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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online