Codomain

Definition

The codomain is the set of possible output values declared for a function.

Example:

f(x) = x + 1
Domain: {1, 2, 3}
Codomain: {1, 2, 3, 4, 5}

The function outputs:

2, 3, 4

All of these outputs are inside the codomain.

Codomain = where outputs are allowed to land.

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Codomain Is Declared

The codomain is part of how the function is described.

Example:

f: {1, 2, 3} -> {1, 2, 3, 4, 5}

This says:

inputs come from {1, 2, 3}
outputs are expected to be in {1, 2, 3, 4, 5}

The codomain gives the output space of the function.

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Codomain vs Actual Outputs

The codomain may include values that never appear as outputs.

Example:

f(x) = x + 1
Domain: {1, 2, 3}
Codomain: {1, 2, 3, 4, 5}

Actual outputs:

{2, 3, 4}

The codomain contains 1 and 5, but the function never produces them.

The codomain can be larger than the actual outputs.

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Why Codomain Matters

The codomain lets us ask whether the function reaches all intended outputs.

Example:

Domain: {1, 2, 3}
Codomain: {2, 3, 4}
f(x) = x + 1

Here every codomain value is reached.

But if the codomain is:

{1, 2, 3, 4, 5}

then not every codomain value is reached.

Codomain matters when studying coverage of outputs.

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Final Summary

• The codomain is the declared output set.
• Every actual output must belong to the codomain.
• The codomain can contain values the function never produces.
• The codomain is different from the actual set of outputs.
• Codomain helps define whether a function covers its intended output space.