I am working problem 16 in section 3.5 from \"bridge to abstract book\" and I th

Question

I am working problem 16 in section 3.5 from "bridge to abstract book" and I think this is a more challenging problem in this section. I thought I'd post what I did so far to get some feedback.

16a. The hint in the textbook answer is to use parts of Theorem 2.2.1 and Exercise 9 of Section 2.2. Let C A and let D A .

From Exercise 9f, if C A and D A ., then C D A .

Because C is the ordering, it appears I can immediately say the greatest lower bound of {C,D} is C D .

I am wondering if this is acceptable for proving the greatest lower bound by just citing Exercise 9f, if C A and D A ., then C D A , or should I prove this explicitly?

I'm looking for some hints to how to finish it, not the whole answer. Please try to write the answer with the keyboard not with taking a picture because I need to copy and paste the answer. Thanks

Explanation / Answer

I see that your argument works (i.e., citing 9f works if your relation is 'reverse inclusion'. (i .e., if I denote < as your relation, then X < Y iff X contains Y as subset)

But if your relation is actually ' set inclusion', your argument (i.e., citing 9f) is not a correct argument for proving greatest lower bound. but you argument is correct for proving CunionD as least upper bound.

So for your proof of greatest bound(considering the relation to be set inclusion), first observe C intersection is a subset of both C and D. Further observe that, if A is a subset of both C and D, then A is a sunset of C intersection D. Hope this helps.

Related Questions

Navigate

• Browse (All)
• Browse I
• Subjects

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