MIT6_042JS10_lec29

MIT6_042JS10_lec29 - Sum Rule Mathematics for Computer...

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1 Albert R Meyer, April 16, 2010 Mathematics for Computer Science MIT 6.042J/18.062J Inclusion-exclusion Counting practice lec 10F.1 Albert R Meyer, April 16, 2010 Sum Rule |A " B| = |A| + |B| A B for disjoint sets A, B lec 10F.2 Albert R Meyer, April 16, 2010 A B What if not disjoint? Sum Rule |A " B| = ? lec 10F.3 Albert R Meyer, April 16, 2010 Inclusion-Exclusion A B What if not disjoint? lec 10F.4 Albert R Meyer, April 16, 2010 Inclusion-Exclusion (3 Sets) |A " B " C| = |A|+|B|+|C| – |A ! B| – |A ! C| – |B ! C| + |A ! B ! C| A B C lec 10F.8 Albert R Meyer, April 16, 2010 A town has n clubs. Each club S i has a secretary M i who knows if person x is a club member: M i ( x ) = 1 if x in S i , = 0 if x not in S i . lec 10F.10 Incl-Excl Formula: Proof
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2 Albert R Meyer, April 16, 2010 M i ( x )M j ( x ) is sec’y for S i ! S j , so Incl-Excl Formula: Proof lec 10F.11 So A = M A ( x ) x people S i S j = M i ( x ) x M j ( x ) S i S j S k = M i ( x ) x M j ( x ) M k ( x ) etc. Albert R Meyer, April 16, 2010 Let D ::= S 1 " S 2 " " S n sec’y M D ( x ) =0 iff M i ( x ) =0 for all n clubs . So 1- M D ( x ) = (1- M 1 ( x ) )(1- M 2 ( x ) ) (1- M n ( x ) ) lec 10F.12 Incl-Excl Formula: Proof Albert R Meyer, April 16, 2010
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