lecture07

lecture07 - COMPSCI 102 Discrete Mathematics for Computer...

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COMPSCI 102 Discrete Mathematics for Computer Science
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+ + ( ) + ( ) = ? Counting II: Recurring Problems and Correspondences Lecture7
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A B 1-1 onto Correspondence (just “correspondence” for short)
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Correspondence Principle If two finite sets can be placed into 1-1 onto correspondence, then they have the same size
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If a finite set A has a k-to-1 correspondence to finite set B, then |B| = |A|/k
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The number of subsets of an n-element set is 2 n .
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Sometimes it is easiest to count the number of objects with property Q, by counting the number of objects that do not have property Q.
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The number of subsets of size r that can be formed from an n-element set is: n! r!(n-r)! = n r
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A choice tree provides a “choice tree representation” of a set S, if 1. Each leaf label is in S, and each element of S is some leaf label 2. No two leaf labels are the same
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Product Rule (Rephrased) Suppose every object of a set S can be constructed by a sequence of choices with P 1 possibilities for the first choice, P 2 for the second, and so on. IF 1. Each sequence of choices constructs an object of type S 2. No two different sequences create the same object There are P 1 P 2 P 3 …P n objects of type S AND THEN
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How Many Different Orderings of Deck With 52 Cards? What object are we making? Ordering of a deck Construct an ordering of a deck by a sequence Construct an ordering of a deck by a sequence of 52 choices: of 52 choices: 52 possible choices for the first card; 51 possible choices for the second card; : : 1 possible choice for the 52 nd card. By product rule: 52 × 51 × 50 × … × 2 × 1 = 52!
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The Sleuth’s Criterion There should be a unique way to create an object in S. In other words: For any object in S, it should be possible to reconstruct the (unique) sequence of choices which lead to it.
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The three big mistakes people make in associating a choice tree with a set S are: 1. Creating objects not in S 2. Leaving out some objects from the set S 3. Creating the same object two different ways
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DEFENSIVE THINKING ask yourself: Am I creating objects of the right type?
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lecture07 - COMPSCI 102 Discrete Mathematics for Computer...

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