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Boolean Algebra

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Boolean Algebra. Boolean Algebra 2. Basic Definitions. Boolean ... Duality. interchange OR and AND. interchange 0 and 1. eg. x 1 = x. x 0 = x. see table 2-1 ... – PowerPoint PPT presentation

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Title: Boolean Algebra


1
Boolean Algebra
2
Basic Definitions
  • Boolean algebra set of elements, set of
    operators, and axioms
  • Axioms
  • Closure
  • Associative Law
  • Commutative Law
  • Identity Element
  • Inverse
  • Distributive Law

3
Axiomatic Definition of Boolean Algebra
  • A set B with operators and
  • 1) closure and
  • 2) identity
  • x0 0x x
  • x1 1x x
  • 3) commutative
  • x y y x
  • xy yx
  • 4) distributive
  • x(y z) (xy) (xz)
  • x (yz) (x y)(x z)
  • 5) for x B there exist x B (complement)
  • x x 1 and xx 0
  • 6) at least two element x,y B such that x ? y

4
  • Boolean algebra requires
  • elements of the set B
  • rules of operation for and
  • they satisfy the six postulates
  • Two-Valued Boolean Algebra
  • B 0,1
  • AND, OR, NOT operations
  • check postulates

5
Basic Theorems of Boolean Algebra
  • Duality
  • interchange OR and AND
  • interchange 0 and 1
  • eg
  • x1 x
  • x 0 x
  • see table 2-1
  • operator precedence
  • ()
  • NOT
  • AND
  • OR
  • Venn Diagrams

6
Boolean functions
  • consider the functions
  • F1 xyz
  • F2 z xy
  • F3 xyz xz xyz
  • F4 xy yz
  • show truth table (like table 2-2)
  • note F3 F4
  • obtain F4 by manipulating F3

7
Algebraic Manipulation
  • literal gt primed or unprimed variable
  • simplify (minimize number of literals)
  • x xy
  • x(xy)
  • xyz xyz xz
  • xy xz yz
  • (x y)(x z)(y z)

8
Solution
  • x xy x1 xy
  • x(y y) xy
  • xy xy xy
  • xy xy xy xy
  • x(y y) y(x x)
  • x y
  • x(xy) xx x y 0 xy xy
  • xyz xyz xz yz(x x) xz
  • yz xz
  • xy xz yz
  • xyz xyz xyz xyz xyz xyz
  • xyz xyz xyz xyz (eliminate
    duplicates)
  • xy(z z) xz(y y)
  • xy xz
  • (x y)(x z)(y z)
  • (x y)(x z) (dual of previous example)

9
Canonical and Standard Forms
  • minterms
  • how can we represent a 1 in the truth table?

10
Canonical and Standard Forms
  • maxterms
  • how can we represent a 0 in the truth table?

11
Other Logic Operators
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