Testwiki:Reference desk/Archives/Mathematics/2011 November 27

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November 27

Linear ordered set and well ordered set

Does every subset of a linear ordered set has a least element? If yes, then there's no difference between Linear ordered set and well ordered set. If no, how to prove it?(or can you give me a example? I know the answer should be 'no', but my intuition tells me 'yes').TUOYUTSENG (talk) 08:27, 27 November 2011 (UTC)

Very easy counterexample — the positive real numbers ("positive" in the English-language sense meaning "strictly greater than zero"). This set is linearly ordered by the usual order on the reals, but it has no least element, because for any ε in the set, ε/2 is strictly smaller. --Trovatore (talk) 08:33, 27 November 2011 (UTC)

Thank you!TUOYUTSENG (talk) 10:23, 27 November 2011 (UTC)

Even simpler example exists: the whole set of real numbers $ℝ$ has no least element. --CiaPan (talk) 06:20, 28 November 2011 (UTC)