A nonempty subset S of the set of real numbers is said to be well-ordered if eve

Question

A nonempty subset S of the set of real numbers is said to be well-ordered if every nonempty subset of S contains a least element.

(a) Use this definition to concisely restate the Well-Ordering Principle.

(b) Is the set of Real Numbers well ordered? Why or why not?

(c) Is the set of real numbers={x an element of real numbers:x>=0}

(d)Is{-9,-7,-5,...} well ordered? Why or why not?

(e) Prove or disprove: If a set S is well-ordered, then S conatains a least element.

(f) Prove or disprove: If a set S conatains a least element then S is well-ordered.

Explanation / Answer

(a). The well ordering principle states that every non empty set of positive integers contains a least element. In other words, the set of positive integers is well-ordered.

(b). The set of real number is not well ordered, since for example, the open interval (0,1) does not contain a least element.

(c).the set of real numbers={x an element of real numbers:x>=0} is not well ordered.

(d). Set {-9.-7,-5....} is well ordered because each set contains a least number.

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