Boolean Algebra

• Boolean algebra


  A branch of mathematics in which algebraic expressions are made up of variables, constants, and operators
• Components of Boolean algebra

  • Variables (can represent 1 and 0)
  • Constants (1 and 0)
  • Operators (AND, OR, NOT)
• Boolean laws


  Statements of equivalence (called identities) between two Boolean expressions
• Boolean laws generally (but not always) follow rules that you will be familiar with from the standard rules of algebra
• AND (⋅) can be considered as multiplication and OR (+) as addition in the context of Boolean algebra
• Manipulating Boolean expressions


  Algebraically to form simpler expressions that implement the same logic
• Benefit of manipulating Boolean expressions

  • Allows a simpler circuit to be produced with fewer logic gates
  • Reduces the cost of the circuit
  • Reduces the amount of heat generated
  • Reduces the processing time
• Digital circuits can have many inputs, usually labelled with letters
• Digital circuits can also have inputs that are permanently on (1) or off (0)
• Boolean expressions are made up of variables and constants
• The value of each variable in a Boolean expression can only be 0 or 1
• Fundamental functions of Boolean logic

  • AND
  • OR
  • NOT
• Digital circuits can include many types of logic gate, which combine the fundamental functions of Boolean logic
• A ⋅ 1 = A
• A + 0 = A
• A ⋅ A = A
• A + A = A
• A + /A = 1
• A ⋅ /A = 0
• //A = A
• A ⋅ 0 = 0
• A + 1 = 1
• Commutative law


  X ⋅ Y = Y ⋅ X

  X + Y = Y + X
• Associative law


  X ⋅ (Y ⋅ Z) = ( X ⋅ Y) ⋅ Z

  X + (Y + Z) = (X + Y) + Z
• Absorption law


  X + (X ⋅ Y) = X

  X ⋅ (X + Y) = X
• Distributive law


  The distributive law states that:

  X ⋅ (Y + Z) = (X ⋅ Y) + (X ⋅ Z)

  If you consider the ⋅ operator as multiplication, and the + operator as addition, this is equivalent to expanding the brackets or factorisation (depending on which way you are working) in normal algebra.

  It is also true that:

  X + (Y ⋅ Z) = (X + Y) . (X + Z)

  This is not like in normal algebra!
• De Morgan's Laws


  De Morgan’s laws are named after Augustus De Morgan, a 19th-century British mathematician. De Morgan proved that:

  X⋅Y = /(/X + /Y)​

  X + Y = /(/X . /Y)