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Zorn’s Lemma Part 2

Hello, infinite-set reader. Set theory is back.

Finally. Today we will discuss about Zorn’s Lemma. This lemma is also often referred to as Kuratowski’s Lemma. Lemma’s name is taken ‘the inventor’, i.e Max Zorn and Kazimierz Kuratowski. May be sometime I will post about the history of Zorn’s Lemma. The previous post is on this link. Happy reading.

(x) Theorem : (Zorn’s Lemma / Kuratowski’s Lemma)

For any non-empty poset P that each simply ordered subset has a upper bound then P has at least one maximum element.

Proof :

Since each well ordered set is simply ordered set then the proof is analog like a proof of Theorem in the previous post.

At Theorem (x) above and Theorem (y) at this link, the theorem is equivalent to saying that for any non-empty poset P that the every well/simply ordered subset has an upper bound then P has at least one maximum element. In other words, for any x\in P there is maximum element of m\in P such that x\leq m.

Theorem :

If S is the set of all simply ordered subset of poset \left(P,\leq\right) which are partially ordered by \subseteq then every simply ordered subset of S has the smallest upper bound.

Proof :

Take any R=\left\{ u,v,\ldots\right\} simply ordered subset of poset \left(S,\subseteq\right). If R=\emptyset then sup\, R=\emptyset. Next, for R\neq\emptyset, consider :

    \[ r=\cup R=\cup\left\{ u,v,\ldots\right\} =\bigcup_{u\in S}u\]

It will be proved first that r is simply ordered subset of \left(P,\leq\right). Take any a,b\in r then there is u,v\in R such that a\in u and b\in v. Since \left(R,\subseteq\right) is simply ordered set then apply u\subseteq v or v\subseteq u. Consequently a and b are both elements of u or v. Since \left(u,\leq\right) and \left(v,\leq\right) is simply ordered set then apply a\leq b or b\leq a. Proved r is simply ordered subset of P. Consequently r is the smallest upper bound of \left(R,\subseteq\right) cause for any u\in R, it is true that u\subseteq r.

Based on the above description, then simply ordered subset r of poset P above is called largest simply ordered subset of P. So r is called largest simply ordered subset of P if r is not proper subset of the other simply ordered subset of P.

Thanks for reading. See you in the next post. May be useful.

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