Understanding the Divisibility Relation in Natural Numbers: Proving Partial Order

In the realm of mathematics, particularly in the theory of relations, the concept of a partial ordering is pivotal. A relation R on a set is a partial order if it satisfies three fundamental properties: reflexivity, antisymmetry, and transitivity. This article will delve into the specific case of the divisibility relation over the set of natural numbers, ?, where xRy means x divides y. We will rigorously prove the three properties that define a partial order to solidify our understanding.

Defining the Divisibility Relation

The divisibility relation on the natural numbers is defined as xy, meaning that x divides y. This notation is a shorthand for stating that there exists an integer k such that yk. Formally, for any two natural numbers x and y, xRy if and only if there exists a natural number k (possibly k 1) such that yk x.

Proof of Reflexivity

To prove the reflexivity property, we must show that for every natural number x, x divides x. That is, for all x in ?, xRxx. This is straightforward due to the following reasoning: If x and x are both natural numbers, then we can write x 1 * x, where 1 is the natural number 1. Therefore, xRx is true because we have expressed x as the product of 1 and x, which is a natural number. Thus, the reflexive property is satisfied:

For all n in ?, we have nn.

This is further confirmed by the fact that 1n/n 1, which is a natural number, ensuring the reflexive property holds.

Proof of Antisymmetry

Next, we prove the antisymmetry property. For this, we need to show that if x divides y and y divides x, then x y. This can be mathematically written as:

For all n, m in ?, if nm and mn, then n m.

Let's start by assuming that nm and mn. By definition of divisibility, there exist natural numbers k and l such that: y kx x ly

Substituting x in the first equation, we get:

y k(ly) (kl)y

To satisfy the equality y y, it must be that:

kl 1

Since x and y are natural numbers, the only way this can hold is if k 1 and l 1. Therefore, x y, satisfying the antisymmetry condition.

Proof of Transitivity

Finally, we prove the transitive property. This requires showing that if x divides y and y divides z, then x divides z. That is, for all n, m, l in ?, if ny and myl, then nxl. Formally, this can be written as:

For all n, m, l in ?, if ny and myl, then nxl

Let's assume ny and myl. By the property of divisibility, there exist natural numbers k and l such that:

Substituting y kx into the second equation:

z l(kx) (lk)z

Therefore, we can express z as a product of lk and x. Since lk is a natural number, it follows that x divides z, satisfying the transitive condition.

Conclusion

The divisibility relation over the natural numbers, defined by xy, is a partial order because it satisfies the three fundamental properties: reflexivity, antisymmetry, and transitivity. This means, for any natural number x, x divides x; if x divides y and y divides x, then x y; and if x divides y and y divides z, then x divides z. Understanding these properties is crucial for comprehending the structure of the divisibility relation and its role in discrete mathematics and number theory.