A relation shows how two things or people are connected — like a father and son, or a brother and sister. In mathematics, a relation is used to describe how elements of one set are connected to elements of another set (or the same set). This idea comes from set theory, which is the part of math that deals with sets — or collections of objects, like numbers or letters.

Set theory helps us understand how sets work, how we can combine or compare them, and how big they are. It is the base for many other areas in math, such as algebra and probability. Relations are one of the key parts of set theory. They help us show connections between items in a clear way using ordered pairs, like (a, b), meaning "a is related to b." These relations can be of many types and follow different rules.

What are Relations?

In mathematics, relations and functions help us understand how elements from one set are connected to elements of another set. Think of a relation as a rule that links items from Set X (called the domain) to items in Set Y (called the range). These connections are written as ordered pairs like (x, y), where “x” is the input and “y” is the output.

A function is a special kind of relation. In a function, each input is linked to only one output. So, if you give the same input twice, you’ll always get the same result.

Relations come from something called the Cartesian product, which is the set of all possible pairs from two sets. For example, if X = {1, 2} and Y = {a, b}, then the Cartesian product X × Y = {(1, a), (1, b), (2, a), (2, b)}.

What are Properties of Relations?

The properties of relations are given below:

1. Identity Relation
2. Empty Relation
3. Reflexive Relation
4. Irreflexive Relation
5. Inverse Relation
6. Symmetric Relation
7. Transitive Relation
8. Equivalence Relation
9. Universal Relation

Identity Relation

Each element only maps to itself in an identity relationship. The identity relation rule is shown below.

“Each element will only have one relationship with itself,”. If R signifies an identity connection, and R symbolizes the relation stated on Set A, then, then

\( R=\text{ }\{\left( a,\text{ }a \right)/\text{ }for\text{ }all\text{ }a\in A\} \)

That is to say, each member of A must only be connected to itself. If R contains an ordered list (a, b), therefore R is indeed not identity. So, because the set of points (a, b) does not meet the identity relation condition stated above. For example, \( P=\left\{5,\ 9,\ 11\right\} \) then \( I=\left\{\left(5,\ 5\right),\ \left(9,9\right),\ \left(11,\ 11\right)\right\} \)

Empty Relation

An empty relation is one where no element of a set is mapped to another set’s element or to itself. It is denoted as \( R=\varnothing \), Let’s consider an example, \( P=\left\{7,\ 9,\ 11\right\} \) and the relation on \( P,\ R=\left\{\left(x,\ y\right)\ where\ x+y=96\right\} \) Because no two elements of P sum up to 96, it would be an empty relation, i.e. R is an empty set, \( R=\varnothing \)

Reflexive Relation

Every element in a reflexive relation maps back to itself. The reflexive relation rule is listed below. “Every element has a relationship with itself”. It sounds similar to identity relation, but it varies. Identity relation maps an element of a set ‘only’ to itself whereas a reflexive relation maps an element to itself and possibly other elements. Consider the relation R, which is specified on the set A. If R denotes a reflexive relationship,

\( R=\text{ }\{\left( a,\text{ }a \right)/\text{ }for\text{ }all\text{ }a\in A\} \)

That is, each element of A must have a relationship with itself. For example, if \( x\in X \) then this reflexive relation is defined by \( \left(x,\ x\right)\in R \), if \( P=\left\{8,\ 9\right\} \) then \( R=\left\{\left\{8,\ 9\right\},\ \left\{9,\ 9\right\}\right\} \) is the reflexive relation

Irreflexive Relation

If for a relation R defined on A. \({\left(x,\ x\right)\notin R\right\}\) for each and every element x in A, the relation R on set A is considered irreflexive.

The cartesian product of a set of N elements with itself contains N pairs of (x, x) that must not be used in an irreflexive relationship.

For instance, if set \( A=\left\{2,\ 4\right\} \) then \( R=\left\{\left\{2,\ 4\right\}\left\{4,\ 2\right\}\right\} \) is irreflexive relation

Inverse Relation

An inverse relation of any given relation R is the set of ordered pairs of elements obtained by interchanging the first and second element in the ordered pair connection exists when the members with one set are indeed the inverse pair of the elements of another set. In other words, a relation’s inverse is also a relation. The inverse of a Relation R is denoted as \( R^{-1} \)

Each ordered pair of R has a first element that is equal to the second element of the corresponding ordered pair of\( R^{-1}\) and a second element that is equal to the first element of the same ordered pair of\( R^{-1}\).

Symmetric Relation

A relation R on a set or from a set to another set is said to be symmetric if, for any\( \left(x,\ y\right)\in R \), \( \left(y,\ x\right)\in R \). For instance, let us assume \( P=\left\{1,\ 2\right\} \), then its symmetric relation is said to be \( R=\left\{\left(1,\ 2\right),\ \left(2,\ 1\right)\right\} \)

Transitive Relation

Binary relationships on a set called “transitive relations” require that if the first element is connected to the second element and the second element is related to the third element, then the first element must also be related to the third element. Let’s have a look at set A, which is shown below. \( A=\left\{x,\ y,\ z\right\} \)

Assume R is a transitive relation on the set A. Then \( R=\left\{\left(x,\ y\right),\ \left(y,\ z\right),\ \left(x,\ z\right)\right\} \)v, That instance, if “x” is connected to “y” and “y” is connected to “z,” “x” must be connected to “z.”

