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I have to prove or disprove that every non-empty set on non-negative rational numbers has a least element. I thought this may be pretty straight forward but I thought what if the non-empty set contains only one element. For example; A = {2}. Does this mean that the least element is 2? Or the highest element is 2? Or both or neither? Would love to hear from everybody. PS: I'm second year math major so I'm still new to this.

Andrés E. Caicedo
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Chance Gordon
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  • If the set contains only one element, then that element is the least element. For example, in the set ${ 2}$, the least element and the biggest element is $2$. Now, the point is that your statement is false. For example, you could just take ${ \frac 1n}, n \in \mathbb N$ as a counterexample: – Sarvesh Ravichandran Iyer Nov 09 '16 at 23:32

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It's both; the point is that it has not necessarily a least element; for example the set $A=\{\frac 1 n|n\in\mathbb N \}$ is a non empty set of rational numbers but it hasn't a least element, so the statement is false.

Bargabbiati
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