Math 370 Solutions to Assignment 1 5, 8.1, 8.2, 18, 22, 25, 28. 5. Let A = f n 2 N : n > 3 g : Then A has no maximal element. Proof. Suppose that A has a greatest element m: Then m 2 A so 3 < m < m +1 2 A: This contradicts the property that m is maximmal in A: Thus our supposition is false so A has no maximal element. 8.1 Let F x 2 F: Then x >0 i/ x <0 : Proof. Suppose x >0 : Then by O 2 , x = 0 + ( x ) < x + ( x ) = 0 : Conversely, suppose x <0 : Then 0 = x + x < 0 + x = x: 8.2 Let e F: Then e >0 : Proof. We are given that e 6 = 0 : By the trichotomy law there remain only two mutually exclusive possiblities:0 < e and e <0 : Suppose that e <0 : Adding e , we get0 < e: Thus0 < ( e ) ( e ) = e: This contradiction shows that0 < e: 18 Let S F . The b = inf S i/ the following conditions hold ( i ) b ± x for all x 2 S and ( ii ) " >0 ) b + " > x for some x 2 S: Proof. (= ) ) Suppose
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