## Smallest/Largest Element – Lower/Upper Bound In Set Theory

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

In the previous post we’ve learned about partial ordered set (poset), ordered set, minimum element, and maximum element. In the previous post, I said that the unity existence of the maximum elements and minimal elements are also not guaranteed. You can reopen the post on this link. This is done in order to avoid chaos in reading the this post. Ok, let’s start the discussion.

**Definition :**

Given any poset .

(i) said the smallest element (minimum, first) of if for all .

(ii) said the largest element (maximum, last) of if for all .

**Example :**

I will take the same example as yesterday.

In the example above, has the smallest element as well as the largest element . For they do not have the smallest element and the largest element.

As another example, poset has the smallest element and has the largest element . (more…)

## Ordered Set

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

Last post we discuss about poset or partially ordered set. As per my promise, today we will learn about the ordered set. Also will be discuss about minimal element and maximal element. Happy reading.

**Definition :**

Given any set . A subset is said to be order in set if for any apply :

(i) (irreflexive)

(ii) if and then (transitive)

**Example :**

is an order in the set .

Furthermore any order is denoted by . Just like a poset, it says an ordered set if is an order in the set .

**Theorem :**

If is an ordered set then for any it’s valid statement that then .

**Proof :**

Given . Suppose . Since transitive then . Contradictions with are irreflexive.

**Theorem :**

Given any poset . If for any , defined :

then is an ordered set.

**Proof :** (more…)

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