Difference between Relation and Function

Relation defines how elements of one set relate to elements of another set whereas a Function is a special type of relation in which each element in the domain (input) is related to exactly one element in the codomain (output).

This article explores relations and functions, highlighting their definitions, properties, differences, and applications in mathematics.

A relation is a collection of ordered pairs, pairing elements from distinct sets. These pairs signify a relationship between the elements, without prescribing a definite output for each input. Relations can exhibit various properties, influencing how elements interact across sets, forming the basis of mathematical analysis and modeling.

• Domain: Set of all input values in the relation.
• Codomain: Set of all possible output values.
• Range: Set of all actual output values obtained from the relation.
• Properties: Includes reflexivity, symmetry, transitivity, etc., influencing its behavior and characteristics.

A function is a specific type of relation where each input value (domain) aligns with precisely one output value (range). It ensures a unique assignment of output to each input, designated as f: A → B, where A signifies the domain and B denotes the range of potential output values.

This functional mapping characterizes various phenomena across mathematics, science, and beyond, providing a fundamental tool for understanding and analyzing diverse systems and processes.

• Domain: Set of all possible input values that the function can accept.
• Codomain: Set of all potential output values that the function can produce.
• Range: Set of actual output values obtained from the function by applying it to elements in the domain.
• Properties: May include characteristics like injectivity, surjectivity, bijectivity, continuity, differentiability, etc., defining its behavior and functionality.

Also Check, Types of Function

The difference between relation and function is given below:

Question 1: Given the set A = {1, 2, 3, 4} and set B = {a, b, c}, define a relation from set A to set B where each element of set A is related to each element of set B.

Solution:

To define a relation from set A to set B, we can create a relation where each element of set A is related to each element of set B. This is essentially a Cartesian product of A and B. So, the relation R can be defined as follows:

R = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c), (3, a), (3, b), (3, c), (4, a), (4, b), (4, c)}

Question 2: Let f: ℝ → ℝ be defined as f(x) = x^2 + 1. Determine whether the function f is injective, surjective, or bijective.

Solution:

To determine if f is injective, we need to check if every element in the co-domain has at most one pre-image. To check if f is surjective, we need to verify if every element in the co-domain has at least one pre-image. Finally, if f is both injective and surjective, it’s bijective.

1. Injective (One-to-One): Assume f(x₁) = f(x₂) for some x₁, x₂ in the domain. f(x₁) = x₁² + 1 and f(x₂) = x₂² + 1. If x₁ ≠ x₂, then f(x₁) ≠ f(x₂), as squaring a real number always results in a non-negative value and adding 1 makes it strictly greater. So, f is not injective.
2. Surjective (Onto): To check if f is surjective, we need to verify if for every y in the co-domain, there exists an x in the domain such that f(x) = y. Let’s take y = 0. Solving x² + 1 = 0 does not yield any real solutions. Hence, f is not surjective.

Question 3: Given the set A = {1, 2, 3} and set B = {x, y, z}, define a relation from set A to set B where each element of set A is related to exactly one element of set B.

Solution:

To define a relation from set A to set B where each element of set A is related to exactly one element of set B, we can simply pair each element of A with an element of B in a one-to-one manner. So, the relation R can be defined as follows:

R = {(1, x), (2, y), (3, z)}

Question 4: Let g: ℝ → ℝ be defined as g(x) = 2x – 3. Determine whether the function g is injective, surjective, or bijective.

Solution:

1. Injective (One-to-One): Assume g(x₁) = g(x₂) for some x₁, x₂ in the domain. 2x₁ – 3 = 2x₂ – 3 2x₁ = 2x₂ Dividing by 2, x₁ = x₂. Since every element in the co-domain has at most one pre-image, g is injective.
2. Surjective (Onto): To check if g is surjective, we need to verify if for every y in the co-domain, there exists an x in the domain such that g(x) = y. Let’s take any y in ℝ, say y = 0. Solving 2x – 3 = 0, we get x = 3/2. So, g(3/2) = 0. Since for any y in ℝ, there exists x = 3/2 such that g(x) = y, g is surjective.

Since g is both injective and surjective, it’s bijective.

In conclusion, relations and functions are fundamental concepts in mathematics, both involving sets of ordered pairs that establish connections between elements. While relations allow for multiple outputs per input, functions enforce a one-to-one correspondence. Understanding their distinctions is essential for various mathematical applications and problem-solving.

Also Check,

• Relation and Function
• Domain and Range of a Relation
• Domain and Range of a Function

What is a relation?

A relation shows how elements in one set connect to elements in another set.

What is a function?

A function is a relation where each input corresponds to exactly one output.

What are the domain and codomain?

The domain is the set of input values, and the codomain is the set of possible output values.

What types of functions are there?

Functions include linear, quadratic, exponential, and trigonometric, each with unique properties.

What’s the difference between a relation and a function?

A function is a type of relation where each input matches exactly one output.