$(a)\quad R^+ \cup \{0\}, <$

$(b)\quad [0,1], >$

$(c)\quad \text{The set of integers divisible by 5}, <$

$(d)\quad \{\{0,1,...,n\}|n ∈ N\},⊆$

I believe that:

(a) Is not well-ordered because of the fact that rational numbers would be in a continuous flow and would be infinitely smaller.

(b) Is not well-ordered, because (0,1) as a subset would break the ability of it.

(c) Is not well-ordered, because the set of integers could go to negative infinity, making the set not have a value that can satisfy the less than requirement.

(d) Is well ordered, as 0 is the smallest element of subsets in the set of {0,1,...,n}

Are my beliefs correct on these scenarios, and how would I go about trying to actually prove them?


1 Answer 1


Nice, they're correct. For (a)-(c) you've essentially already proved them through contradiction. Try to write them out in more detail.

Here's a sketch of (a): Take the rationals in $(0,1)$. If it had a minimum $q$, then take the midpoint of $0$ and $q$, contradiction.


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