m347_e2pracSpring05

m347_e2pracSpring05 - k | m . Prove that a b (mod k ). 7....

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Math 347 Practice Exam #2 Name All problems except # 2 require proofs and/or explanations. 1. Let f : Z Z be defined by f ( n ) = 3 n + 1. (a) Is f an injection ? Explain: (b) Is f a surjection ? Explain: 2. Let f, g : R R . Answer each as True or False (no explanation required). (a) if f is increasing, then f is surjective. (b) if f is surjective, then f is unbounded. (c) if f and g are bijections, then g - 1 f - 1 is injective. 3. Define a relation on the set R by x y if | x - y | ≤ 1. Is an equivalence relation? 4. Suppose n N and n 5 (mod 6). Prove that n 2 + 2 is composite. 5. Show that if a is even and b is odd, then gcd( a, b ) = gcd( a/ 2 , b ). 6. Suppose that a b (mod m ) and than
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Unformatted text preview: k | m . Prove that a b (mod k ). 7. Suppose gcd( a, 21) = 1. Use Fermats Theorem to prove that a 6 1 (mod 21). 8. A drawer contains 7 red socks, 10 white socks and 8 blue socks. You grab three of the socks at random. What is the probability that all three are the same color? 9. Let S be a square of side length 1. Let P 1 , . . . , P 101 be 101 points inside of S . Prove that two of the points are a distance 2 / 10 apart. Hint: partition S into little squares of side 1 10 , and use the pigeonhole principle....
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