Table of contents
Basic Logical Operations
1. "AND" operation (logic multiplication)
Expressions:
Truth Tables:
Logic Graphical Symbols:
2. "OR" operation (logical addition)
Expressions:
Truth Tables:
Logic Graphical Symbols:
3. "Not" operation (logical NOT)
Expressions:
Truth Tables:
Logic Graphical Symbols:
Several common compound logic operations
Basic formulas and rules of logical algebra
Representation of Logical Functions
For example, the expression is
1. Truth table method
2. Logic diagram method
Basic Formulas for Logical Functions
1. Basic operations
The relationship between value and value
The relationship between value and variable (0-1 law)
and or
The relationship between variable and variable (overlapping law)
and or not
2. Formulas similar to ordinary algebra
commutative law
A+B=B+A
associative law
A+(B+C)=(A+B)+C
distributive law
A(B+C)=AB+AC, A+BC=(A+B)(A+ C)
Because: (A+B)(A+C)=AA+AC+AB+BC=A(1+C+B)+BC=A+BC
3. Special formulas of logic algebra
4. Four commonly used operations
(1) XOR If
the variable is different, it is 1, otherwise it is 0
(2) Same or If
the variable is the same, it is 1, otherwise it is 0
Important formula
5. Proof of logical algebraic formulas
1. Truth table method: check whether the truth tables of the functions on both sides of the equation are equal
2. Algebraic method: use proven formulas (especially the basic operations of logic functions) and theorems to derive
Basic Rules for Logical Functions
1. Inversion rules
If all "*" in the logic function F are changed into "+"; "+" becomes "*"; "0" becomes "1"; "1" becomes "0"; ;The inverse variable becomes the original variable; the resulting new function is the inverse of the original function.
For example:
After the inversion:
Note:
(1) Keep the operation order of the original function (brackets first, then AND, then OR)
(2) The negative sign that does not belong to a single variable remains unchanged (long not sign)
2. Duality rule
If all "*" in the logic function F are changed into "+"; "+" becomes "*"; "0" becomes "1"; "1" becomes "0"; then the obtained new logic The function is the dual F' of F. (i.e. with less variable variation than the inversion rule)
eg:
3. Substitution rules
For any logic equation containing a variable A, if all occurrences of A are replaced by the same logic function F, the equation still holds true.
For example: Given a logical equation A(B+C)=AB+AC, if A+BC is used instead of A, the equation is still valid
: (A+BC)(B+C)=(A+BC) B+(A+BC)C
Simplification of Logical Functions
According to the characteristics of the product term in the logical function expression and the relationship between the product terms, it can be divided into five general forms. Function expressions are generally simplified into **AND or formula,** conditions are:
1. There are at least AND items in the expression
2. Under the premise of satisfying 1), the number of variables in each "AND item" is at least
For example:
Simplified method
1. Algebraic simplification method (formula method)
A and B in the basic formula can be any complex logical formula
(1) Parallel method
Use the formula 
for example:
(2) Absorption method
Use the formula A+AB=A
For example:
(3) Elimination method
Eliminate A non-
use formula 
For example:
(4) Matching item method
Use the formula A+A=A
for example:
In addition, you can also use the formula A+A=1 to simplify other formulas.
When simplifying more complex formulas, first use the distributive law , and then use the formula to simplify
2. Karnaugh map simplification method
The number of the minimum term
The expression of the minimum term is written as
Method of writing the expression of the minimum term
(1) Simplify the general expression (complex and not recommended)
(2) Write directly (see which variable is missing in the minimum term, according to the variable Two possibilities can be directly written)
(3) Use the truth table to find the (recommended)
Karnaugh map of the minimum term.
Each of the 2 n minimum terms of n input variables is represented by a small square, and they are arranged in a matrix, and guarantee Adjacent minterms differ only by one variable, and the resulting array map is a Karnaugh map of n variables.
Two-variable Karnaugh map
A and B can change the upper and lower positions
Three-variable Karnaugh map
Two or more variables, arranged according to Gray code rules ↓
Four-variable Karnaugh map
** Use Karnaugh map to represent logic functions**
Convert the function into a minimum term In the form of the sum, fill in the positions corresponding to these minimum items in the Karnaugh map with "1", and fill in the rest of the positions with "0".
Truth tables, expressions, logic diagrams, waveform diagrams, and Karnaugh diagrams can all express a logic function.
Method: truth table method, expression method truth table method expression
method Karnaugh map
simplification Simplification steps: (1) fill in the Karnaugh map according to the expression or truth table (2) draw adjacent items Carnot circle (3) merges each circle to eliminate complementary variables into one item, and the addition of these items is the simplified result . For example:
