What is the proof of the following: Let < be a linear orders on A. Show that < i

Question

What is the proof of the following: Let < be a linear orders on A. Show that < is a well ordering on A if and only if there does not exist an infinite decreasing sequence in A. (Hint: Use contrapositive to prove in both directions: induction may be needed to prove by contrapositive in the second direction)

Explanation / Answer

I have answered it before. Linear order means that, any two elements can be compared. Well order means that any set has a smallest element. Well ordered => does not exist infinite decreasing sequence. Contrapositive is used - if there exists an infinite decreasing sequence, then it has no smallest element, if it had a smallest element a_n, choose a_(n+1), it would be smaller than the element. the other way -> No infinite decreasing sequence => well ordered So, consider any subset S, if S has a smallest element, done. Else, consider any element a_1 in S, there is a smaller element a_2 in A a_2

Related Questions

Navigate

• Browse (All)
• Browse W
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.