Unformatted text preview: Submit your solutions to the following problems: 1. Solve Exercise 4.36 from the textbook. Note : The identity function on a set X is the function f : X → X , which sends each element to itself (i.e., f ( x ) = x for all x ∈ X ). 2. Prove that R and the interval (0 , ∞ ) have the same cardinality. 3. Let X = { , 1 } . List all the elements in P ( P ( X )) . 4. TRUE or FALSE? Justify with a proof or a counterexample. (a) For any two sets A,B we have P ( A ∩ B ) = P ( A ) ∩ P ( B ) . (b) For any two sets A,B we have P ( A ∪ B ) = P ( A ) ∪ P ( B ) . 5. Let A and B be disjoint sets, which are both countable (i.e., have the same cardinality as N ). Prove that A ∪ B is also countable. 6. Use the SchroederBernstein Theorem (page 91) to prove that the intervals (0 , 1) and [0 , 1] have the same cardinality. 7. Solve Exercises 5.4 and 5.17 from the textbook....
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 Spring '10
 Math, TA, Georg Cantor, Mathematical Proofs, UTM Problem Set

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