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# Boolean Algebra - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about ' Boolean Algebra' - quinn-beach

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Presentation Transcript

### Boolean Algebra

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

Axioms:

• Closure

• Associative Law

• Commutative Law

• Identity Element

• Inverse

• Distributive Law

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 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

• 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

• 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

• 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)

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)

minterms

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

maxterms

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