# a1 - Solutions to Suggested Problems from Appendix 1 A1(i(a...

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Unformatted text preview: Solutions to Suggested Problems from Appendix 1 A1.(i) (a) Both are equal to n ! k !( n- k )! . (b) A one-to-one correspondence between two sets is a function that associates each el- ement in one set to exactly one element in the other set, in a way that all elements of both sets are used. The idea is to show that both functions must count the same number of objects, so they must be equal. This is called a combinatorial proof of the equation, and we will be seeing a lot of this kind of argument this quarter. Here, let A be the set of all k-element subsets of { 1 ,...,n } , and let B be the set of all n- k-element subsets of { 1 ,...,n } . For a set S ∈ A , define f ( S ) = S c . For each S , it is easy to see that f ( S ) ∈ B , since the complement of a k-element set is an n- k element set. This function is one-to-one because no two sets have the same complement. The function maps onto because B because every element of B is the complement of a set in A ....
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a1 - Solutions to Suggested Problems from Appendix 1 A1(i(a...

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