Proving Injectivity of the Greatest Integer Function

Understanding the concept of injectivity is crucial in various fields of mathematics, including analysis and algebra. In this article, we will explore how to prove that a given function is not injective using the greatest integer function as an example. If by fx [x], you mean the integer part (or floor) of x, it is important to note that this function is not injective. This article will walk you through the necessary steps and provide practical examples to illustrate this point.

What is the Greatest Integer Function?

The greatest integer function, denoted as [x], is a function that returns the largest integer less than or equal to x. In mathematical terms, it can be represented as:

[x] n, where n is the largest integer satisfying x ≥ n.

For example:

[1.2] 1 [1.7] 1 [-1.2] -2 [-1.5] -2

How to Prove a Function is Not Injective

To verify if a function is not injective, we need to find at least two distinct inputs that yield the same output. Let's consider the function fx [x] with the domain .

Step 1: Define the Domain

The represents the set of all real numbers, which is an infinite set. This makes selecting pairs of distinct real numbers relatively straightforward.

Step 2: Identify Distinct Inputs with the Same Output

Let's take the example where x 1.2 and x 1.5. Both of these inputs are distinct and belong to the set of real numbers. Now, let's evaluate the floor function for each:

For x 1.2: [1.2] 1 For x 1.5: [1.5] 1

As we can see, both 1.2 and 1.5 have the same floor value of 1. This demonstrates that the function fx [x] is not injective.

Step 3: Generalize the Result

The above example can be generalized. Given any two real numbers a and b such that a , both ?a? and ?b? will be equal to a. This is because the floor function returns the largest integer less than or equal to the input.

For instance, take 0.5 and 0.7 as inputs:

For x 0.5 [0.5] 0 For x 0.7 [0.7] 0

Again, both inputs yield the same output, confirming that the function fx [x] is not injective.

Further Considerations

It is worth noting that although the function fx [x] is not injective, it is surjective over the set of integers. This means that every integer is the output of the function for at least one input. However, the absence of injectivity makes this function less useful in contexts where uniqueness of mapping is required.

Conclusion

In summary, to prove that a function is not injective, we need to find at least two distinct inputs that produce the same output. For the greatest integer function fx [x], we demonstrated that inputs such as 1.2 and 1.5 both yield the same floor value of 1. This proof applies to any pair of inputs that are not consecutive integers and lie within the same integer interval.