Surjective Function

Definition

A function is surjective when every value in the codomain is reached by at least one input.

Basic form:

for every output y in the codomain
there is some input x such that f(x) = y

Surjective function = every codomain value is hit.

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Simple Example

Suppose:

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

Then:

1 -> 2
2 -> 3
3 -> 4

Every codomain value is reached.

So the function is surjective.

Surjectivity means the range equals the codomain.

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Non-Example

Suppose:

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

Actual outputs:

{2, 3, 4}

The function does not output:

1
5

So it is not surjective.

A function is not surjective if any codomain value is missed.

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Surjectivity Depends On Codomain

The same formula can be surjective or not surjective depending on the codomain.

Example:

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

With codomain:

{2, 3, 4}

it is surjective.

With codomain:

{1, 2, 3, 4, 5}

it is not surjective.

Surjectivity is about the declared output space, not just the formula.

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Why Surjectivity Matters

Surjectivity tells us whether a function covers all intended outputs.

It answers:

can every codomain value be produced?

If yes, the function reaches the whole codomain.

If no, some declared outputs are impossible.

Surjectivity is about output coverage.

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Final Summary

• A surjective function reaches every codomain value.
• Surjectivity means the range equals the codomain.
• A function can miss some codomain values.
• Surjectivity depends on the chosen codomain.
• Surjectivity describes output coverage.