Relations in mathematics becomes easier when we look at real-life examples. A relation, in simple terms, is a set of ordered pairs that show how elements from one set are connected to elements of another set. These connections are all around us in everyday situations.

For instance, think about the relationship between students and their grades in a school. Each student (element of one set) is associated with a grade (element of another set). Similarly, the cost of fuel at a gas station is another example. The cost is related to the amount of fuel; as the quantity of fuel increases, so does the cost.

What is Relation in Maths?

In mathematics, a relation is a connection between sets of values. Specifically, it refers to a set of ordered pairs, where the first elements of the pairs come from one set (called the domain) and the second elements come from another set (called the range). Relations are used to show how elements from these sets are associated with each other.

Relation Definition

A relation R from a set A to a set B is a subset of the Cartesian product A × B. That is, R ⊆ A × B, where each element in R is an ordered pair (a, b) with a ∈ A and b ∈ B.

Real-Life Examples of Relations

Here are several instances where relations are evident in everyday life:

Student and Grades: The relationship between students and their grades is a common example. Each student (input) is linked to a specific grade (output). For instance, in a class, we might have a relation like {(Alice, A), (Bob, B), (Charlie, A)}.

Temperature and Time: The temperature at different times of the day can be considered a relation. For example, the relation can be expressed as {(8 AM, 70°F), (12 PM, 85°F), (6 PM, 75°F)}, showing how the temperature varies over time​​.

Cost of Fuel: The cost of filling a car’s fuel tank is related to the amount of fuel purchased. This can be represented as {(10 liters, $20), (20 liters, $40)}, where the cost is directly proportional to the amount of fuel​​.

Family Relationships: Family relationships such as parent-child or siblings are also relations. For instance, {(John, Mike), (Sarah, Anna)} where each pair shows the parent-child relationship.

Library Book Lending: The relationship between books and borrowers in a library system. Each book (input) can be related to one or more borrowers (output). For example, {(Book 1, Borrower1), (Book 2, Borrower2), (Book 3, Borrower1)}.

Sports Teams and Players: The relation between sports teams and their players. For example, in a soccer league, we might have a relation like {(Team A, Player 1), (Team B, Player 2), (Team A, Player 3)}.

Shopping: The relationship between items and their prices in a store. Each item (input) is associated with a specific price (output). For example, {(Milk, $2.50), (Bread, $1.50), (Eggs, $3.00)}.

Read More,

• Relation and Function
• Difference between Relation and Function
• Domain and Range of a Relation
• Composite Function

FAQ: Relations

What is a relation in mathematics?

A relation in mathematics is a set of ordered pairs where each element from the first set (called the domain) is paired with one or more elements from the second set (called the range). This pairing shows how elements from these two sets are connected.

What are some examples of relations in real life?

Some real-life examples of relations include:

• Student and Grades: Each student is associated with their respective grades.
• Temperature and Time: The temperature varies with the time of day.
• Cost of Fuel: The cost changes depending on the amount of fuel purchased.
• Family Relationships: Parent-child or sibling relationships.
• Library Book Lending: Books are related to the borrowers.

How do you determine if a relation is a function?

To determine if a relation is a function, each element in the domain must be associated with exactly one element in the range. If any element in the domain is associated with more than one element in the range, the relation is not a function.

What is the difference between a relation and a function?

While all functions are relations, not all relations are functions. A function is a special type of relation where each element in the domain is paired with exactly one element in the range. In a relation, an element in the domain can be paired with multiple elements in the range.

What are some types of relations?

Some types of relations include:

• Reflexive Relation: Every element is related to itself.
• Symmetric Relation: If (a,b) is in the relation, then (b,a) is also in the relation.
• Transitive Relation: If (a,b) and (b,c) are in the relation, then (a,c) is also in the relation.
• Equivalence Relation: A relation that is reflexive, symmetric, and transitive

What are some practical applications of relations?

Relations are used in various fields such as:

• Database Management: To link tables and retrieve information.
• Economics: To show the relationship between variables like supply and demand.
• Computer Science: In algorithms and data structures.
• Social Networks: To represent connections between people.