1 votes 1 votes Which of the following are Well ordered set $\left [ Z^{+},\leq \right ]$ $\left [ Z^{-},\leq \right ]$ $\left [ Z^{+},\geq \right ]$ $\left [ Z^{-},\geq \right ]$ Set Theory & Algebra made-easy-test-series set-theory&algebra + – jatin khachane 1 asked Jan 28, 2019 • edited Mar 3, 2019 by ajaysoni1924 jatin khachane 1 1.2k views answer comment Share Follow See all 23 Comments See all 23 23 Comments reply aambazinga commented Jan 28, 2019 reply Follow Share I think 1st one and last one are w ordered(definition: if every non empty subset of the given set has a least element, the set is well ordered). Keeping this in mind, only 1st option and last is satisfying the condition. Why? Because 1st will form a linear chain with 1 as the least element, and last one will form a linear chain with -1 as the least element. Rest all doesn't have any finite least element.. because they are either negative infinity as their least element, or positive infinite as their least element, which can't be represented in Haase diagram. 1 votes 1 votes jatin khachane 1 commented Jan 28, 2019 reply Follow Share @aambazinga lets say for this, [Z+,≥] Hasse diagram will be like, 1 2 3 4 5 . . . inf lets take subset { 2 to inf } here least element does not exists right is this right ? My doubt: What least element means : 1) Smallest element in that subset OR 2) Element in subset which is at bottom in hasse dia i,e noone below it in hasse dia 0 votes 0 votes aambazinga commented Jan 28, 2019 reply Follow Share Which is at the bottom in the Haase diagram, and it is unique if exist. 1 votes 1 votes Shaik Masthan commented Jan 29, 2019 reply Follow Share @aambazinga Which is at the bottom in the Haase diagram, and it is unique if exist. the Hasse diagram is correspondence to subset or original set ? 0 votes 0 votes jatin khachane 1 commented Jan 29, 2019 reply Follow Share @Shaik Masthan @aambazinga to find least element..we should see bottom element in hasse diagram of subset .. right? 0 votes 0 votes aambazinga commented Jan 29, 2019 reply Follow Share @jatin khachane 1 If we are talking about well ordered, then we look for the least element for subset( which may or may not be the least element of the lattice as a whole), and in this case, every subset should have a least element, and they may or may not be the same. if we are talking about the lattice, there is a unique least element. 0 votes 0 votes Shaik Masthan commented Jan 29, 2019 reply Follow Share then what about the first statement of this question https://gateoverflow.in/299954/madeeasy#c300331 0 votes 0 votes jatin khachane 1 commented Jan 29, 2019 reply Follow Share First statement is not well ordered acc to me @Shaik Masthan what is your opinion 0 votes 0 votes srestha commented Jan 29, 2019 reply Follow Share well order means, set need to be finite So, 1 st option has possibility of finite but 3rd option has no possibility of finite And $Z^{-}$ has no meaning So, 2nd and 4th option just meaning less, how far I assumed 0 votes 0 votes Shaik Masthan commented Jan 29, 2019 reply Follow Share @jatin khachane 1 yes, it is not well-ordered ! @srestha mam, Z$^-$ means, set of all negative integers ! 0 votes 0 votes srestha commented Jan 29, 2019 reply Follow Share oh, sorry So,I and IV correct right? @Shaik Masthan 0 votes 0 votes Shaik Masthan commented Jan 29, 2019 reply Follow Share yes mam ! 0 votes 0 votes jatin khachane 1 commented Jan 29, 2019 reply Follow Share @Shaik Masthan here if we take subset {2,3,4,5} least element of that subset will be 5 right not 2 https://gateoverflow.in/300668/madeeasy-adv-level?show=300692#c300692 0 votes 0 votes Shaik Masthan commented Jan 29, 2019 reply Follow Share it's depend upon which option this subset is constructed ? 0 votes 0 votes jatin khachane 1 commented Jan 29, 2019 reply Follow Share [Z+,≥] 1 2 3 4 5 . . Inf 0 votes 0 votes Shaik Masthan commented Jan 29, 2019 reply Follow Share yes @jatin 1 votes 1 votes jatin khachane 1 commented Jan 29, 2019 reply Follow Share Thanks :) 0 votes 0 votes jatin khachane 1 commented Jan 29, 2019 reply Follow Share @Shaik Masthan then why it is not called minimal element in that subset instead of least ? 0 votes 0 votes Shaik Masthan commented Jan 29, 2019 reply Follow Share i didn't get you ! but this may help you " minimal element always not unique in Hasse diagram, but it is unique in Lattice. " 0 votes 0 votes jatin khachane 1 commented Jan 29, 2019 reply Follow Share @Shaik Masthan No what i mean to say In def of well ordered it is " every non empty subset should have least element". I have seen many people taking least as smallest element(acc to value) in that subset....which is wrong .. right? It depends on relation Can't we say like this Def of well ordered as " in hasse dia of every non empty subset there should exist a MINIMAL element" and as we are dealing with TOS ..it will be unique only 0 votes 0 votes Shaik Masthan commented Jan 29, 2019 reply Follow Share Hmm... i don't why you didn't respond on the answer ? He makes answer for you only, right ? atleast you have to comment like " got it " 0 votes 0 votes Shaik Masthan commented Jan 29, 2019 reply Follow Share sry but dnt know why u always point some mistake in me :) :) actually, the positive comments made by others, give some boost to help more ! if you doesn't want i will stop ! 1 votes 1 votes jatin khachane 1 commented Jan 29, 2019 reply Follow Share Dnt get me wrong bro @Shaik Masthan m sry :) 0 votes 0 votes Please log in or register to add a comment.
2 votes 2 votes z+= 1,2,3,4,5,6.... if orderd then 1 should be related to 2 and 2 should be related to 3 so on in ordered fashion. so, only one possibility is 1<=2<=3<=4..... similarly for z- we have -1>= -2>= -3....... hence ans. should be (z+,<=) and z-(>=). _sonu answered Jan 28, 2019 _sonu comment Share Follow See all 2 Comments See all 2 2 Comments reply jatin khachane 1 commented Jan 29, 2019 reply Follow Share 👍 @sonu 0 votes 0 votes Raj Kumar 7 commented Jan 29, 2019 reply Follow Share I think answer is only (Z+, <=) and least element in ordering is 0. But in case of (Z-, >=) there is no any fixed least element in this ordering. If i am wrong explain this 0 votes 0 votes Please log in or register to add a comment.