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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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