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Boolean Algebra and Logic - PowerPoint Presentation

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									Boolean Algebra & Logic

        Prepared by
        Dr P Marais
   (Modified by D Burford)
Boolean Algebra & Logic

 Modern computing devices are digital
  – Use two states to represent all entities: 1 and 0
  – Call these two logical states TRUE and FALSE


 All operations are on such values, and can only
  result in these values
Boolean Algebra & Logic

 George Boole formalised such a logic algebra:
  “Boolean Algebra”

 Modern digital circuits are designed and optimised
  using this theory

 We implement “functions” (such as add, compare etc)
  in hardware, using corresponding Boolean expressions
Boolean Operators

 There are 3 basic logic operators
      Operator      Usage             Notation
      AND           A AND B           A.B
      OR            A OR B            A+B
      NOT           NOT A             A
 A, B are variables that can be TRUE or FALSE
 TRUE represented by 1; FALSE by 0
Truth Table: AND, OR, NOT
 To show the value of each operator we use a Truth Table
   – AND:      only if both are TRUE
   – OR:       if either is TRUE
   – NOT:      inverts value


         A B F=A.B F=A+B F = A F=B
         0 0 0     0     1     1
         0 1 0     1     1     0
         1 0 0           1             0   1
         1 1 1           1             0   0
NAND, NOR and XOR
 – NAND:   if either are FALSE [NOT (A AND B)]
 – NOR:    if both are FALSE [NOT (A OR B)]
 – XOR:    if either is TRUE, but not both


           A B F=A.B F=A+B F=AB
           0 0 1     1     0
           0 1 1     0     1
           1 0 1          0         1
           1 1 0          0         0
Logic Gates

 These operators have symbolic
  representations: “logic gates”

 Building blocks for computer circuit desgin
Logic Gates
Finding a Boolean Representation
                                             A B C F
 F = F(A,B,C); F called “output variable”
                                             0 0 0 0
                                             0 0 1 0
 Find F values which are TRUE:
   – If A=0, B=1, C=0, then F = 1.           0   1   0   1
                                             0   1   1   1
   –   F1 =A.B.C                             1   0   0   0
   –   F2 =
                                             1   0   1   0
   –   F3 =
   –   F=                                    1   1   0   1
                                             1   1   1   0
Finding a Boolean Representation
                                             A B C F
 F = F(A,B,C); F called “output variable”
                                             0 0 0 0
                                             0 0 1 0
 Find F values which are TRUE:
   – If A=0, B=1, C=0, then F = 1.           0   1   0   1
                                             0   1   1   1
   –   F1 =A.B.C                             1   0   0   0
   –   F2 = A.B.C
                                             1   0   1   0
   –   F3 = A.B.C
   –   F = F 1 + F2 + F 3                    1   1   0   1
                                             1   1   1   0
Algebraic Identities

 Commutative: A.B = B.A and A+B = B+A

 Distributive:
    A.(B+C) = (A.B) + (A.C)
    A+(B.C) = (A+B).(A+C)

 Associative:
    A.(B.C) = (A.B).C and A+(B+C) = (A+B)+C
Algebraic Identities

 Identitiy Elements: 1.A = A and 0 + A = A

 Inverse: A.A = 0 and A + A = 1

 DeMorgan's Laws:
    A.B = A + B and
    A+B = A.B

								
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