Exploring the Correct Method in Determining Relations Between Integers

When it comes to defining and analyzing mathematical relations, clarity and precision are paramount. This article delves into the specific relation where one integer is a multiple of another, and provides a rigorous examination of the properties of reflexivity, symmetry, antisymmetry, and transitivity.

Introduction

Consider the relation defined as follows:

xy Leftrightarrow exists k in mathbb{Z} : y k cdot x

This definition asserts that integer y is a multiple of integer x. This relation is central to many areas of mathematics and computer science, underpinning countless algorithms and theoretical frameworks. Our focus will be on exploring the properties of this relation: specifically, its reflexivity, symmetry, antisymmetry, and transitivity.

Reflexivity

First, we address the property of reflexivity: a relation is reflexive if for every integer x, we have xx. Given our definition:

xx Leftrightarrow exists k in mathbb{Z} : x k cdot x

Clearly, we can choose k 1, since x 1 cdot x. Therefore, xx holds for all x in mathbb{Z}. Hence, the relation is reflexive.

Symmetry

To investigate the symmetry property, we start by assuming xy. This means:

exists k in mathbb{Z} : y k cdot x

We need to determine if yx also holds. This would mean:

exists l in mathbb{Z} : x l cdot y

However, from the initial condition, we have y kx. Substituting this into the second condition:

x l cdot (kx) x (lk) cdot x

This implies:

lk 1

The only integer solutions to this equation are l 1 and k 1. This means that if xy, then we must have yx. Hence, the relation is symmetric.

Antisymmetry

The property of antisymmetry states that if both xy and yx hold, then x y. According to our definition:

xy Leftrightarrow exists k in mathbb{Z} : y k cdot x yx Leftrightarrow exists l in mathbb{Z} : x l cdot y

From xy, we have y kx. Substituting y kx into yx, we get:

x l (kx) x (lk) cdot x

This implies:

lk 1

Thus, the only integer solutions satisfy l 1 and k 1. Therefore, if both xy and yx hold, then x y. Hence, the relation is antisymmetric.

Transitivity

Finally, we consider the property of transitivity: if xy and yz, then xz. Given our definition:

xy Leftrightarrow exists k in mathbb{Z} : y k cdot x yz Leftrightarrow exists l in mathbb{Z} : z l cdot y

We need to show that xz holds. From xy, we have y kx. Substituting this into yz, we get:

z l (kx) z (lk) cdot x

This implies:

z m cdot x exists m in mathbb{Z}

Therefore, xz holds. Hence, the relation is transitive.

Conclusion

By examining the properties of the relation where one integer is a multiple of another, we have determined that it is reflexive, symmetric, antisymmetric, and transitive. These properties are crucial for understanding the behavior and implications of the relation in various mathematical contexts.

References

For further reading, refer to the following resources:

Axler, S. (2015). Linear Algebra Done Right. Springer. Fraleigh, J. B. (2023). A First Course in Abstract Algebra. Pearson.