Is the identity function on a non empty set A invertible?

The identity function on any non-empty set A is indeed invertible. To understand why, let's break down the concept of the identity function and what it means for a function to be invertible.

What is the Identity Function?

The identity function, often denoted as id_A, is a function that maps every element in a set A to itself. In mathematical terms, for any element x in A, the identity function is defined as:

• id_A(x) = x for all x in A

This means that the output of the identity function is exactly the same as the input, which is a straightforward and intuitive concept.

Understanding Invertibility

A function is considered invertible if there exists another function, called the inverse function, that can "reverse" the action of the original function. For a function f to have an inverse, it must satisfy two main conditions:

• The function must be one-to-one (injective), meaning no two different inputs produce the same output.
• The function must be onto (surjective), meaning every possible output in the codomain is produced by some input from the domain.

Analyzing the Identity Function

Now, let's apply these criteria to the identity function:

Injectivity

The identity function is injective because if id_A(x1) = id_A(x2), then x1 must equal x2. Since each element maps to itself, there are no two distinct elements that can yield the same output.

Surjectivity

It is also surjective because for every element y in A, there exists an element x in A such that id_A(x) = y. In fact, for any y, we can simply choose x = y, and the function will map it correctly.

Finding the Inverse

The inverse of the identity function is itself. That is, if we define a function g: A → A such that g(y) = y for all y in A, then g is the inverse of id_A. We can verify this by checking:

• g(id_A(x)) = g(x) = x for all x in A
• id_A(g(y)) = id_A(y) = y for all y in A

Conclusion

Since the identity function is both injective and surjective, and its inverse is the function itself, we can confidently say that the identity function on a non-empty set A is indeed invertible. This property holds true for any set, regardless of its size or the nature of its elements.