For example,P ={a,b,c} , the relation R={(a,b),(b,c),(a,c)}, here a,b,c ∈ P

Equivalence Relation

Consider the relation R, which is defined on set A. R is an equivalence relation if the relation R is reflexive, symmetric, and transitive. Thus, to check for equivalence, we must see if the relation is reflexive, symmetric, and transitive.

Universal Relation

A universal relation is one in which all of the elements from one set were related to all of the elements of some other set or to themselves. \( R=X\times Y \) denotes a universal relation as each element of X is connected to each and every element of Y. For example, let \( P=\left\{1,\ 2,\ 3\right\},\ Q=\left\{4,\ 5,\ 6\right\}\ and\ R=\left\{\left(x,\ y\right)\ where\ x<y\right\} \)

Application of Properties of Relations

• Relations are two given sets’ subsets. For instance, R of A and B is demonstrated. This is an illustration of a full relation. Now, there are a number of applications of set relations specifically or even set theory generally:
• Sets and set relations can be used to describe languages (such as compiler grammar or a universal Turing computer).
• To keep track of node visits, graph traversal needs sets.
• Set-based data structures are a given. In terms of table operations, relational databases are completely based on set theory.
• Some of the notable applications include relational management systems, functional analysis etc.

Also, learn about the Difference Between Relation and Function.

Summary and Review of Properties of Relations

A relation on a set A means that the elements of the set are being related to each other in some way. This relation can have some important properties, which help us understand how the elements are connected.

Here are the main five properties:

1. Reflexive Relation
A relation is reflexive if every element in the set is related to itself.
Example: If A = {1, 2, 3}, then (1,1), (2,2), and (3,3) must all be in the relation for it to be reflexive.

2. Irreflexive Relation
A relation is irreflexive if no element in the set is related to itself.
In the same example, if (1,1), (2,2), and (3,3) are not in the relation at all, then it is irreflexive. 

• Note: A relation can’t be both reflexive and irreflexive. But it is possible that it’s neither.

3. Symmetric Relation
A relation is symmetric if for every pair (a, b) in the relation, the pair (b, a) is also in the relation.
In short, the relation goes both ways.

4. Antisymmetric Relation
A relation is antisymmetric if for all pairs (a, b) and (b, a) in the relation, it must be true that a = b.
This means: if two different elements are related in both directions, it’s not antisymmetric. So, for antisymmetry, either:

• (a, b) is in R, but (b, a) is not, or
• if both are in R, then a and b must be the same.

5. Transitive Relation
A relation is transitive if whenever an element a is related to b, and b is related to c, then a must be related to c.
This is like a chain: if a → b and b → c, then it must follow that a → c.

Solved Examples of Properties of Relation

A few examples which will help you understand the concept of the above properties of relations.

Example 1: Let \( A=\left\{2,\ 3,\ 4\right\} \) and R be relation defined as set A, \( R=\left\{\left(2,\ 2\right),\ \left(3,\ 3\right),\ \left(4,\ 4\right)\right\} \), Verify R is identity.

Solution:

The elements in the above question are 2,3,4 and the ordered pairs of relation R, we identify the associations.\( \left(2,\ 2\right) \) where 2 is related to 2, and every element of A is related to itself only. Thus, R is identity

Example 2: Check if the relation R is symmetric

Given:
Set A = {2, 3, 4}
Relation R = {(2, 2), (3, 3), (4, 4), (2, 3)}

To check:
A relation is symmetric if for every pair (a, b) in R, the reverse pair (b, a) is also in R.

Solution:
The pairs (2, 2), (3, 3), and (4, 4) are symmetric because the reverse is the same.
But (2, 3) is in R, while the reverse (3, 2) is not in R.

Conclusion:
Since (3, 2) is missing, R is not symmetric.

Example 3:  Check if the relation R is transitive

Given:
Set A = {2, 3, 4}
Relation R = {(2, 2), (3, 3), (4, 4), (2, 3)}

To check:
A relation is transitive if for all (a, b) and (b, c) in R, the pair (a, c) is also in R.

Solution:
The only non-identity pair is (2, 3).
To test transitivity, we would need a pair like (3, x) in R to combine with (2, 3), forming (2, x).
The only pair starting with 3 is (3, 3).
So we check if (2, 3) and (3, 3) imply (2, 3), which is already in R.

Conclusion:
All required conditions for transitivity are satisfied.
So, R is transitive.

Example 4: Analyze the graph to determine the characteristics of the binary relation R.

• Due to the fact that not all set items have loops on the graph, the relation is not reflexive.
• R is also not irreflexive since certain set elements in the digraph have self-loops.
• Since some edges only move in one direction, the relationship is not symmetric.
• Because there are no edges that run in the opposite direction from each other, the relation R is antisymmetric.
• R cannot be transitive.

Example 5: A relation R’s matrix MR defines it on a set A. Find out the relationship’s characteristics.

M_{R}=\begin{bmatrix} 1& 0& 0& 1 \\ 0& 1& 1& 0 \\ 0& 1& 1& 0 \\ 1& 0& 0& 1 \end{bmatrix}

= Given that there are 1s on the main diagonal, the relation R is reflexive.

R cannot be irreflexive because it is reflexive.

The matrix MR and its transpose, MTR, coincide, making the relationship R symmetric.

M_{R}=M_{R}^{T}=\begin{bmatrix} 1& 0& 0& 1 \\0& 1& 1& 0 \\0& 1& 1& 0 \\1& 0& 0& 1 \\\end{bmatrix}

R is a transitive relation. The transitivity property is true for all pairs that overlap.

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