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Unformatted text preview: CS2603 Applied Logic for Hardware and Software Rex Page University of Oklahoma 1 CS 2603 Applied Logic for Hardware and Software Review Predicate Calculus, Inductive Definitions, Mathematical Induction, Sets, and Circuit Minimization CS2603 Applied Logic for Hardware and Software Rex Page University of Oklahoma 2 + a b Circuit Minimization with Karnaugh Maps 1. Group together the maximal, contiguous, rectangular regions with 2 k adjacent cells containing True (1) values 2. Because of the Graycode ordering, each group of minterms puts k variables through all possible combinations of values and the remaining (n k) literals are the same in all of the minterms of the group 3. Use distributive law to factor out the (n k) identical literals w = w nk literals (w 1 + w 2 + + w 2 k ) 4. Note that w 1 + w 2 + + w 2 k = True ( null law), then apply identity law wraparound = a d + b c d + a c b=1 00 01 11 10 00 01 11 10 c = 1 b a d c a = 1 1 1 1 1 1 1 1 d = 1 1 F(a,b,c,d) = a b c d + a b c d + a b c d + a b c d + a b c d + a b c d + a b c d + a b c d CS2603 Applied Logic for Hardware and Software Rex Page University of Oklahoma 3 Notation for Sets Explicit Enumeration {2, 3, 5, 7, 11} 2 {2, 3, 5, 7, 11} stylized epsilon denotes element of 5 {2, 3, 5, 7, 11} 1 {2, 3, 5, 7, 11} x A (x A) Long Tall Sally {2, 3, 5, 7, 11} {{Whad I Say, Nadine}, {Peer Gynt, Moonlight Sonata, Finlandia}} = A {Whad I Say, Nadine} A Moonlight Sonata A { } the empty set, which has no elements none nada the number of elements in { } is zero 3 { } x { } no matter what x stands for { } stylized Greek letter phi denotes empty set ( { }) = False, or ( { }) = True ? False There is NOTHING in { } CS2603 Applied Logic for Hardware and Software Rex Page University of Oklahoma 4 Notation for Sets Set Comprehension { x  x {2, 3, 5, 7, 11} x 4} {5, 7, 11} { x + x  x {2, 3, 5, 7, 11} x 3 x < 11} {6, 10, 14} {f x  x A, p x} Denotes set whose elements have the form (f x), where x comes from A and (p x) has the value True To avoid contradictions Predicate (or context) specifies universe of discourse Examples of invalid set comprehensions 9 {X  X is a set} 9 {X  X X} 9 Universe of discourse is missing in these examples CS2603 Applied Logic for Hardware and Software Rex Page University of Oklahoma 5 Set Operations Power Set Power set of A Definition: P(A) = {S  S A} Examples 9 P({2, 3, 5}) = { , {2}, {3}, {5}, {2,3}, {2,5}, {3,5}, {2,3,5}} 9 P( ) = { } 9 Theorem: If A has n elements, P(A) has 2 n elements This will be proved later, using mathematical induction Example: has 0 elements, and P( ) has 2 = 1 element 9 How many elements in P(P(...
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This note was uploaded on 04/08/2008 for the course CS 2603 taught by Professor Rexpage during the Spring '08 term at The University of Oklahoma.
 Spring '08
 RexPage

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