I am supposed to know how to use Zorn's Lemma on a final exam of the course Discrete Math, but I only had a few chances to practice it (and only about 1 of them were by me, only).
Where can I find questions about Zorn's Lemma, or ways of using it in order to solve problems?
I tried to google it - but didn't really find anything useful, and thought that some of you might know some nice questions that require the use of Zorn's lemma..
Thank you very much!
Let be the set of integers. An ideal is a subset such: (i) if , (ii) if and then . So for example, are two such ideals. We say an ideal is maximal if it NOT contained properly in another ideal.
You must prove that given any ideal , you can find a maximal idea such that . Notice, this problem can be solved without Zorn's Lemma, but your exercise is to try to solve it with Zorn's Lemma.
Does it make sense now?
You may try to solve this using Zorn's Lemma:
Every set can be well ordered (this is also known as Zermelo's Theorem or the Well-Ordering Theorem).
You should start by proving that given a poset where there exist some such that they are not comparable in , there exists a coset such that for any or After you prove this, you may proceed to prove the initial theorem using Zorn's Lemma.