Bijective Function

Definition

A function is bijective when it is both injective and surjective.

That means:

different inputs produce different outputs
and every codomain value is reached

Bijective function = perfect pairing between domain and codomain.

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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

Each input has a different output.

Every codomain value is reached.

So the function is bijective.

Bijection creates one-to-one coverage.

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Non-Example: Not Injective

Suppose:

f(x) = x * x
Domain: {-1, 0, 1}
Codomain: {0, 1}

Then:

-1 -> 1
0 -> 0
1 -> 1

Every codomain value is reached, but output 1 is used by two different inputs.

So the function is not bijective.

A bijection cannot merge different inputs into one output.

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Non-Example: Not Surjective

Suppose:

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

Then:

1 -> 2
2 -> 3

The codomain value 4 is never reached.

So the function is not bijective.

A bijection cannot leave codomain values unused.

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Why Bijectivity Matters

A bijective function preserves distinction and covers the whole target set.

This means each output corresponds to exactly one input.

That makes reversal possible.

Example:

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

Can be reversed as:

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

Bijection supports clean reversibility.

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Final Summary

• A bijective function is both injective and surjective.
• Different inputs produce different outputs.
• Every codomain value is reached.
• Each output corresponds to exactly one input.
• Bijective functions create perfect pairing.
• Bijectivity makes reversal possible.