# Introduction

### Ordered sets

#### Totally ordered sets

A totally ordered set is one where the relation is defined on all pairs:

$$\forall a \forall b (a\le b)\lor (b\le a)$$

Note that totality implies reflexivity.

#### Partially ordered sets (poset)

A partially ordered set, or poset, is one where the relation is defined between each element and itself.

$$\forall a (a\le a)$$

That is, every element is related to itself.

These are also called posets.

#### Well-ordering

A well-ordering on a set is a total order on the set where the set contains a minimum number. For example the relation $$\le$$ on the natural numbers is a well-ordering because $$0$$ is the minimum.

The relation $$\le$$ on the integers however is not a well-ordering, as there is no minimum number in the set.

### Intervals

For a totally ordered set we can define a subset as being all elements with a relationship to a number. For example:

$$[a,b]=\{x:a\le x \land x\le b\}$$

This denotes a closed interval. Using the definition above we can also define an open interval:

$$(a,b)=\{x:a< x \land x< b\}$$

### Infinitum and supremum

#### Infinitum

Consider a subset $$S$$ of a partially ordered set $$T$$.

The infinitum of $$S$$ is the greatest element in $$T$$ that is less than or equal to all elements in $$S$$.

For example:

$$\inf [0,1]=0$$

$$\inf (0,1)=0$$

#### Supremum

The supremum is the opposite: the smallest element in $$T$$ which is greater than or equal to all elements in $$S$$.

$$\sup [0,1]=1$$

$$\sup (0,1)=1$$

#### Max and min

If the infinitum of a set $$S$$ is in $$S$$, then the infinimum is the minimum of set $$S$$. Otherwise, the minimum is not defined.

$$\min [0,1]=0$$

$$\min (0,1)$$ isn’t defined.

Similarly:

$$\max [0,1]=1$$

$$\max (0,1)$$ isn’t defined.