## Poset – Partially Ordered Set – Part 2

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

We return to the previous post that any partial order is denoted by and often denoted by . Today we will discuss about the partially ordered set. If you forget the previous post, you can go to this link. Happy reading.

**Definition :**

The pair is said to be partially ordered set if is the partial order in the set .

If is a partially ordered set then it is said that partial order by . Furthermore to further abbreviate the mention, partially ordered set will be called a poset. Suppose given form :

then the intent of the above form is a partially ordered set with :

and is partial order with if is on the left under the condition are in the same area / in the same brackets.

**Example :** (more…)

## Poset – Partially Ordered Set – Part 1

Hallo, infinite-set readers. Set theory is back. Already reading the about relation? Today we will use that chapter. I hope you already know about relation.

In this post I will actually discuss about Zorn’s Lemma. But before that, it will be discussed first about the partially ordered set, simply ordered set, and well ordered set. For this post will be given first about the opening of the partially ordered set. Happy reading.

**Definition :**

Given any set . A subset is called \emph{partial order} in set if for any apply :

(i) (reflexive)

(ii) if and then (antisymmetric)

(iii) if and then (transitive)

**Example :**

(a) Given then is partial order in .

**Proof :**

(i) Reflexive

Therefore for any applies then is proven reflexive.

(ii) Antisymmetric

Look at the above. Therefore if applies then for any statement is true. Proven P is antisymmetric.

(iii) Transitive

Look at the above. For if known and then apply one of the following two things, i.e or . Consequently, for any if and then apply . Proved is transitive.

So, P is partial order in S.

**Example :** (more…)

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