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lecture06 - COMPSCI 102 Discrete Mathematics for Computer...

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COMPSCI 102 Discrete Mathematics for Computer Science
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Lecture 6 Counting I: One-To-One Correspondence and Choice Trees
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If I have 14 teeth on the top and 12 teeth on the bottom, how many teeth do I have in all?
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A B A B = + Addition Rule Let A and B be two disjoint finite sets The size of (A B) is the sum of the size of A and the size of B
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Addition Rule (2 possibly overlapping sets) Let A and B be two finite sets |A B| = |A| + |B| - |A B|
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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
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Partition Method 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| = A A i i i=1 n i n = = 1
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S = all possible outcomes of one white die and one black die. Partition Method
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S = all possible outcomes of one white die and one black die. Partition S into 6 sets: 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. Each of 6 disjoint sets have size 6 = 36 outcomes
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S = all possible outcomes where the white die and the black die have different values Partition Method
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A i set of outcomes where black die says i and the white die says something else. S A A 5 3 0 i i i=1 i=1 = = = = = i 1 6 6 6 S Set of all outcomes where the dice show different values. S = ?
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| S T | = # of outcomes = 36 |S| + |T| = 36 |T| = 6 |S| = 36 – 6 = 30 S Set of all outcomes where the dice show different values. S = ? T set of outcomes where dice agree. = { <1,1>, <2,2>, <3,3>,<4,4>,<5,5>,<6,6>}
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How many seats in this auditorium? Count without Counting: The auditorium can be Partitioned into n rows with k seats each Thus, we have nk seat in the room
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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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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 = ? L set of all outcomes where the black die shows a larger number than the white die.
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“It is clear by symmetry that |S| = |L|?”
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S L Pinning Down the Idea of Symmetry by Exhibiting a Correspondence Put each outcome in S in correspondence with an outcome in L by swapping color of the dice. Thus: S = L Each outcome in S gets matched with exactly one outcome in L, with none left over.
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f is 1-1 if and only if 2200 x,y A, x y f(x) f(y) For Every There Exists f is onto if and only if 2200 z B 5 x A f(x) = z Let f : A B Be a Function From a Set A to a Set B
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A B Let’s Restrict Our Attention to Finite Sets 5 1-1 f : A B | A | ≤ | B |
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B A 5 onto f : A B | A | ≥ | B |
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