automata-1 - Automata Chapter1Introductionto...

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Automata Chapter 1   Introduction to  Theory of Computation
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2 Sets Set: a collection of elements Notations:  Universal set U, empty set    Element: S={0,1,2},  1 S, 3 S Operations: Union( ), Intersection( ), Difference(-),  Complement(Ā), Subset( ), Proper  Subset( )
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3 Sets Disjoint  Power set(2 S DeMorgan’s law Cartesian product of two sets S 1  = {2, 4}, S 2  = {2,3,5,6} S 1  X S 2  = ? 
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4 Functions f : S 1 S 2      dom(f)   S 1 , range(f)   S 2        If dom(f) = S 1 , total o.w. partial (cf) f : S 1 S 2        S 1 :domain, S 2  :codomain
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5 Relations More general than functions (one in  domain to multiple in range) Equivalence Relation(  ) on S Reflexive    x   x for all x in S Symmetric  if x  y then y  x Transitive   if x  y and y  z then x  z R b a R b a ) , ( R b a R b a ) , (
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6 Ex : Relation on the set of nonnegative integers   defined by x~y iff x mod 3 = y mod 3 (2~5, 3~0).    Show that ~ is an equivalence relation    ans)       x~x iff x mod 3 = x mod 3       x~y iff x mod 3 = y mod 3 =>                  y mod 3 = x mod 3 iff y~x       x~y and y~z iff x mod 3 = y mod 3 and                              y mod 3 = z mod 3 =>                              x mod 3 = z mod 3 iff x~z 1.1 Math Preliminaries & Notation 
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7 Equivalence Relation(  ) partitions S into nonempty  equivalence classes       (1)          (2) if a, b       then (a,b)       (3) if a      , b       ,        then (a,b) Ex) Previous Example      {0, 3, 6, 9, 12,  …… }      {1, 4, 7, 10, 13, ……}      {2, 5, 8, 11, 14, ……}    ,...... , 2 1 S S ...... 2 1 3 3 S S S = i S ∈≡ i S j S j i ∉≡
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8 Graph 2 finite sets, vertices and edges Vertices V = {v 1 , …, v n },  Edges E = {e 1 , …, e m } Digraph: directed graph,  visualized by diagram   example: www web pages  vertices files,  a directed edge  a link from a file to another  path, simple path, cycle, loop v 1 v 2 v 3
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9 Tree  Directed graph without cycles  root, leaves, parent, child  level, height
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automata-1 - Automata Chapter1Introductionto...

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