Difference Between Relation and Function

Difference between Relation and Function

In mathematics, relations and functions are fundamental concepts that describe how elements from two sets are related to each other. They play a crucial role in various mathematical fields and have wide applications in different areas, including algebra, calculus, discrete mathematics, and computer science.

A relation is a general concept that defines the connection between elements of two sets. It establishes a correspondence between elements of one set (called the domain) and elements of another set (called the codomain). This correspondence may or may not be well-defined and can vary in nature. Relations can be represented in various ways, such as through tables, graphs, or explicit rules.

A function is a specific type of relation that has an additional property: each element in the domain is uniquely mapped to exactly one element in the codomain. This means that for every input in the domain, there exists exactly one output in the codomain. In other words, a function assigns a single, definite value to each element in the domain.

What is a Relation?

In mathematics, a relation is a set of ordered pairs that establish a connection between elements of two sets. It defines how the elements from one set are related to the elements of another set. The term “relation” is quite general and can be used to describe various mathematical concepts that involve relationships between elements.

Formally, let’s say we have two sets, A and B. A relation R from set A to set B is represented as a subset of the Cartesian product of A and B. The Cartesian product of two sets A and B is denoted as A × B and is the set of all possible ordered pairs (a, b) where a is an element of A and b is an element of B.

The relation R is defined as a subset of A × B, meaning it consists of certain selected ordered pairs (a, b) that have a specific relationship between the elements a from A and b from B. This relationship could be any kind of association or condition between the elements, such as equality, inequality, divisibility, similarity, etc.

For example, consider two sets:

A = {1, 2, 3}

B = {4, 5, 6}

Now, a relation R from A to B could be defined as follows:

R = {(1, 4), (2, 5), (3, 6)}

In this case, the relation R establishes a connection between elements in A and B, indicating that 1 is related to 4, 2 is related to 5, and 3 is related to 6.

Relations are a fundamental concept in various areas of mathematics, including algebra, calculus, discrete mathematics, and graph theory. They play a crucial role in defining

What is a Function?

In mathematics, a function is a fundamental concept that describes the relationship between two sets of elements, typically called the domain and the codomain. It is a rule or a process that assigns to each element from the domain exactly one element from the codomain.

A function is denoted by the notation f: A → B, where:

f represents the function name.

• A is the domain of the function, which is the set of all input values for which the function is defined.
• B is the codomain of the function, which is the set of all possible output values.

For each element “x” in the domain A, the function f assigns a unique element “y” in the codomain B. This is typically written as f(x) = y, indicating that applying the function to the input “x” produces the output “y.”

It’s important to note that a function should satisfy the property that each element in the domain is mapped to exactly one element in the codomain. In other words, there should be no ambiguity in the output for a given input.

Functions are used in various mathematical fields and have broad applications in science, engineering, economics, computer science, and many other disciplines. They play a crucial role in modelling and solving problems, analyzing data, and understanding the relationships between different quantities.

What is the Difference between a Relation and a Function?

In mathematics, both relations and functions are concepts that describe how elements from two sets are related. However, there are fundamental differences between the two:

Relation:

A relation is a set of ordered pairs that represent the connections or associations between elements from two different sets. These ordered pairs consist of one element from the first set (called the domain) and one element from the second set (called the codomain). Relations can be represented as sets of ordered pairs, tables, graphs, or mapping diagrams.

Relations can be classified based on their properties:

• Reflexive: If every element in the domain is related to itself.
• Symmetric: If for every ordered pair (a, b) in the relation, the pair (b, a) is also in the relation.
• Transitive: If for every ordered pair (a, b) and (b, c) in the relation, the pair (a, c) is also in the relation.

An example of a relation would be a set of ordered pairs representing the parent-child relationship in a family: {(parent, child)}.

Function:

A function is a special type of relation in which each element in the domain is related to exactly one element in the codomain. In other words, for every input (domain element), there is a unique output (codomain element). Functions can be represented using various methods such as graphs, tables, equations, or mapping diagrams.

Functions are often denoted as f(x), where “x” represents the input (domain element), and “f(x)” represents the corresponding output (codomain element).

For a relation to be a function, it must satisfy the following condition:

Each element in the domain should be related to exactly one element in the codomain.

An example of a function would be a rule that doubles the input value: f(x) = 2x.

In summary, the main difference between a relation and a function lies in the uniqueness of the outputs. A relation can relate elements in any way, possibly allowing multiple outputs for a given input, while a function must have a unique output for each input. Every function is a relation, but not every relation is a function.

Distinguish Between Relation and Function

Below is a tabular column that highlights the key differences between a relation and a function:

In summary, a relation is a general concept that establishes a relationship between two sets, whereas a function is a specific type of relation with the added condition that each input has only one corresponding output. The vertical line test and the uniqueness of inputs in a function are crucial distinctions between the two concepts.

Solved Problems on Relations and Functions

Let’s go through some solved problems on relations and functions. These problems cover different aspects of relations and functions to help you better understand the concepts.

Problem 1:

Let’s define two sets, A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, and a relation R from set A to set B as follows: R = {(1, 4), (2, 3), (3, 5), (4, 6)}.

a) Determine if this relation R is a function or not.

b) If it’s a function, find the domain and range of the function.

Solution:

a) To check if the relation is a function, we need to ensure that each element of set A is related to exactly one element in set B. In other words, for every element “a” in A, there should be only one “b” in B such that (a, b) belongs to R.

In this case, the relation R satisfies this condition since each element of set A is related to exactly one element in set B. For example, (1, 4) means 1 is related to 4, and (2, 3) means 2 is related to 3, and so on.

So, the relation R is a function.

b) The domain of the function is the set of all input elements (i.e., set A), and the range is the set of all output elements (i.e., set B).

Domain: {1, 2, 3, 4}

Range: {4, 3, 5, 6}

Problem 2:

Consider the function f(x) = 2x + 1. Find the inverse function f^(-1)(x) and check if it is a valid function.

Solution:

To find the inverse function, we need to interchange the roles of x and y in the original function and then solve for y.

Let y = 2x + 1

Interchange x and y:

x = 2y + 1

Now, solve for y:

2y = x – 1

y = (x – 1) / 2

So, the inverse function is f^(-1)(x) = (x – 1) / 2.

To check if it is a valid function, we need to ensure that for every input x, there exists only one corresponding output y in the inverse function.

The inverse function seems valid since it passes the horizontal line test. Each input x maps to a unique output y in the inverse function, and thus, it is also a valid function.

Problem 3:

Let R be the relation on set A = {1, 2, 3, 4, 5}, defined as follows: R = {(1, 1), (1, 4), (2, 2), (3, 3), (4, 1)}. Determine if this relation is reflexive, symmetric, and transitive.

Solution:

a) Reflexive: A relation R on set A is reflexive if every element in A is related to itself.

In this case, (1, 1), (2, 2), and (3, 3) are present in the relation, so it is reflexive because every element is related to itself.

b) Symmetric: A relation R on set A is symmetric if for every (a, b) in R, (b, a) is also in R.

Here, (1, 4) is present in the relation, but (4, 1) is not. Thus, the relation is not symmetric.

c) Transitive: A relation R on set A is transitive if for every (a, b) and (b, c) in R, (a, c) is also in R.

In this case, (1, 4) and (4, 1) are not in the relation, so the condition for transitivity cannot be evaluated. Thus, we cannot determine whether the relation is transitive or not.

Summary:

The relation R is reflexive but not symmetric. The transitivity cannot be determined with the given information.