Problem 2.5: For this question, use the definition of interval given above might

Question

Problem 2.5: For this question, use the definition of interval given above might or might not be the same interval, or it might or might not be that al-a2 etc. None of these possibilities are ruled out.) 9 In the following three parts, suppose the intersection [ai, bil n [a2, b2l of the intervals [ai, bl and [a2, b2] is nonempty Part 1) Is [a1, bi] n [a2, b2] an interval? Prove or disprove. Part 2) Does [ai,bi]nla2, bal have a nonempty subset that is an interval? Prove or disprove. Part 3) If R is also reflexive, does (a1, bi]nla2, ba] have a nonempty subset that is an interval? Prove or disprove.

Explanation / Answer

Since the intersection of [a,b] and [c,d] is nonempty

Let x belongs to the intersection.

Hence x is in [a,b] and x is in [c,d]

Hence a<=x<=b. And c<=x<=d.

max(a,c)<=x<=min(b,d)

Hence the intersection of the closed intervals is nothing but [max(a,c) , min(b,d)] which is again an interval and contained in both the closed intervals.

If max(a,c)= min(b,d) then the intersection is a singleton set and hence not an interval. That can happen when b1=a2.

In that case the intersection does not have a nonempty subset that is an interval.

Unless otherwise , the intersection is always an interval.

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

---

---

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