Codomain

60. Codomain

For a function $𝑓 : 𝑋 \to 𝑌$ , the codomain is the set $𝑌$ where outputs are declared to live — though outputs don’t necessarily fill it.

𝑌 = codomain (𝑓)

The actual subset of $𝑌$ that gets hit by $𝑓$ is the image (or range) — always a subset of the codomain, sometimes strict.

Example

$𝑓 : ℝ \to ℝ$ , $𝑓 (𝑥) = 𝑥^{2}$ .

Negative reals like $- 3$ are in the codomain (they’re “allowed” outputs by the type signature) but not in the image (no real $𝑥$ produces them).

The choice of codomain is partly conventional. Two functions can be the “same” with different codomains:

\begin{matrix} 𝑓_{1} : ℝ \to ℝ, 𝑓_{1} (𝑥) = 𝑥^{2} \\ 𝑓_{2} : ℝ \to [0, \infty), 𝑓_{2} (𝑥) = 𝑥^{2} \end{matrix}

$𝑓_{2}$ is surjective (image = codomain); $𝑓_{1}$ is not.

If $𝑇 (\vec{𝑥}) = 𝐴 \vec{𝑥}$ with $𝐴$ an $𝑚 \times 𝑛$ matrix: