MATH 301 Homework 2 Solutions

# MATH 301 Homework 2 Solutions - Math 301 Solutions for...

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Math 301, Solutions for Homework 2 1. A librarian decides to make a catalogue of all the books in his library that do not contain the name of the book in the text (except for the title pages of course!). He names this catalogue " Complete list of all books in my library that do not contain their names ", publishes it and gets one copy in his library. Will this catalogue include " Complete list of all books in my library that do not contain their names " in its list? Solution 1 Suppose it does not contain its name in its list. Then accord- ing to the rule, it must contain itself in the list. Conversely, if its name is contained in the list, then according to the rule, it can not be in the list. Thus we get a contradiction, same as Russel°s paradox. This means, such a book can not claim to be a complete list. 2. For a °xed positive integer k; let P k ( N ) be the set (collection) of all subsets of N with exactly k elements. Prove that P k ( N ) is countable.

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