Boolean algebra
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Boolean Algebra. Basic Definitions. Boolean algebra: set of elements, set of operators, and axioms Axioms: Closure Associative Law Commutative Law Identity Element Inverse Distributive Law. Axiomatic Definition of Boolean Algebra. A set B with operators + and • 1)closure + and •

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Boolean Algebra

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Boolean algebra

Boolean Algebra


Basic definitions

Basic Definitions

Boolean algebra: set of elements, set of operators, and axioms

Axioms:

  • Closure

  • Associative Law

  • Commutative Law

  • Identity Element

  • Inverse

  • Distributive Law


Axiomatic definition of boolean algebra

Axiomatic Definition of Boolean Algebra

A set B with operators + and •

1)closure + and •

2)identity

x+0 = 0+x = x

x•1 = 1•x = x

3)commutative

x + y = y + x

x•y = y•x

4)distributive

x•(y + z) = (x•y) + (x•z)

x + (y•z) = (x + y)•(x + z)

5)for x Œ B there exist x’ Œ B (complement)

x + x’ = 1 and x•x’ = 0

6) at least two element x,y Œ B such that x ≠ y


Boolean algebra

  • 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


Basic theorems of boolean algebra

Basic Theorems of Boolean Algebra

  • Duality

    • interchange OR and AND

    • interchange 0 and 1

    • eg

      • x•1 = x

      • x + 0 = x

  • see table 2-1

  • operator precedence

    • ()

    • NOT

    • AND

    • OR

  • Venn Diagrams


Boolean functions

Boolean functions

  • consider the functions:

    F1 = x’yz’

    F2 = z + x’y’

    F3 = x’yz’ + x’z + xy’z

    F4 = x’y + y’z

  • show truth table (like table 2-2)

  • note: F3 = F4

  • obtain F4 by manipulating F3


Algebraic manipulation

Algebraic Manipulation

  • literal ==> primed or unprimed variable

  • simplify (minimize number of literals)

    x’ + xy’

    x(x’+y)

    xy’z + x’y’z + xz’

    xy + x’z + yz

    (x + y)(x’ + z)(y + z)


Solution

Solution

x’ + xy’ = x’1 + xy’

= x’(y + y’) + xy’

= x’y + x’y’ + xy’

= x’y + x’y’ + x’y’ + xy’

= x’(y + y’) + y’(x’ + x)

= x’ + y’

x(x’+y) = xx’ + x y’ = 0 + xy’ = xy’

xy’z + x’y’z + xz’ = y’z(x + x’) + xz’

= y’z + xz’

xy + x’z + yz

= xyz’ + xyz + x’y’z + x’yz + xyz + x’yz

= xyz’ + xyz + x’y’z + x’yz (eliminate duplicates)

= xy(z + z’) + x’z(y + y’)

= xy + x’z

(x + y)(x’ + z)(y + z)

= (x + y)(x’ + z) (dual of previous example)


Canonical and standard forms

Canonical and Standard Forms

minterms

  • how can we represent a 1 in the truth table?


Canonical and standard forms1

Canonical and Standard Forms

maxterms

  • how can we represent a 0 in the truth table?


Other logic operators

Other Logic Operators


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