What is a Surjection in Mathematics?

A surjection is a function that maps every element of the domain to an element of the codomain. In other words, it is a function that "covers" all the elements of the domain, and does not leave any gaps or unused elements in the codomain.

For example, consider the function f(x) = 2x + 1, which maps the real numbers to themselves. This function is a surjection because for every real number x, there exists a unique real number y such that f(x) = y. In other words, the range of f contains all the real numbers, and there are no gaps or unused elements in the range.

On the other hand, the function g(x) = x^2 is not a surjection, because it only maps the non-negative real numbers to the positive real numbers. There are many real numbers that are not in the image of g, such as negative numbers and zero.

It's important to note that surjections are not necessarily injective or bijective. A surjective function can have multiple outputs for a single input, and it may not be one-to-one or onto.