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Discussion 4

# Discussion 4 - 30-15 2 2 2 Given two sets A and B let m = |...

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CSE 21: Problems in Discussion 4 (April 30, 2007) ( 1 ) Using bijections to count: to count elements in set A , if there is a bijection between A and another set B , we may count elements in B instead. Example 1: Given set { 1 , 2 , · , n } . How many sub-multisets of size k are there? Sol 1. This is a Stars and Bars problem. Number of stars: k , number of bars: n - 1. ( k + n - 1 n - 1 ) . Sol 2. Let A be the collection of all sub-multisets of size k . For each multiset { a 1 , a 2 , · · · , a k } ∈ A , assume a i ’s are in ascending order. Define a function f on A so that f ( { a 1 , a 2 , · · · , a k } ) = { a 1 , a 2 + 1 , · · · , a k + k - 1 } . Then f is a bijection from A to B , where B is the collection of all k -subsets of { 1 , 2 , · · · , n + k - 1 } . ( n + k - 1 k ) = ( k + n - 1 n - 1 ) . Example 2: Given equation x 1 + x 2 + x 3 = 15, (a) How many non-negative integer solutions are there to the ? ( 15+2 2 ) (b) How many positive integer solutions? ( x 1 - 1) + ( x 2 - 1) + ( x 3 - 1) = 15 - 3 . ( 15 - 3+2 2 ) (c) How many integer solutions with all variables at least 2? ( x 1 - 2) + ( x 2 - 2) + ( x 3 - 2) = 15 - 6 . ( 15 - 6+2 2 ) (d) How many integer solutions with all variables at most 10? (10 - x 1 ) + (10 - x
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Unformatted text preview: . ( 30-15+2 2 ) ( 2 ) Given two sets A and B , let m = | A | , n = | B | . Type Deﬁnition Number correspondence no restrictions (2 n ) m = 2 mn function ∀ a ∈ A, ∃ unique b ∈ B, s.t., f ( a ) = b n m surjection ( onto ) function + ∀ b ∈ B, ∃ a ∈ A, s.t., f ( a ) = b n ! S ( m, n ) injection ( one-to-one ) function + ∀ x 1 , x 2 ∈ A, x 1 6 = x 2 → f ( x 1 ) 6 = f ( x 2 ) n ( n-1) ··· ( n-m + 1) = n ! ( n-m )! ( 3 ) Cardinality. Two sets have the same cardinality if there is a bijection between them. (a) Does a line segment of 2 inches contain more points than a line segment of 1 inch? (b) Are there more integers than even integers? (c) Is there a bijection between natural numbers and real numbers? (d) For any set A (ﬁnite or inﬁnite), is there a bijection between A and P ( A ), the power set of A ? 1...
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