## Introduction to Discrete Mathematics and Partially Ordered Sets

Discrete mathematics is a branch of mathematics that deals with objects that can only take on distinct, separate values. It is a fundamental area of study in mathematics and has applications in computer science, cryptography, and other fields. One important concept in discrete mathematics is that of partially ordered sets.

## What is a Partially Ordered Set?

A partially ordered set, also known as a poset, is a set equipped with a binary relation that is reflexive, antisymmetric, and transitive. In simpler terms, it is a set of elements with a relation that establishes a partial order among them.

Let’s break down the three properties of a partially ordered set:

## Reflexivity

A relation is reflexive if every element of the set is related to itself. In a partially ordered set, this means that every element is related to itself. For example, consider the set of natural numbers {1, 2, 3, 4, …} with the relation “less than or equal to” (≤). In this case, every natural number is less than or equal to itself, so the relation is reflexive.

## Antisymmetry

A relation is antisymmetric if, for every pair of distinct elements, if one element is related to the other, then the other element is not related to the first. In a partially ordered set, this means that if two elements are related, they must be equal. For example, consider the set {a, b, c} with the relation “is a subset of” (⊆). If set A is a subset of set B and set B is a subset of set A, then set A and set B must be equal. Therefore, the relation is antisymmetric.

## Transitivity

A relation is transitive if, whenever element A is related to element B and element B is related to element C, then element A is also related to element C. In a partially ordered set, this means that if one element is related to another and the second element is related to a third element, then the first element must be related to the third element. For example, consider the set {1, 2, 3} with the relation “divides” (|). If 1 divides 2 and 2 divides 3, then 1 must divide 3. Therefore, the relation is transitive.

## Examples of Partially Ordered Sets

Now that we have a better understanding of partially ordered sets, let’s explore some examples:

## Example 1: The Set of Integers with the Relation “Less Than or Equal To”

Consider the set of integers {…, -3, -2, -1, 0, 1, 2, 3, …} with the relation “less than or equal to” (≤). In this case, the relation is reflexive because every integer is less than or equal to itself. It is also antisymmetric because if one integer is less than or equal to another, they must be equal. Finally, it is transitive because if one integer is less than or equal to another and the second integer is less than or equal to a third integer, then the first integer must be less than or equal to the third integer. Therefore, the set of integers with the relation “less than or equal to” forms a partially ordered set.

## Example 2: The Power Set of a Set with the Relation “Is a Subset Of”

Let’s consider a set A = {1, 2}. The power set of A is the set of all possible subsets of A, including the empty set and the set itself. The power set of A is {{}, {1}, {2}, {1, 2}}. If we define the relation “is a subset of” (⊆) on the power set of A, we can see that it forms a partially ordered set. The relation is reflexive because every subset is a subset of itself. It is antisymmetric because if one subset is a subset of another, they must be equal. It is also transitive because if one subset is a subset of another and the second subset is a subset of a third subset, then the first subset must be a subset of the third subset.

## Example 3: The Set of Real Numbers with the Relation “Divides”

Consider the set of real numbers with the relation “divides” (|). In this case, the relation is not reflexive because a real number does not divide itself unless it is 1 or -1. It is antisymmetric because if one real number divides another, they must be equal. However, it is transitive because if one real number divides another and the second real number divides a third real number, then the first real number must divide the third real number. Therefore, the set of real numbers with the relation “divides” does not form a partially ordered set.

## Conclusion

Partially ordered sets are a fundamental concept in discrete mathematics. They provide a way to establish a partial order among a set of elements using a binary relation. Understanding the properties of reflexivity, antisymmetry, and transitivity is crucial in determining whether a set with a given relation forms a partially ordered set. Through examples, we have seen how different sets with different relations can either form or not form partially ordered sets. Partially ordered sets have applications in various fields and are essential in understanding the foundations of discrete mathematics.