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

BenoguigoliB 2021-08-22 Answered

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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komunidadO
Answered 2021-08-23 Author has 19559 answers

(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.

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