60. Codomain
For a function , the codomain is the set where outputs are declared to live — though outputs don’t necessarily fill it.
The actual subset of that gets hit by is the image (or range) — always a subset of the codomain, sometimes strict.
Example
, .
- Codomain:
- Image:
Negative reals like are in the codomain (they’re “allowed” outputs by the type signature) but not in the image (no real produces them).
60.1. Codomain vs image
The choice of codomain is partly conventional. Two functions can be the “same” with different codomains:
is surjective (image = codomain); is not.
60.2. For matrices
If with an matrix:
- Codomain: (set by the number of rows)
- Image: column space of , generally a subspace of
- Surjective iff the columns of span — i.e.,