test4-fall92

# test4-fall92 - A has a least upper bound and we let w:= lub...

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Name: S.S.N: MAS 3300 – Test 4 Fall 1992 In giving proofs make sure your reasoning is clear. The questions have equal value. Write on ONE side of your paper. 1. Exhibit a rational number between 19 and 21. 2. Show that Q [ 3] := { a + b 3 : a,b Q } is closed under multiplication. 3. (i) Complete the following multiplication table in Z 6 . × 1 2 3 4 5 1 2 3 4 5 (ii) Find the elements of Z 24 that have multiplicative inverses. 4. Show that if a Z n and a and n are relatively prime then a has a multiplicative inverse in Z n . 5. Let α , β R with α < 0 < β. Prove that there is an irrational γ such that α < γ < β. You may assume that between any two reals there is a rational number. 1

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6. Let c R with c > 0. Let A = { a R : a 0 and a 2 c } . It can be shown that A is non-empty and bounded above. So, by LUB,
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Unformatted text preview: A has a least upper bound and we let w := lub A. It can be shown that w > 0. PROVE that w 2 = c. You may assume the Lemma given below. Lemma . Let u , a ∈ R with u > and a > . (i) If u 2 > a then there is an integer n ≥ 1 such that u-1 n > and ± u-1 n ² 2 > a. (ii) If u 2 < a then there is an integer n ≥ 1 such that ± u + 1 n ² 2 > a. 7. PROVE the following. (i) If n is prime and a · b = 0 in Z n then either a = 0 or b = 0 (in Z n ). (ii) If n is not prime then there exists a , b ∈ Z n with a 6 = 0 and b 6 = 0 such that a · b = 0 in Z n . 2...
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test4-fall92 - A has a least upper bound and we let w:= lub...

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