# What Are Relations in Set Theory?

I am a PhD student of mathematics. I have complete MS in math from the University of Pakistan and have been writing online since 2020.

## What Are Relations in Set Theory?

Mathematical relations offer a way of establishing a link between any two items or things. The relationship between two items is described by a relation, which is often written as an ordered pair (input, output) or (x, y). The relation explains how two sets are interrelated. When two sets are given, we use relations to see if there is a relationship between them. For example, an empty relation means that there are no identical components in the two sets.

Many relations, like "less than," "is parallel to," "is a subset of," and others, are well known to the observer. These relations, in a sense, take the existence or absence of a particular relationship between pairs of things taken in a specific order into consideration. We define a relation in terms of these "ordered pairs" formally.

## Definition of Ordered Pair

If x and y are two elements, then the ordered pair of x and y are defined as (x, y), where x is listed as the first element, and y is listed as the second element.

Specifically, (x, y) = (k, l)

If and only if x = k and y = l. Thus (x, y) is not equal to (x, y) unless x = y. In contrast, there are sets where the order of the elements doesn't matter, like {3, 5} = {5, 3}.

## Definition of Product Sets

Consider any two sets K and L. The product, or Cartesian product, of K and L is the collection of all ordered pairs (x, y) where x ∈ K and y ∈ L. This product's symbol is K × L, which may be read as "K cross L."

Accordingly, K × L = {(x, y) | x ∈ K and y ∈ L}

### Example

Let K = {2, 1} and L = {a, b, c}. Then

K × L = {(2, a), (2, b), (2, c), (1, a), (1, b), (1, c)}

L × K = {(a, 2), (b, 2), (c, 2), (a, 1), (b, 1), (c, 1)}

Also, K × K = {(2, 2), (2, 1), (1, 2), (1, 1)}

## Relation Definition

Let K and L be sets. A binary relation or, simply, relation from K to L is a subset of K × L.

Suppose R is a relation from K to L. Then R is a set of ordered pairs where each ﬁrst element comes from K and each second element comes from L. That is, for each pair x ∈ K and y ∈ L, exactly one of the following is true:

(i) (x, y) ∈ R; we then say, “x is R-related to y”, written x R y.

(ii) (x, y) is not belong to R; we then say “x is not R-related to y”

Please Note: If a relation R is defined from a set K to itself, then we say that R is a relation on K.

K^2 = K× K

## Domain and Range of Sets

The first elements of a relation are called Domain, and the second elements of a relation is called the range of relation.

### Example

Consider the relation {(1 ,0), (3, 9), (6, 8), (4, k), (j, 9)}

Domain is = {1, 3, 6, 4, j}.

Range is = {0, 9, 8, k, 9}

## Types of Relations

There are many types of relations. Here we will discuss four of the most important types of relations.

## 1. Reﬂexive Relation

If a relation of the elements of sets is to itself, then the relation is said to be a reflexive relation.

### Example

If we consider the element x of a set K the reflexive relation is represented as, (x, x) K.

Let K = {(1, 5)} then relation R is defined on set K as, K × K {(1, 1), (1, 5), (5, 1), (5, 5)}.

## 2. Symmetric Relation

A relation R is said to be symmetric if there is a relation of one element to another element.

### Example

If we consider (x, y) ∈ K and (y, x) ∈ K then the symmetric relation is represented as, if xRy then yRx.

If we consider a set K = {(5, 1)} then symmetric relation of K is {(5, 1), (1, 5)}

## 3. Transitive Relation

A relation is said to be transitive if xRy, yRz and zRx for (x, y) ∈ K, (y, z) ∈ K and (z, x) ∈ K.

### Example

R= {(a, b), (b, c), (a, d)} on set A= {a, b, c} is transitive.

## 4. Equivalence Relation

A relation is said to be equivalent if the relation is reflexive, symmetric and transitive.

### Example

R= {(a, a), (b,b), (c,c), (a,b), (b,a), (b,c), (c,b), (a,c), (c,a)} on set K={a,b,c} is an equivalence relation since it is reflexive, symmetric, and transitive.

## Types of Relations in Set Theory

Relation CategoryFunctionality

Reﬂexive Relations

(x, x) ∈ K

Empty Relation

R = φ ⊂ K× K

Symmetric Relation

(x, y) ∈ K and (y, x) ∈ K ,if xRy then yRx.

Transitive Relation

if xRy, yRz and zRx for (x, y) ∈ K, (y, z) ∈ K and (z, x) ∈ K.

Equivalence Relation

Reflexive+Symmetric+Transitive