Injective Function

Injective function is a function with relates an element of a given set with a distinct element of another set. An injective function is also referred to as a one-to-one function. There are numerous examples of injective functions. The name of a student in a class, and his roll number, the person, and his shadow, are all examples of injective function.

Let us learn more about the definition, properties, examples of injective functions.

In an injective function, every element of a given set is related to a distinct element of another set. A function f : X → Y is defined to be one-one (or injective), if the images of distinct elements of X under f are distinct, i.e., for every x1, x2 ∈ X, there exists distinct y1, y2 ∈ Y, such that f(x1) = y1, and f(x2) = y2.

The injective function can be represented in the form of an equation or a set of elements. The function f(x) = x + 5, is a one-to-one function. This can be understood by taking the first five natural numbers as domain elements for the function. The function f = {(1, 6), (2, 7), (3, 8), (4, 9), (5, 10)} is an injective function.

The following images in Venn diagram format helpss in easily finding and understanding the injective function. We can observe that every element of set A is mapped to a unique element in set B. Further, if any element is set B is an image of more than one element of set A, then it is not a one-to-one or injective function.

The following are the few important properties of injective functions.

• The domain and the range of an injective function are equivalent sets.
• The sets representing the domain and range set of the injective function have an equal cardinal number.
• Injective functions if represented as a graph is always a straight line.
• The injective function follows a reflexive, symmetric, and transitive property.

Related Topics

The following topics help in a better understanding of injective function.

• Relations and Functions
• Identity Function
• Zero Function
• Even and Odd Functions
• Onto Function
• Periodic Function
• Signum Function
• Continuous Function

What Is Injective Function?

The function in which every element of a given set is related to a distinct element of another set is called an injective function. A function f : X → Y is defined to be one-one (or injective), if the images of distinct elements of X under f are distinct, i.e., for every x1, x2 ∈ X, there exists distinct y1, y2 ∈ Y, such that f(x1) = y1, and f(x2) = y2.

How Do You Know If a Function is an Injective Function?

A function can be identified as an injective function if every element of a set is related to a distinct element of another set. The codomain element is distinctly related to different elements of a given set. If this is not possible, then it is not an injective function.

What Is the Difference Between Injective and Surjective Function?

The injective function related every element of a given set, with a distinct element of another set, and is also called a one-to-one function. The subjective function relates every element in the range with a distinct element in the domain of the given set. A subjective function is also called an onto function. The injective function and subjective function can appear together, and such a function is called a Bijective Function.

Give a Few Real-Life Examples of Injective Function?

The following are a few real-life examples of injective function.

• The name of the student in a class and the roll number of the class.
• The person and the shadow of the person, for a single light source.
• The traveller and his reserved ticket, for traveling by train, from one destination to another.