The Relationship Between Numbers

We can find relationships between numbers using properties of numbers such as divisibility or using the basics of number relationships such as odd or even numbers. But mathematically, when two collections of numbers are given and if properties of numbers or basics of number relationships don’t find any connection between them, we can lean on sets and relations. Let us first understand what we mean by sets and relations and then how to find the relationship between numbers using sets.

Sets are a collection of organised objects that can be represented in Roster or Set-builder form, for example, A = {1,2,3}. Here, A is a set that contains the organised elements 1, 2, and 3. It is usually represented with a capital letter. When an element is present in a set, we can represent it with the symbol ∊, which means ‘to belong’. 

How to find the relationship between numbers?

Consider, A = {2,5,3,7,11,19,17}. To find a relation between the numbers in this set of A using basic properties of numbers, we can observe that all these numbers fulfill the following conditions:

1. Less than 20
2. Prime numbers
3. Multiple of 1

Now, consider two sets, A and B, and let us try to find the relationship between them using relations. Let x represent the elements of A and y of that of B. The relation between the two sets is that x=2y. 

A = { 2, -2, -3, 4}

B = {4, -4, -6, 8}

So, we can represent this relation as, R = { (2,4), (-2,-4), (-3,-6), (4,8) }, where R is the relation between A and B. 

Types of relations in mathematics

There are mainly seven types of relations in mathematics. They are as follows:

• Empty Relation

When there is no relation between the elements of a set, it is called an empty relation. It can be represented by 

R = ф ∊ A x A, where ф = NULL 

• Universal Relation

When every element of a set is related to each other, that set has a universal relation. For example, 

 A = {1,2}

R = { (1,2), (2,1) }

• Trivial Relation 

Trivial relation is a combination of an empty and universal relation. 

• Reflexive Relation 

When every element relates to itself, it is called a reflexive relation. For example,

 A = {1,2}

R = { (2,2), (1,2), (2,1), (1,1) }

• Symmetric Relation 

In a set A, where (a, b) is an element in R and (b, a) is also an element in R, then relation R on set A is a symmetric relation. For example, Set A = {1,3} and R = {(1,3), (3, 1)} then R is a symmetric relation. 

• Transitive Relation 

In a set A, where (a, b), (b,c) and (c,a) are elements of A and (a, b), (b,c) and (c,a) ∊ R, then this relationship is called a transitive relation.

• Equivalence Relation 

When relation R in set A is reflexive, symmetric, and transitive, then relation R is an equivalence relation. 

Types of relations in discrete mathematics 

Types of relations in discrete mathematics refer to all kinds of relations that we have discussed in the previous section and more. Therefore, the following are some other types of relations in discrete mathematics with which we can find the relationship between discrete numbers. There are mainly five types of relations in discrete mathematics. They are as follows:

• Complement Relation 

A relation R in set A will be considered a complement relation if the relation contains pairs of elements that belong to the cartesian product and not the relation. 

For example, if set A = {5, 6}, set B = {7, 8} and Relation R = {(5, 7), (6, 8)}, then the complement of relation R will be {(5, 8), (6, 7)}. 

• Inverse of Relation 

If set A = {(1, 3), (3, 4)}, then inverse relation R, which will be denoted as R-1, will be {(3, 1), (4, 3)}. Hence, in inverse relation, a relationship is obtained from interchanging the first and second elements of each pair in the set. 

• Composite Relation 

Suppose there are three sets A, B, and C. Let R be the relation of sets A and B, and S be the relation of sets B and C. Since R is considered a subset of A and B, and S is considered a subset of relations B and C, the composition of R and S will be RS, a composite relation. 

• Asymmetric Relation 

Let there be a set, A. A relation a R on set A will be considered an asymmetric relation if for every (a, b) R, (b, a) R. 

Let’s take A = {(2, 3)} and R = {(2,2), (3,3), (3,2)} as an example. Here. R is an asymmetric relation because (3, 2) belongs to R but (2, 3) does not. 

• Irreflexive Relation 

In an irreflexive relation, no element is connected to itself. Therefore, for every a A, when (a, a) R, then relation R is irreflexive.

Let’s take A = {1, 4, 6} and R = {(4, 6), (4, 1), (1, 4), (6, 1), (1, 6)} as an example. 

Relation R is an irreflexive relation because for every a A, (a, a) R. It is because (1, 1), (4, 4), and (6, 6) do not belong to the relation. 

Conclusion 

A well-defined collection of objects, in mathematics, is called a set. As the name suggests, relation refers to the correspondence or relationship between two sets of numbers. There are various types of relationships between numbers that can be formed. Other than relations, there are different relations used in discrete mathematics as well. Thus, relations are how to find the relationship between numbers which we may not find using the basics of number relationships and properties of numbers.