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lecture06

# lecture06 - 15-251 15-251 Great Theoretical Ideas in...

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9/15/2011 1 15-251 Great Theoretical Ideas in Computer Science 15-251 Counting for Computer Scientists Lecture 6 (September 14, 2011) Counting I: One-To-One Correspondence and Choice Trees In the next three lectures we will learn some fundamental counting methods. Addition and Product Rules The Principle of Inclusion-Exclusion Choice Trees Permutations and Combinations The Binomial Theorem The Pigeonhole Principle Diophantine Equations Generating Functions If I have 14 teeth on the top and 12 teeth on the bottom, how many teeth do I have in all? A B A B = + Addition Rule Let A and B be two disjoint finite sets

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9/15/2011 2 Addition of Multiple Disjoint Sets: Let A 1 , A 2 , A 3 , …, A n be disjoint, finite sets: A A i i i=1 n i n = = 1 Addition Rule (2 Possibly Overlapping Sets) Let A and B be two finite sets: |A| + |B| - |A B| |A B| = Inclusion-Exclusion If A, B, C are three finite sets, what is the size of (A B C) ? |A| + |B| + |C| - |A B| - |A C| - |B C| + |A B C| Inclusion-Exclusion If A 1 , A 2 , …, A n are n finite sets, what is the size of (A 1 A 2 A n ) ? i |A i | - i < j |A i A j | + i < j < k |A i A j A k | + (-1) n-1 |A 1 A 2 A n | Partition Method A A i i i=1 n i n = = 1 To count the elements of a finite set S, partition the elements into non-overlapping subsets A 1 , A 2 , A 3 , …, A n . S = all possible outcomes of one white die and one black die. Partition Method
9/15/2011 3 Each of 6 disjoint set have size 6 = 36 outcomes Partition S into 6 sets: S = all possible outcomes of one white die and one black die. Partition Method A 1 = the set of outcomes where the white die is 1. A 2 = the set of outcomes where the white die is 2. A 3 = the set of outcomes where the white die is 3. A 4 = the set of outcomes where the white die is 4. A 5 = the set of outcomes where the white die is 5. A 6 = the set of outcomes where the white die is 6. S = all possible outcomes where the white die and the black die have different values Partition Method A i set of outcomes where black die says i and the white die says something else. S Set of all outcomes where the dice show different values. S = ? | S | = i = 1 6 | A i | = i = 1 6 5 = 30 | S B | = # of outcomes = 36 |S| + |B| = 36 |B| = 6 |S| = 36 – 6 = 30 B set of outcomes where dice agree. S Set of all outcomes where the dice show different values. S = ? Difference Method To count the elements of a finite set S, find two sets A and B such that S and B are disjoint and S B = A then |S| = |A| - |B| S Set of all outcomes where the black die shows a smaller number than the white die. S = ? A i set of outcomes where the black die says i and the white die says something larger. S = A 1 A 2 A 3 A 4 A 5 A 6 |S| = 5 + 4 + 3 + 2 + 1 + 0 = 15

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9/15/2011 4 It is clear by symmetry that | S | = | L |. S + L = 30 Therefore | S | = 15 S Set of all outcomes where the black die shows a smaller number than the white die. S = ?
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lecture06 - 15-251 15-251 Great Theoretical Ideas in...

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