
2
NOT
3
AND
4
OR
From the priority list of acts that one expression of Boole Algebra acts performed first in parentheses, calculated after the supplements, then performed the acts AND and finally executed transactions OR.
