Boolean Algebra

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Boolean Algebra
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Basic Definitions

• Boolean algebra set of elements, set of
  operators, and axioms
• Axioms
• Closure
• Associative Law
• Commutative Law
• Identity Element
• Inverse
• Distributive Law

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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

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• 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

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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

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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

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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)

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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)

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Canonical and Standard Forms

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

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Canonical and Standard Forms

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

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Other Logic Operators