Number systems and codes

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Combinational logic
--- outputs logical functions of inputs --- new
outputs appear shortly after changed
inputs (propagation delay) --- no feedback
loops --- no clock
Sequential logic --- outputs logical functions
of inputs and previous history of circuit
(memory) --- after changed inputs, new outputs
appear in the next clock cycle --- frequent
feedback loops
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Fundamentals of Boolean algebra

• Named after George Boole
• He presented an algebraic formulation of the
  process of logical thought and reason
• This formulation come to be known as Boolean
  Algebra

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Postulates of Boolean algebra

• Definition
• A Boolean algebra is a closed algebraic system
  containing a set K of two or more elements and
  the two operators or /\ or ?, called AND,
  and or \/ or ?, called OR
• Closed system for every a and b in set K, ab
  belongs to K and ab belongs to K.

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Postulates of Boolean algebra

• Existence of 1 and 0
• There exist unique elements 1 (one) and 0 (zero)
  in set A" such that for every a in K
• a) a 0 a,
• b) a 1 a,
• where 0 is the identity element for the
  operation and 1 is the identity element for the
  operation.

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Postulates of Boolean algebra

• Commutativity of the and operations
• For every a and b in K
• a) a b b a.
• b) a b b a
• Associativity of the and operations
• For every a, b, and c in K
• a) a (b c) (a b) c.
• b) a (b c) (a b) c.

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Postulates of Boolean algebra

• Distributivity of over and over
• For every a, b, and c in K
• a) a (b c) (a b) (a c),
• b) a (b c) (a b) (a c).
• Existence of the complement
• For every a in K there exists a unique element
  called a (complement of a) in K such that
• a) a a 1.
• b) a a 0.

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Venn diagrams for the postulates

• Operations on sets
• Sets ? closed regions
• Sets correspond to elements
• Intersection ? corresponds to
• Union ? corresponds to

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Venn diagrams for the postulates

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Venn diagrams

• Examples of Venn diagrams

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Venn diagrams

• a b c (a b) (a c)

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Venn diagrams

• a b c (a b) (a c)

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Boolean algebra

• Duality
• If an expression
• f(x1, x2, xn, , , 0, 1)
• is valid, then
• f(x1, x2, xn, , , 1, 0)
• obtained by interchanging and , 0 and 1 is
  also valid
• a (b c) (a b) (a c)
• a (b c) (a b) (a c)
• Postulates 2 6 are stated in dual form

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Fundamental theorems of Boolean algebra

• Prove part (b) by exchanging with , and use
  the dual form of the postulates

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Fundamental theorems of Boolean algebra
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Fundamental theorems of Boolean algebra

• a a 0 P6(b)
• a a 1 P6(a)
• Therefore, a is the complement of a, and also a
  is the complement of a. Because the complement of
  a is unique, it must be equal to a.

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Fundamental theorems of Boolean algebra
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Fundamental theorems of Boolean algebra

• Why a ab a

a
ab
b
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Fundamental theorems of Boolean algebra
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Fundamental theorems of Boolean algebra
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Fundamental theorems of Boolean algebra
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Fundamental theorems of Boolean algebra
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Fundamental theorems of Boolean algebra
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Fundamental theorems of Boolean algebra

• Example using DeMorgans theorem

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Fundamental theorems of Boolean algebra
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Boolean algebra postulates and theorems
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Theorems

• Proofs by perfect induction

Proofs by exhaustion Let variables assume all
possible values and show validity of result in
all cases
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Example Show X 0 X
(b) Elaborate cases if X 0, have X 0 0
0 0 X if X 1, have X 0 1 0 1
X
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More Theorems
Can prove by exhaustion....but have more
cases For distributive laws, T8 looks like
ordinary algebra T8 also true (swap operators,
factor, swap back) T9, T10 for logic minimization
- drop irrelevant terms
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T9, T10, T11 for logic minimization - drop
superfluous terms
T9 (Covering) X X?Y X and X?(XY)X
Proof X X?Y X?1 X?Y X?(1Y) X?1 X
X?(XY) (X0)?(XY) X(0?Y) X0 X
T10 (Combining) X?Y X?Y X and (X Y) ? (X
Y) X Proof X?Y X?Y X?(Y Y) X?1
X (X Y)?(X Y) X (Y?Y) X
0 X
T11 (Consensus) X?YX?ZY?Z X?YX?Z and
(XY)?(XZ)?(YZ) (XY)?(XZ) Proof If Y?Z
0 X?YX?ZY?Z X?YX?Z 0 X?YX?Z
else Y Z 1 left side X?YX?ZYZ
something YZ something 1 1 right side
X?YX?Z X X 1 So, in either
case, X?YX?ZYZ X?YX?Z If YZ
1 (XY)?(XZ)?(YZ) (XY)?(XZ)?1
(XY)?(XZ) else Y Z 0 left
side (XY)?(XZ)?(YZ) something ? (Y Z)
something ? 0 0 right side (XY)?(XZ)
(X0)?(X0) X?X 0 So, in either
case, (XY)?(XZ)?(YZ) (XY)?(XZ)
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Duality

• De Morgans Theorems (X Y) X ? Y (X ?
  Y) X Y
• Dual Swap 0 1, AND OR, but leave variables
  unchanged
• Result Theorems still true
• Why?
• f(X, Y) g(X, Y)
• complementf(X, Y) complementg(X, Y)
• dualf(X, Y) dualg(X, Y)
• but X, Y just dummy variables, replace with
  originals
• Counterexample?X X Y X (T9)X X Y X
  (dual)X Y X (T3)!! error ?

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N-variable Theorems

• Prove via induction
• Most important DeMorgan theorems

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DeMorgan Symbol Equivalence
Bubble-pushing...
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Likewise for OR