Question

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)

Vectors and spaces
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asked 2021-05-04

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)\(\displaystyle{\left\lbrace{\left\lbrace{0},{1},\ldots,{n}\right\rbrace}{\mid}{n}​∈​{N}\right\rbrace},⊆\)

Answers (1)

2021-05-05

(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
\(S = \{0,1,2,\cdots4\}\)
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 \(\displaystyle{T}⊆{N}.{\sin{{c}}}{e}{P}{S}{K}{T}⊆{N}\), 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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