Mathematical Thinking: Problem-Solving and Proofs (2nd Edition)

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MAT102S - Introduction to Mathematical Proofs - Spring 2010 - UTM Problem Set I - TO BE SUBMITTED TO YOUR TA Due : Monday, March 15, in tutorials. This assignment must be submitted to your TA at the beginning of the tutorial on the above date. Marking scheme : 8 marks for your solutions to the problems assigned (only some of the questions will be marked) and 2 marks for presentation. To receive the 2 marks allotted to presentation, you must submit at least half of the problems assigned and pay attention to the following: Your full name, student number, tutorial section and the name of your TA appear in the top of the first page. The section and question number (including a,b,c. .. if the question has several parts) are clearly indicated at the beginning of each of your answers. Your assignment is stapled in the top left corner (if it is more than one page in length). You are using a clean paper (no ripped paper), that is not folded or rolled. Your writing is clear and organized (use full sentences), and your reasoning is easy to follow.
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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 Schroeder-Bernstein 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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