11A Week 11 - CMPUT 272(Stewart University of Alberta...

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CMPUT 272 (Stewart) University of Alberta Lecture 19 Reading: Epp 9.1-9.3 Counting Recall: A set is finite if it has no elements or there is a one-to-one corre- spondence from { 1 , 2 , . . . , n } to it for some n Z + . If A is a finite set, then the number of elements in A , denoted | A | or N ( A ), is | A | = 0 if A = n if there is a bijection from { 1 , 2 , . . . , n } to A, for some n Z + Example: If m and n are integers, m n , then the number of integers from m to n inclusive is n - m + 1. To justify this, we find a bijection from { 1 , 2 , . . . , n - m + 1 } to { m, . . . , n } . Consider matching elements of { 1 , 2 , . . . , n - m +1 } with elements of { m, . . . , n } : m m + 1 m + 2 . . . n l l l l 1 2 3 n - m + 1 m m + 1 m + 2 . . . n = = = = ( m - 1) + 1 ( m - 1) + 2 ( m - 1) + 3 ( m - 1) + n - ( m - 1) l l l l 1 2 3 n - m + 1 Thus f : { 1 , . . . , n - m + 1 } 7→ { m, . . . , n } where f ( x ) = ( m - 1) + x is a bijection. Therefore, |{ m, . . . , n }| = n - m + 1, that is, the number of integers from m to n inclusive is n - m + 1. 1
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Example: The number of 3-digit positive integers that are multiples of 5 is 180. We find a bijection from { 1 , 2 , . . . , 180 } to { 100 , 105 , . . . , 995 } . 100 101 . . . 105 106 . . . 995 . . . 999 = = = 20 × 5 21 × 5 199 × 5 = = = (19 + 1) × 5 (19 + 2) × 5 (19 + 180) × 5 l l l 1 2 180 f : { 1 , 2 , . . . , 180 } 7→ { 100 , 105 , . . . , 995 } where f ( x ) = (19 + x ) × 5 is a bijection. Therefore, |{ 100 , 105 , . . . , 995 }| = 180, that is, the number of 3-digit positive integers that are multiples of 5 is 180. 2
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Rules for Counting The Addition Rule [Epp Theorem 9.3.1] Suppose a finite set A equals the union of k distinct mutually disjoint subsets A 1 , A 2 , . . . , A k . Then | A | = | A 1 | + | A 2 | + · · · + | A k | . The Difference Rule [Epp Theorem 9.3.2] If A is a finite set and B is a subset of A then | A - B | = | A | - | B | . The Inclusion/Exclusion Rule [Epp Theorem 9.3.3] If A, B , and C are finite sets then | A B | = | A | + | B | - | A B | , and | A B C | = | A | + | B | + | C | - | A B | - | A C | - | B C | + | A B C | . (Generalizes to any number of sets.) The Multiplication Rule [Epp Theorem 9.2.1] If an operation consists of k steps and the first step can be performed in n 1 ways, the second step can be performed in n 2
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