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

BenoguigoliB

BenoguigoliB

Answered question

2021-08-22

Which of the following sets are well ordered under the specified operation? Justify why they are/ are not well-ordered
(a) R+U0<
(b) [0,1],>
(c)The set of integers divisible by 5, <
(d)0,1,...,n|nN,

Answer & Explanation

komunidadO

komunidadO

Skilled2021-08-23Added 86 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 0<y<=, and so y(0,1) 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, T={SiT},
where TN. Since TNand N is well-ordered, it has a minimal element in. Now we see that
Si0T and TSi0,TT
so Si0 is a minimal element of T.

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