Boolean Algebra-數位邏輯設計筆記

一定愛配王俊堯教授開放課程食用:課程連結

Reference:連結

Boolean Algebra

Basic Operations and Truth Tables

• • Ex:

    

Basic Theorems

OR operations:

AND operations:

XOR operations:

Advanced Theroems

Commutative Law

Associative Law

Distributive Law

DeMorgan’s Law

• Can be proved with truth table.

• Ex:

  

• One-Step Rule(For DeMorgan's Law)

  • Do all transforms at a time.

  • 

    • Invert every variable.
    • Invert 0s and 1s.(0 <=> 1)
    • Transform every AND and OR logic gates.(AND <=> OR)

  • Ex:

    

Duality

• 
  • Invert 0s and 1s. (0 <=> 1)
  • Transform every AND and OR logic gates. (AND <=> OR)
• Ex:
• Important Property: If =>

Consensus Theorem

• • Proof:

    

• Dual form:

• Ex:

  

Important Equations

Example

Multiplying Out and Factoring

Sum of Product and Mutiplying Out

• Sum of Product

  • Two or more products (AND) are summed (OR) together.

  • One single variable can also be called a term of product.

  • When exisit in a function, we can remove , since is dominant for .

  • Ex:

    

• Mutiplying Out

  • To get the expressions to be SOP form. => 展開

  • Ex:

    

Product of Sum and Factoring

• Product of Sum

  • Two or more sums (OR) are producted (AND) together.

  • One single variable can also be called a term of Sum.

  • When exisit in a function, we can remove , since is dominant for .

  • Ex:

    

• Factoring

  • To get the expressions to be POS form. => 展開

  • Ex:

    

2-Level Logic Realization

• Every variable will only pass 2 logic gates.
• Both POS and SOP are 2-level realization.
• Inverter is not included.

Common Multoplying Out and Factoring Expressions

• • Proof:

    

• Ex:

  

Simplification of Switching Expression

Combining terms

• Ex:

  

Eliminating terms

• Consensus Theorem =>

• Ex:

  

Eliminating literals

• Ex:

  

Adding redundant terms

• Add
• Adding to (according to consensus theorem)
• Adding to
• Multiply (x+x')

Example

Proving the Validity of an Equation

• Construct a Truth Table for LHS and RHS.

• Simplify LHS and RHS.

• Ex: