2.1 Overview
1. Only two opposing logic states of logic becomes binary logic.
2. The so-called logic, in this case refers to the causal relationship between things. When two binary numbers represent different logic states, they can be inference operation between the specified causality. We call this operation a logical operation.
3.1849 British mathematician George Boole first proposed a mathematical method Boolean logic operations. Also known as Boolean algebra logic switching algebra or algebra. Chapter is spoken logical algebra Boolean algebra applied in binary logic circuits.
4. Logical Algebra also denote variable, this variable is called a logical variable.
The logical operation is represented by a logical inference operation between the logical state variables and constants, rather than between the number of calculation.
Three basic operations 2.2 Logical Algebra and (AND), or (OR), non (NOT) three kinds of
1. Introduction and non concepts:
The picture shows three LEDs as a control circuit.
FIG. (A)
Only when both switches are closed, the LED lights up. Show: all conditions only decide things along with the results, the result will occur. Causality and logic become, also known as logical multiplication.
FIG. (B)
If any of the switch is closed, the indicator lit. Shows that: as long as there is any meet, the results will take place in various conditions determine the outcome of things. This is called a logical or causal relationship, also known as logical addition.
FIG. (C)
When the switch is turned off lights, but the switch is closed the lamp does not light. It showed that: as long as the conditions are met, the result would not have happened; but when the conditions are not met, the result will occur. This logic is called non-causal relationship, also known as logical negation.
2. Introduction of truth table:
In terms of A, B represents a state of the switch, the switch is closed and is represented by 1, 0 indicates the switch is open; represented by Y status indicators and lights to represent 1, 0 indicates no light, it can be listed as 0 , aND, oR, NAND logic relationship graph represented. As shown below. This chart is called a logical truth table, the truth table for short.
3. The introduction of graphical symbols and operation symbols
(1) arithmetic sign:
Logical Algebra, the AND, OR, logical variable is regarded as non-A, three basic logical operation between B.
When A and B can be written with the logic operation:
can be written as A and B or when a logic operation:
can be written when A non-logic operation:
while the arithmetic and logic unit implemented referred to the AND gate circuit;
be implemented or arithmetic logic unit circuit or gate is called;
unit circuit will implement logical operations called non NAND gate (also referred to as an inverter).
(2) graphic symbols:
With a specific shape symbol more of this, we will use this again. Personal feeling better remember better looking!
4. Advanced Edition: See compound logical operation:
The actual logic problems more often than with, or non-complex, but they can be used, or a combination of Africa to achieve.
(1) and non (the NAND)
Definition: The A, B with the first operation, then the results negated, the resulting is the A, B and the NAND operation result.
1. Truth table:
2. graphic symbols and operation symbols:
(2) or (NOR)
Definition: The A, B to perform an OR operation, then the results negated, the resulting is the A, B or the result of calculation.
1. Truth table:
2. graphic symbols and operation symbols:
(3) or (AND-NOR)
Definition: the NOR logic, and the relation between all C, D between the A, B, as long as the A, B or C, D of any one group are 1, the output Y is 0; only when each is not a full set of inputs, the output Y is 1.
1. Truth table:
2. The operation symbols and graphic symbols:
（4）异或（EXCLUSIVE OR）
Definitions: When A, B are different, the output Y is 1; and when A, B are the same, the output Y is 0. It can also be used with, or combination of non-representation:
. ##### 1 truth table:
2. graphic symbols and operation symbols:
(5) or with (EXCLUSIVE NOR)
Definition: the same or different and or the contrary, if A, B are the same, Y is equal to 1; if A, B are not simultaneously, Y is equal to 0. or may be written with the same, or a combination of non-:
##### 1. Truth table:
2. graphic symbols and operation symbols:
Small knowledge:
XOR and with each other or the inverse operation, namely:
The basic formula used and 2.3 of logic algebra formula
1. Basic formula:
(1), 2, 11 and 12 shows operation rules between variables and constants.
(2) 3 and 13 are the same variable operation rules, also referred to as overlapping law.
(3) 4 and 14 represents a variable and its operation rules between the anti-variable, also termed complementarity law.
(4) Formula 5 to 15 and the commutative law; Formula 6 and 16 in conjunction with the law; Formula 7 and 17 the distributive law.
(5) formula 8 and 18 are known DeMorgan's theorem, also known as inversion law. In exchange simplification and logic functions in regular use of this formula.
(6) 9 show a variable for itself after reducing twice negates, which is called the so reducing law.
(7) 10 Formula 1 is 0 and negates rule, it indicates 0 and 1 results mutually negated.
Small knowledge:
the correctness 1. The formula can be verified using the method of truth table column. If equality holds, then any of the values of a set of variables into the results obtained should be equal on both sides of the equation. Thus, the corresponding sides of the equation are bound to the same truth table.
2. Carefully Observe that: Equation 1 and 11; 12 and 2; 3 and 13; 14 and 4; 5 and 15; 16 and 6; 17 and 7; 8 and 18 are all dual type. Therefore, if proven formulas 1-8 set up, the formula is no longer subject to doing 11-18 proved.
2. A number of commonly used formula :( These formulas are derived using the basic formula!)
(1) 21 proof:
Description formula: When two product terms are added to one another when a factor, it is unnecessary that can be deleted.
(2) Proof of formula 22:
It indicates that the formula: When adding two product terms, if a negated is another factor, then this factor is redundant, can be eliminated.
(3) 23 proof:
Indicates that the formula: When the two product terms are added, if they contain the B and B 'and the same two factors other factors, will be able to merge the two, and B and B can be' erased two factors.
(4) Proof of formula 24:
Indicates that the formula: A variable containing the time A and the multiplied result is equal to A, and which can be eliminate.
(5) demonstrated 25;
It indicates that the formula: If the two product terms included in each of A and A 'two factors, and the two remaining factors of product terms form a third product terms, the third product term is redundant and can be eliminated.
Proof (6) of formula 26:
Description formula: When the product term A and a non-multiplied, and the product term A is a factor, the factor A that can be eliminated.
It indicates that the formula: When A 'and a non-product term by multiplying the product term and A is a factor, the result is equal to A'.
Fundamental theorem of algebra logic 2.4
1. assignment theorem
Substituting Theorem: In any logical variable A in equation contains, if the logical formula is substituted into another position wherein all of A, the equation still holds.
(1) understand:
Since only 0 and 1 two possible state variables A, whether all the A = 0 or A = 1 is substituted into logical equations, equations are necessarily true. And any type of logical value 0 and 1 do not go beyond two kinds, so that when substituted with the formula A, equation established naturally. So, this may be no need to prove the theorems as axioms.
(2) Examples Example :( to DeMorgan theorems)
Small knowledge:
for when product term logic or negate formula, product term in parentheses should be logical expression or the outside, and negate the entire contents of the brackets.
Further, when a complex logical equation calculation, need to comply with normal algebra operation priority, i.e. the first count parentheses content, followed by the multiplication operator, and finally the addition operator.
2. Inverse Theorem
Example 1:
Example 2:
3. duality theorem
Note: and inversion theorems difference: There inversion theorems original variables into a variable anti-anti-variable becomes the original variable! ! !
example:
