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Subset of real numbers with properties

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November 2, 2010
algebra unsolvedalgebra

Problem Statement

Let SS be a subset of the real numbers with the following properties: (i)(i) If x∈Sx \in S and y∈Sy \in S, then x−y∈Sx - y \in S; (ii)(ii) If x∈Sx \in S and y∈Sy \in S, then xy∈Sxy \in S; (iii)(iii) SS contains an exceptional number x′x' such that there is no number yy in SS satisfying x′y+x′+y=0x'y + x' + y = 0; (iv)(iv) If x∈Sx \in S and x≠x′x \neq x' , there is a number yy in SS such that xy+x+y=0xy+x+y = 0. Show that (a)(a) SS has more than one number in it; (b)(b) x′≠−1x' \neq -1 leads to a contradiction; (c)(c) x∈Sx \in S and x≠0x \neq 0 implies 1/x∈S1/x \in S.
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