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inclass0826 - In-class exercises, Aug. 26 1. Prove that √...

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Unformatted text preview: In-class exercises, Aug. 26 1. Prove that √ 2 is irrational. 2. Use induction to prove that n i=1 ∑ i3 = n2 (n + 1)2 . 4 3. List the axioms for the real numbers (commutative and associative laws, etc.) 4. Use the axioms to prove the following results about real numbers: (the ﬁrst two are done as examples): (a) The number 0 is unique. Proof. Suppose that 0 and 0 have the property that, for all real numbers x, x + 0 = x + 0 = x. Then, in particular, if x = 0, we have 0 + 0 = 0 + 0 . By the identity axiom for addition, 0 = 0 . (b) For all real numbers x, x · 0 = 0. Proof. By the closure axiom, x · 0 is a real number, and by the additive inverse axiom, its additive inverse, −(x · 0), exists. By the identity and associative axioms, x · 0 = x · (0 + 0) = x · 0 + x · 0. Now 0 = −(x · 0) + x · 0 = [−(x · 0) + x · 0] + x · 0, so 0 = x · 0. (c) The additive inverse, −x, of the real number x is unique. (d) (−1) · (−1) = 1. (e) (−1) · x is the additive inverse of x. (f) For all real numbers x and y, (−x) · (−y) = x · y. (g) If x = 0 then x2 > 0. (h) If x = 0, then x−1 = 0. (i) 0 < 1. 1 ...
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This note was uploaded on 04/02/2009 for the course MAT 371 taught by Professor Thieme during the Spring '07 term at ASU.

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