Relations in Mathematics

Relation in Mathematics is defined as the relationship between two sets. If we are given two sets set A and set B and set A has a relation with set B then each value of set A is related to a value of set B through some unique relation. Here, set A is called the domain of the relation, and set B is called the range of the relation.

For example, if we are given two sets,

• Set A = {1, 2, 3, 4} and Set B = {1, 4, 9, 16}

The relation R from set A to set B is defined by the rule: y = x²: y ϵ B, x ϵ A}

Ordered Pairs:

• R = {(1, 1), (2, 4), (3, 9), (4, 16)}

Relation is defined as the relation between two different sets of information. Suppose we are given two sets containing two different values then a relation defined such that it connects the value of the first set with the value of the second set is called the relation.

Suppose we are given a set A that contains the name of girls of a class and another set B that includes the height of girls then a relation connects set A with set B. In mathematical terms, we can say that,

"A set of ordered pairs is a relation"

Example of a Relation

Examples of relation in mathematics include,

Suppose there are two sets X = {4, 36, 49, 50} and Y = {1, -2, -6, -7, 7, 6, 2}. A relation R states that

"(x, y) is in the relation R if x is a square of y" can be represented using ordered pairs,

• R = {(4, -2), (4, 2), (36, -6), (36, 6), (49, -7), (49, 7)}

Also, the image added below shows two sets A and B, and the relation between them,

• Set A = {x, y, z}
• Set B = {1, 2, 3}

Representation of Relations

In mathematics or set theory we can represent the relation using different techniques and the two important ways to represent the set are,

• Set Builder Notation
• Roaster Notation

Let's study them in detail in the article below,

Set Builder Notation

if are relation between two sets is represented using the logical formula then this type of representation is called the set builder notation.

For example, if we are given two sets set X = {2, 4, 6} and set Y = {4, 8, 12}. Then after observing clearly, we can see that each element of set Y is twice each element of set X the relation between them is,

• R {(a, b): b is twice of a, a ∈ X, b ∈ Y}

Roaster Form

Roaster form is another way of representing a relation. In roaster form, we use ordered pairs to represent the relation.

For example, if we are given two sets set X = {2, 4, 6} and set Y = {4, 8, 12}. Then the relation between set X and set Y is represented using the relation R such that,

• R = {(2, 4), (4, 8), (6, 12)}

Types of Relation

Various types of relations defined in mathematics are,

• Empty Relation
• Reflexive Relation
• Symmetric Relation
• Transitive Relation
• Equivalence Relation
• Universal Relation
• Identity Relation
• Inverse Relation

Now let's learn about them in detail.

Empty Relation

A relation R on a set A is called Empty if the set A is an empty set, i.e. any relation where no element of set A is not related to the element of set B then it is called an empty relation. For example, A = {1, 2, 3} and B = {5, 6, 7} where, R = {(x, y) where x + y = 22}, then it is an empty relation.

Reflexive Relation

A relation R on a set A is called reflexive if (a, a) ∈ R holds for every element a∈ A . i.e. if set A = {a, b} then R = {(a, a), (b, b)} is reflexive relation.

For example, A = {2, 3} then the reflexive relation R on A is,

• R = {(2, 2), (3, 3)}

Symmetric Relation

A relation R on a set A is called symmetric if (b, a) ∈ R holds when (a, b) ∈ R i.e. The relation R = {(a, b), (b, a)} is a reflexive relation on (a, b)

For example, A = {2, 3} then symmetric relation R on A is,

• R = {(2, 3), (3, 2)}

Transitive Relation

A relation R on a set A is called transitive if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R for all a,b,c ∈  A i.e.

For example, set A = {1, 2, 3} then transitive relation R on A is,

• R = {(1, 2), (2, 3), (1, 3)}

Equivalence Relation

A relation is an Equivalence Relation if it is reflexive, symmetric, and transitive. i.e. relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2), (1, 3), (3, 1)} on set A = {1, 2, 3} is equivalence relation as it is reflexive, symmetric, and transitive.

Universal Relation

A universal relation is a relation in which all elements of the set are mapped to another element of the set then it is called universal relation. For example, A = {4, 8, 12} and B = {1, 2, 3} then universal relation is, R = {(x, y) where x > y}

Identity Relation

An identity relation is a relation defined such all elements in a set are related to itself. It is defined as, I = {(x, x): for all x ∈ X}.

For example P = {1, 2, 3} then Identity Relation(I) = {(1, 1), (2, 2), (3, 3)}

Inverse Relation

A relation is called the inverse of any relation if elements of one set are inverse pairs of another set. The inverse of a relation R is denoted as R^-1. i.e., R^-1 = {(y, x): (x, y) ∈ R}.

Graphing Relations

Relations can be easily represented on the graphs and representing them on graphs is an easy way of explaining them. The ordered pair in a relation represents a coordinate that can be plotted on the cartesian coordinate system. We can easily graph the relation by following the steps added below,

• Substitute x with random numerical values in the relation.
• Find the corresponding y value of the respective x value.
• Write the ordered pair such that, {(x, y)}
• Plot these points and join them to find the required curve.

The graph of the relation y = x² is added below,

Articles related to Relations in Mathematics:

• Types Of Sets
• Relation and Function
• Types of Functions
• Representation of Relation in Graphs and Matrices
• Relations in Mathematics

Solved Examples of Relations

Question 1: Find the inverse relation of R = {(1, 3), (2, 4), (3, 5)}

Solution:

Inverse relation is defined as R^-1 = {(y, x): (x, y) ∈ R}

Given,
R = {(1, 3), (2, 4), (3, 5)}

Inverse relation of R is R^-1 then,
R^-1 = {(3, 1), (4, 2), (5, 3)}

Question 2: Find the inverse relation of R = {(a, x), (b, y), (c, z)}

Solution:

Inverse relation is defined as R^-1 = {(y, x): (x, y) ∈ R}

Given,
R = {(a, x), (b, y), (c, z)}

Inverse relation of R is R-1 then,
R^-1 = {(x, a), (y, b), (z, c)}

Question 3: Is the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)} on set A = {1, 2, 3} an equivalence relation?

Solution:

No, it's not an equivalence relation. It's reflexive and symmetric, but not transitive.

Question 4: Find the domain and range of the relation R = {(1, 2), (2, 3), (3, 4), (4, 5)}.

Solution:

Domain = {1, 2, 3, 4}, Range = {2, 3, 4, 5}

Question 5: Is the relation R = {(x, y) | x² + y² = 25} on the set of real numbers a function?

Solution:

No, it's not a function. For example, (0, 5) and (0, -5) are both in the relation.

Question 6: Find the inverse of the relation R = {(1,1), (2,4), (3,9), (4,16)}.

Solution:

R⁻¹ = {(1, 1), (4, 2), (9, 3), (16, 4)}

Question 7: Determine if the relation R = {(x, y) | x - y is even} on the set of integers is an equivalence relation.

Solution:

Yes, it's an equivalence relation. It's reflexive, symmetric, and transitive.

Question 8: Find the composition R ∘ S for R = {(1, 2), (2, 3), (3, 4)} and S = {(1, 1), (2, 2), (3, 3), (4, 4)}.

Solution:

R ∘ S = {(1, 2), (2, 3), (3, 4)}

Question 9: Is the relation R = {(a, b) | a is a multiple of b} on the set of positive integers transitive?

Solution:

Yes, it's transitive.

Question 10: Find the reflexive closure of R = {(1, 2), (2, 3), (3, 1)} on set A = {1, 2, 3}.

Solution:

Reflexive closure = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (3, 1)}