Boolean Algebra Pt.2

• Other Logical Operators


  • Transfer / Buffer
  • NOR
  • NAND
  • XOR
  • XNOR
• Transfer / Buffer


  No operation, x = y
• NOR


  Negated OR / OR Complement, x ↓ y = x + y'
• NAND


  Negated AND / AND Complement, x ↑ y = (xy)'
• XOR


  Exclusive OR, x⊕y
• XNOR


  Exclusive NOR, (x⊕y)'
• Boolean Functions can be expressed in its canonical form
• Canonical form


  Unique representation of a Boolean function
• Equivalent Boolean functions have the same canonical form
• Canonical forms of Boolean Expressions

  • Sum of Products (SoP) aka Disjunctive Form or Sum of Minterms
  • Product of Sums (PoS) aka Conjunctive Form or Product of Maxterms
• Minterms


  Combination of AND operations that can be performed between n number of Boolean variables and its' complements
• There are 2^n minterms for n number of variables
• Minterms for 2 Boolean variables x and y


  • xy
  • x'y
  • xy'
  • x'y'
• Maxterms


  Combination of OR operations that can be performed between n number of Boolean variables and its' complements
• There are 2^n maxterms for n number of variables
• Maxterms for 2 Boolean variables x and y

  • x+y
  • x'+y
  • x+y'
  • x'+y'
• The objective is to write down a Boolean function in terms of minterm or maxterm notation (aka. its Canonical Form)
• Every minterm and maxterm is obtained under the assumption that every term has all n variables being considered in the Boolean function
• The challenge is to convert every term in the Boolean function into one that contains all n variables