I'm learning about set notation. What is the difference between a co-domain and an image?
user
Given a function f: X → Y (Both of these sets can be chosen as wanted):
X is called the domain and Y is called the codomain.

The codomain is were all of the outputs of f are constrained to fall. (f(x) can only be values existing in the set Y)
The image of the function is the subset of the codomain the function can reach.

For an example, take a look at this function:
f: {0, 1, 2, 3, 4, 5} -> N
f(x) = x²

This function does map its domain {0, 1, 2, 3, 4, 5} to its codomain N. But not all of the values in the codomain N can be reached using this function and domain, so the image of the function would be {0, 1, 4, 9, 16, 25}, only a small subset of N.

Formally the image is defined as { f(x) : x ∈ X } where X is the Domain of the function f.
The Codomain can be chosen when writing a function.
assistant
Could you give a real life example to demonstrate this distinction?
user
Sure! Let's consider the function f(x) = x^2, which maps real numbers to non-negative real numbers.

* The co-domain of f is the set of all non-negative real numbers. We can denote it as CoD(f) = [0, ∞). This is the set of all possible outputs that f can produce.

* The image of f is the set of all non-negative real numbers that are actually produced by the function. In other words, it's the set of all values that f(x) takes on for some input x. We can denote it as Im(f) = {y | y = f(x) for some x}.

For example, if we consider the input set {1, 2, 3}, the outputs of f(x) for these inputs are {1, 4, 9}. So the image of f is {1, 4, 9}, and it's a subset of the co-domain [0, ∞).

So in this case, the co-domain is the set of all possible outputs (non-negative real numbers), and the image is the subset of non-negative real numbers that are actually produced by the function.
assistant