# Thread: Zorn's Lemma

1. ## Zorn's Lemma

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!

2. Originally Posted by adam63 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!
Here is an problem involving Zorn's Lemma:
Show that any proper ideal is always contained in a maximal idea.

3. It looks like a nice problem, only I don't understand what 'proper ideal' means :S

4. Originally Posted by adam63 It looks like a nice problem, only I don't understand what 'proper ideal' means :S
Okay, so I make it more specific.

Let $\displaystyle \mathbb{Z}$ be the set of integers. An ideal is a subset $\displaystyle I\subseteq \mathbb{Z}$ such: (i) if $\displaystyle a,b\in I \implies a+b\in I$, (ii) if $\displaystyle n\in \mathbb{Z}$ and $\displaystyle x\in I$ then $\displaystyle nx\in I$. So for example, $\displaystyle I=\{0\},I=\mathbb{Z}$ are two such ideals. We say an ideal $\displaystyle I\not = \mathbb{Z}$ is maximal if it NOT contained properly in another ideal.

You must prove that given any ideal $\displaystyle I\not = \mathbb{Z}$, you can find a maximal idea $\displaystyle J\not = \mathbb{Z}$ such that $\displaystyle I\subseteq J$. 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?

5. Hello!

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 $\displaystyle (X,\preceq)$ where there exist some $\displaystyle x_1,x_2\in X$ such that they are not comparable in $\displaystyle (X,\preceq)$, there exists a coset $\displaystyle (X,\preceq')$ such that for any $\displaystyle x,y\in X, x\preceq' y \Leftrightarrow x \preceq y$ or $\displaystyle x=x_1,y=x_2$ After you prove this, you may proceed to prove the initial theorem using Zorn's Lemma.

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