Laws of logic and complete systems

Содержание и основные понятия

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Chapter 29 Page 29.2
Laws of logic and complete systems
De Morgan · Absorption · NAND · NOR · Functional completeness
Knowing the laws, you can simplify any logical formulas. Knowing the complete systems, you make sure that one is able to express the whole thing. That is the basis of digital electronics.
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He's got a little bit of depth.
01
Laws of logic
Press the law to see details and an example of application.
Table of laws (rapid certificate)
Law
For ♪
For ♪
Commutability
A∧B = B∧A
A∨B = B∨A
Association
(A∧B)∧C = A∧(B∧C)
(A∨B)∨C = A∨(B∨C)
Distributiveness
A∧(B∨C) = (A∧B)∨(A∧C)
A∨(B∧C) = (A∨B)∧(A∨C)
Impotence
A∧A = A
A∨A = A
Absorption
A∧(A∨B) = A
A∨(A∧B) = A
Sticking
(A∧B)∨(A∧¬B) = A
De Morgan.
¬(A∧B) = ¬A∨¬B
De Morgan.
¬(A∨B) = ¬A∧¬B
Constants
A∧0=0, A∧1=A
A∨0=A, A∨1=1
Supplement
A∧¬A = 0
A∨¬A = 1
Double denial
¬¬A = A
02
Interactive: Checking the law through the truth table
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Law Reviewer
You choose the law, you see the truth table confirming the identity, and on the left and right, you have to match all the lines.
03
Complete systems (functional completeness)
Definition
Full system
- A set of logical operations through which to express
Any
This means that of these operations, B, , , and , all 16 functions from 2 variables can be collected.
NAND (Strich Schaeffer)
A | B = ¬(A∧B)
Full system!
¬A = A|A
A∧B = (A|B)|(A|B)
A∨B = (A|A)|(B|B)
NOR (Pierce shooter)
A ↓ B = ¬(A∨B)
Full system!
¬A = A↓A
A∨B = (A↓B)↓(A↓B)
A∧B = (A↓A)↓(B↓B)
NOT COMPLETE
No V
You can't get a BA.
All functions from and
Save 1 (at A=1, B=1)
result = 1.
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System completeness check
Enter the set of operations through a decimal point.
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Mini-test: 5 questions
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Laws and complete systems
5 questions · 20 XP for each correct