(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If a subset (call it B) of a partially ordered (by a relation R) set A has exactly one minimal element, must that element be a smallest element? Give proof or counterexample.

2. Relevant equations

Well, our given, "exactly one minimal element" in PC (pred calc.) translates to:

(∃b)((∀x)((x,b) ∈ R → x = b) ∧ (∀y)((∀z)((z,y) ∈ R → b = y))

i hope...

3. The attempt at a solution

Call b the unique minimal element of B and let k ∈ B. Since (k,k) ∈ R, then using our assumption (∀y)((∀z)((z,y) ∈ R → b = y), b = k. Thus, (b,k) ∈ R.

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# Where is the error in this proof?

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