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# hw1 - 4 Prove that there are an uncountable number of total...

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CS 4114, Assignment 1, due on February 7. Please show all your work! 1. Let S be a set containing n elements. Show that S has 2 n subsets. 2. Find the error in the "proof" of the following assertion. Any set of n elements has the property that all n elements are identical. PROOF: By induction on n . BASIS: n = 1 The set has one element a , and clearly a = a . INDUCTION: Assume true for all sets of up to n - 1 elements and consider any subset of {a 1 , a 2 , ..., a n } . If we form the subset {a 1 , a 2 , ..., a n-1 } , then by assumption a 1 = a 2 = ... = a n-1 . Also form the set {a 2 , a 3 , ..., a n } . Collecting results, we have that a 1 = a 2 = ... = a n-1 = a n . 3. Prove that the set of nonnegative rational numbers is denumerable.
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Unformatted text preview: 4. Prove that there are an uncountable number of total functions from N to {0, 1} . 5. Give a recursive definition of multiplication of natural numbers using the successor function and addition. 6. Prove that if X and Y are countable sets, then so is X Y. 7. Express using a concise mathematical statement the function from N to N N in Example 1.4.2. 8. Prove Theorem 1.4.2 (Schrder-Bernstein). 9. Let R be the relation on N + N + given by (a,b) R (c,d) iff a/b = c/d. Show that R is an equivalence relation. What are its equivalence classes? NOTE: Your problem solutions are due at 5:00 pm of the date....
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