Boolean Algebra and Its Functions

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Boolean Algebra and Its Functions
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Boolean Algebra

• Boolean Algebra is a mathematical Model for
  digital logic circuits.
• Boolean Algebra is a system ltB, V, Pgt
• B0,1 is the set of values
• V is the set of variables
• P, , ? is the set of operators (basic
  functions) defined by the truth tables as follows

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• AND Gate
• Implements AND function

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• OR gate
• Implements OR function

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• NOT gate
• Implements NOT function

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• Basic Laws of Boolean Algebra
• Identities
• x 0 x
• x 1 x
• Compliments
• x x? 1
• x x? 0
• DeMorgan Law
• (x y)? x? y?
• (x y)? x? y?
• Idempotent Law
• x x x
• x x x
• Boundness Laws
• x 1 1
• x 0 0
• Distributive Law
• Associative Law

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• More commonly-used functions
• x XOR y xy xy

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• NAND gate

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• NOR gate

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• XNOR gate

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Simplification of Boolean Functions

• General Boolean functions of n variables can be
  represented by
• Boolean expressions
• Truth tables showing the function values for all
  input combinations
• Boolean functions can be implemented directly
  from their expressions, but
• Complicated expressions may results in circuits
• Using more gates than necessary or
• Having longer accumulative gate delay than
  necesarry

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• Minterms of n variables
• The literals of x is either x or x
• Given n variables, a minterm is a product (result
  of and operations) of n literals, one from each
  variable.
• A mintern is 1 only for one input combination and
  0 for the rest input combinations.
• xyz (i.e. xyz) is 1 only when x1, y0 and
  z1. It is 0 for all other 7 input combinations
  of the three variables x, y, and z.

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• Implementation of Boolean function with minimum
  gate delay
• Obtain the truth table of the function
• Write the minterms corresponding to the input
  combinations for which the function value is 1.
• Form a sum of these minterms using OR operation
• Construct the circuit according to the form
  obtained (maximum 3 gate delays)
• example

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• We have
• f xyz
• xyz
• x yz
• x y z

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• But the sum of minterns can be further simplified
  to reduce
• the number of product terms and
• the number of inputs of the gates
• example
• f xyz xyz xyz xyz
• xz(yy) xz(yy)
• xz xz
• But, how do we reach the simplest form
  systematically?

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Karnaugh Map Simplication

• Karnaugh maps
• three variables and four variables

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• one cell for each minterm
• can be used to represent a function by filling
  1s to the cells corresponding to its minterms
• Adjacent minterns can be grouped (combined) to
  form simpler product terms.
• f xyz xyz xyz xyz

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• Groupings are allowed to be overlapped because
  of idempotent laws, xxx.
• Note the wrap-around adjacency due to the gray
  coding used.
• Two adjacent two-cell grouping can be further
  grouped for form simpler term.

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• Karnaugh map simplification
• Find the minterns of the function from the truth
  table.
• Draw the Karnaugh map for the function.
• Start with the largest groupings possible (8, 4,
  2)
• find all possible groups and mark them with
  corresponding (product) terms (each group should
  contain at least one cell not covered in previous
  groupings).
• All groups obtained are called Prime Implicants.
• Find all the Essential Prime Implicants, each of
  which is a prime implicant that contains at least
  one cell not covered by any other prime
  implicant.
• Find other non-essential prime implicants to
  cover the remaining cells of the function.
• The simplest form (minimum gate delay and least
  number of inputs) is obtained by adding (OR)
• the essential prime implicants and
• non-essential prime implicants
• from above.