Pardon formula: 9 laws of logic algebra

Содержание и основные понятия

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Alterna · Chapter 18 · page 18.1
Pardon formula: 9 laws of logic algebra
Table of laws + interactive formula simplifier.
Boolean algebra
It is a set of laws to simplify the logical expression. Knowledge of the nine key laws saves hours on the EGE No. 2 and No. 15. Most of the tasks are to apply the de Morgan law or to absorb it.
GEF
PIF
OGE No. 2
EEG No. 2.15
Relationship to Chapter 17
Counts in Chapter 17 describe
linkage structure
But what if connections aren't roads between cities, they're the logical relationships of "if A then B"? Algebra logic is a tool for dealing with these connections.
F = A∧B
means that the F exit depends on the A and B entrances through operation I. Any logical scheme (chapter 18.3) is a concrete implementation of the formula. And the algebra laws of logic make the formula simpler so that the scheme becomes simpler and cheaper.
00
Model examples: where this applies
A
B
∧
F=A∧B
Logic valve
AND Vendile: exit signal only if both entrances are active
The Truth Table
1
0
0
1
A B → F
The Truth Table
All connections of entrances and corresponding exit
¬A
¬B
∨
The De Morgan Act
B(A)B = BA\B — Deniability of the situation
A
B
A∧B
=A
=A∨B
Laws of absorption
A (A) (B)=A and A (a)= (B))=A the surplus is removed
01
9 laws of algebra logic
№
Law
For ♪
For ♪
1
Impotence
A∧A = A
A∨A = A
2
Commutability
A∧B = B∧A
A∨B = B∨A
3
Association
(A∧B)∧C = A∧(B∧C)
(A∨B)∨C = A∨(B∨C)
4
Distributiveness
A∧(B∨C) = (A∧B)∨(A∧C)
A∨(B∧C) = (A∨B)∧(A∨C)
5
Absorption
A∧(A∨B) = A
A∨(A∧B) = A
6
Sticking
(A∧B)∨(A∧¬B) = A
(A∨B)∧(A∨¬B) = A
7
Supplement
A∧¬A = 0
A∨¬A = 1
8
Zero element
A∧0 = 0
A∨1 = 1
9
De Morgan.
¬(A∨B) = ¬A∧¬B
¬(A∧B) = ¬A∨¬B
The law of binding
Of particular importance in the EGE No. 2:
(A∧B)∨(A∧¬B) = A
This "clicks" two similar situations that differ only by one variable.
02
Examples of simplification
Example 1: (A)B(A)b)
(A∧B)∨(A∧¬B) = A∧(B∨¬B) = A∧1 =
A
Example 2: B(A)B(A~C)
¬(A∨B) = ¬A∧¬B (де Морган)
(¬A∧¬B)∧(A∨C) = (¬A∧¬B∧A)∨(¬A∧¬B∧C) = 0∨(¬A∧¬B∧C) =
¬A∧¬B∧C
Example 3: A (BA)(B)
A∨(¬A∧B) = (A∨¬A)∧(A∨B) = 1∧(A∨B) =
A∨B
03
Logical calculator
Enter the formula - you get the truth table and step-by-step simplification
🧮
The Truth Table
Formula
📐
Simplified steps
Formula for simplification
Operations:
¬
NOT ·
∧
And ·
∨
OR
→
followed ·
↔
Equivalence
⊕
XOR
04
Mini test
♪ The laws of logic
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