## Homework

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## Homework

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- School:
- University of California, Berkeley
- Course:
- Math 74 - Transition to Upper Division Mathematics

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Math 74 Homework 8 Due Monday October 21st October 12 2008 1 Let X be a set not necessarily finite Show that if there exists a surjection f X Func X X then X 1 2 Let I be a set and suppose that for each i I we are given a set Xi Define Xi x i I such that x Xi i I and similarly define Xi x x Xi for all i I i I Let Y be another set Prove the infinite distributive laws S S a Show that i I Xi Y i I Xi Y and T T b show that i I Xi Y i I Xi Y S 3 For each n N let Xn be a countable set Show that n N Xn is countable This is an extremely useful fact often phrased as a countable union of countable sets is countable 4 Let R be the relation on N 0 given by nRm if n divides m Show that R is a partial order relation Let n m N 0 Describe the least upper bound and greatest lower bound of n m with respect to R 5 Let X be a set and consider the partial order relation on P X S Let Y P X beTa subset of P X Show that A Y A is an upper bound for Y and A Y A is a lower bound for Y with respect to S Note again A Y A means x X x A for some A Y etc 1 6 Give an example of a relation on N which is nonempty and a Reflexive but not transitive b Transitive but not reflexive 2

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