ICS%20241

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ICS 241

• Discrete Mathematics II
• William Albritton, Information and Computer
  Sciences Department at University of Hawaii at
  Manoa
• For use with Kenneth H. Rosens Discrete
  Mathematics Its Applications (5th Edition)
• Based on slides originally created by
• Dr. Michael P. Frank, Department of Computer
  Information Science Engineering at University
  of Florida

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Section 10.2 Representing Boolean Functions

• Sum-of-products Expansions
• A.k.a. Disjunctive Normal Form (DNF)
• Product-of-sums Expansions
• A.k.a. Conjunctive Normal Form (CNF)
• Functional Completeness
• Minimal functionally complete sets of operators.

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Sum-of-Products Expansions

• Theorem Any Boolean function can be represented
  as a sum of products of variables and their
  complements.
• Proof By construction from the functions truth
  table. For each row that is 1, include a term in
  the sum that is a product representing the
  condition that the variables have the values
  given for that row.
• Can also use the Boolean Identities (p. 705)

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Table Representation

• F(a,b,c) (ab)bc abcabcabcabc

a b c ab bc (ab)bc
0000 0011 0101 1100 0001 1101
1111 0011 0101 0000 0001 0001
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Literals, Minterms, DNF

• A literal is a Boolean variable or its
  complement.
• A minterm of Boolean variables x1,,xn is a
  Boolean product of n literals y1yn, where yi is
  either the literal xi or its complement xi
• Note that at most one minterm can have the value
  1.
• The disjunctive normal form (DNF) of a degree-n
  Boolean function f is the unique sum of minterms
  of the variables x1,,xn that represents f
• A.k.a. the sum-of-products expansion of f.

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Class Exercise

• Exercise 1.a. (p. 712)
• Hint Think about the table representation
• Exercise 3.c.
• Use the Boolean Identities on p. 705
• Each pair of students should use only one sheet
  of paper while solving the class exercises

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Functional Completeness

• Since every Boolean function can be expressed in
  terms of ,,, we say that the set of operators
  ,, is functionally complete.
• There are smaller sets of operators that are also
  functionally complete.
• We can eliminate either or using the Law of
  the Double Complement De Morgans Law.
• NAND and NOR ? are also functionally complete,
  each by itself (as a singleton set).
• ?x xx and xy (xy)(xy) and xy
  (xx)(yy)
• ?x x?x and xy (x?x)?(y?y) and xy
  (x?y)?(x?y)

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Class Exercise

• Exercise 13.d. (p. 712)
• Hint Use the Law of the Double Complement De
  Morgans Laws
• Exercise 17. (3.d. only)
• Hint See Exercise 14
• Each pair of students should use only one sheet
  of paper while solving the class exercises