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Well Ordered Set

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

As I said in the previous post, this post will discuss about the well ordered set. Happy reading.

In a poset, not necessarily the poset has the smallest element. So it is with the subsets. This gives rise to a new understanding which is a special occurrence of a poset.

Definition :

A poset is said to be well ordered set if for any non-empty subset of the poset, it has the smallest element.

Example :

(i) Poset \left\{ \left(\ldots,a_{3};a_{2};a_{1}\right),\left(\ldots;b_{3};b_{2};b_{1};\ldots;c_{3};c_{2};c_{1}\right)\right\} does not have the smallest element. In the subset there are also those that do not have the smallest element. For the example , simply ordered subset \left\{ a_{1},a_{2},a_{3},\ldots\right\}, \left\{ b_{1},b_{2},b_{3},\ldots\right\}, and \left\{ c_{1},c_{2},c_{3},\ldots\right\} none of them has the smallest element. So the poset is not a well ordered set.

(ii) Poset \left\{ \left(a_{1};a_{2};a_{3};\ldots\right)\right\} is well ordered set.

Theorem :

Every well ordered set is simply ordered set.

Proof : (more…)

Simply/Linearly/Totally Ordered Set

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

Today we will discuss about simply ordered set. Sometimes also called linearly or totally ordered set. To read this post, you should read this post first. You need the definition of an initial segment to understand the theorem we are going to discuss. Happy reading.

In a poset is not necessarily between one element with another can be compared. For example \left\{ \left(a;b\right),\left(c;d\right)\right\} is a poset but between b and c is incommensurable.

Definition :

Partial order \leq in the set S it says simple (linear, total) order in S if for any x,y\in S apply :

    \[ x\leq y\mbox{ or }y\leq x\]

Then any poset \left(S,\leq\right) it says simply (linearly, totally) ordered set if \leq is a simple (linear, total) order in S.

Example :

(i) \left(\left\{ a,b,c\right\} ,\left\{ \left(a,a\right),\left(b,b\right),\left(c,c\right),\left(a,b\right),\left(b,c\right),\left(a,c\right)\right\} \right) is simply ordered set.

(ii) Poset of the set of all natural numbers\left(\omega,\left(\in\mbox{ or }=\right)\right) it’s also simply ordered set.

Theorem :

Given any simply ordered set \left(S,\leq\right). Then the set of all initial segments of S is simply ordered by \subseteq.

Proof : (more…)

Initial Segment

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

In this post will be introduced about the initial segment. Definition of initial segment will be used in the process of proving Zorn’s Lemma. So you have to make a special mark for this post. It will be important for the future.

Definition :

Given any poset \left(S,\leq\right). For any a\in P defined I\left(a\right) with :

    \[ I\left(a\right)=\left\{ x|\left(x\in S\right)\wedge\left(x<a\right)\right\} \]

Furthermore I\left(a\right) is called the initial segment of P defined by a. If I\left(a\right) is not empty set then I\left(a\right) is called \emph{proper initial segment} of P.

Example :

Given poset \left\{ \left(a;b;c\right),\left(m;n;o\right)\right\} then I\left(c\right)=\left\{ a,b\right\}, I\left(n\right)=\left\{ m\right\}, and I\left(a\right)=\emptyset. (more…)

Bounded Set

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

Before reading this post, you should read the previous post. The previous post concerns the lower and upper bounds of a set. The previous definition became the basic of this post. Happy reading.

If A is a subset of any poset that has a lower bound then A is said to be bounded from below. If it has a upper bound then A is said to be bounded from above. And if it has the upper bound and the lower bound then A is said to be bounded.

For example L is defined as the set of all lower bound of A subsets of any poset S. If the largest element of the set L is g then g single existance. Next g is denoted by inf\, A that is the largest lower bound ofA. With the same understanding, the smallest upper bound of A is denoted by sup\, A. Notice that the sup\, A and inf\, A are in L. So the existence sup\, A and inf\, A not guaranteed in A. To confirm this statement, please refer to the following example.

Example :

Given A=\left\{ b_{1},b_{2},\ldots\right\} is subsets of poset D=\left\{ \left(\ldots;a_{2};a_{1};\ldots;b_{2};b_{1};c_{1}:c_{2};\ldots;d_{2};d_{1}\right),\left(a;b;c\right)\right\}. We get sup\, A=b_{1}\in A but inf\, A=a_{1}\notin A.

Now, it will then be given the theorem of a poset relating to the smallest upper and lower bounds.

Theorem :

(i) Poset \left(S,\leq\right) has the smallest element if and only if the empty set \emptyset has the smallest upper bound in S.

(ii) Poset \left(S,\leq\right) has the largest element if and only if the empty set \emptyset has the largest lower bound in S.

Proof : (more…)