How does CRC work?

Chapter 3 Logical Functions

Chapter 3.1 Introduction
Something similar to the remainder of the division will be used to obtain CRC.
It requires some knowledge about logical functions, about exclusive or functions, especially.
The logical functions will be presented:
– NEGATION
–  AND

– OR
– XOR ( exclusive or)

Chapter 3.2 Logical Function NEGATION

Fig. 3-1
Logical Function NEGATION  ANIMATION
output is a negation of the x input.
Do animation  and create the truth table (Karnaugh table). What is this and how do create it?  See Fig. 3-5a  chapter 3.6.

Chapter 3.3 Logical Function OR

Fig. 3-2
Logical Function OR ANIMATION
C
reate the truth table. See Fig. 3-5b  chapter 3.6.

Chapter 3.4 Logical Function AND

Fig. 3-3 Logical Function AND ANIMATION
Create the truth table. See Fig. 3-5c  chapter 3.6.

Chapter 3.5 Logical Function XOR- other words “EITHER”

Fig. 3-4 Logical Function XOR ANIMATION
y=1 when x1=1 xor x2=1 other words y=1 when gdy x1=1 either  x2=1
Create the truth table. See Fig. 3-5d  chapter 3.6.
By the way. Certain people doesn’t separate the word “or” from the word “either
You can marry Elisabeth either Susan. But the opinion “You can marry Elisabeth or Susan”allows bigamy!

Chapter 3.6 Logical Functions as Truth tables (Karnaugh tables)

Fig. 3-5
Truth tables
Typical and most general logical function presentation is a logical fucnction formula:
y=x1 or x2 for example.
Truth table is the other logical function presentation. It isn’t so general as a formula. But it’s very convenient when you test the unknown device. You construct the table with the cells. Each cell represents inputs cobinations.
The simplest table is for NEGATION see Fig. 3-5a.
When x=0–>y=1, when x=1–>y=0. No comments because it’s the easiest logical function.
Fig. 3-5b is inputs OR logical function. Two inputs x1x2 requires inputs cominations testing. Please associate each table cell with the appropriate combination. The in the cirle says that for inputs combination x1=1 and x2=0 output y=1. Please analize other cells. The table dimension is 2×2
Fig. 3-5c is inputs AND logical function.
Please analize the table 2×2.
Fig. 3-5d is inputs XOR logical function.
Please analize the function as a table 2×2 and as a logical formula function
Fig. 3-5e is inputs logical function.
inputs function requires 8 cells table2×4.  Please analize the function as a table and as a logical formula function.

The undermentioned functions (without figures) require the tables
inputs requires table 4×4
5 inputs requires table 4×8
I am not sure but there aren’t tables for more than 5 inputs.

The truth tables enable easy function simplification–>Karnaugh method.
Fig. 3-5e represents 5 inputs logical function  before simplification and as 2 products formula function after simplification. The elecronical circut after simplification is simpler, cheaper and more sure.  How does obtain  2 simple products from the initial 5 complicated products? Other words. How does simplify the logical functions? Find Karnaugh method in the internet.

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