# Which of the following sets are well ordered under the specified operation? Justify why th

Which of the following sets are well ordered under the specified operation? Justify why they are/ are not well-ordered
(a) $$R​+​ U_{0}<$$
(b) $$[0,1], >$$
(c)The set of integers divisible by 5, <
(d)$${ {0,1,..., n} | n​∈​N}, ⊆$$

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(a)This is not a wellordered set. For example, (0, 1) has no minimal element. Suppose that x is minimal, then we can find some $$\displaystyle{0}{<}{y}{<}=$$, and so $$\displaystyle{y}€{\left({0},{1}\right)}$$ but y (b)This is similar to (a) — (0,1) has no maximal (<-minimal) element.
(c)This set has no minimal element so it cannot be well-ordered.
(a)This is a. well ordered set. Denote this set by S. Denote by 5. the set
S = {0,1,2,....4}
Let T he somenonempty subset of S. We must prove that 7 has a minimal element.
First of all, $$\displaystyle{T}={\left\lbrace{S}{\mid}{i}∈{T}\right\rbrace}$$,
where $$\displaystyle{T}⊆{N}.$$ Since $${T}⊆{N}$$and N is well-ordered, it has a minimal element in. Now we see that
$$\displaystyle{S}{i}_{0}∈{T}$$ and $$\displaystyle{T}⊆{S}{i}_{0},{T}∈{T}$$
so Si0 is a minimal element of T.