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hw9b - Thoerem If X is a ±nite set and there exists a...

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Math 300A&B Introduction to Mathematical Reasoning Fall 2009 Assignment #9: Due Friday, 12/11/09 Part II: For the proofs below, if you’re being asked to prove one of the theorems from the text, you may use any of the theorems stated in class, any theorem stated in the text before the one you’re proving, and any theorems proved in previous homework problems. If the theorem you’re being asked to prove is not one of the theorems from the text, then in addition you may use any theorems from Chapters 1–11 and 14. Exception: The following theorem from class should not be used in the proof of Exercise 11.1, because that would create a circular argument:
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Unformatted text preview: Thoerem: If X is a ±nite set and there exists a surjection X → S , then S is also ±nite and | S | ≤ | X | . 4. Eccles, page 143, Exercise 11.2. 5. Eccles, page 184, Problem 12. 6. If X is a set that contains an uncountable subset, prove that X is uncountable. 7. If X is an uncountable set and A is a countable subset of X , prove that X-A is uncountable. 8. Determine whether each of the following sets is empty, Fnite but nonempty, denumer-able, or uncountable. No proofs necessary. (a) { 1 /n : n ∈ Z + } . (b) R-Q . (c) Z × R . (d) [0 , ∞ ). (e) { x ∈ R : x 2 ∈ Z } . (f) { x ∈ Z : x 2 ∈ R-Z } ....
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