Injective Function

Definition

A function is injective when different inputs always produce different outputs.

Basic form:

if a is not b
then f(a) is not f(b)

Equivalent form:

if f(a) = f(b)
then a = b

Injective function = no output is reused by two different inputs.

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Simple Example

Suppose:

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

Then:

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

Each input produces a different output.

So the function is injective.

Injective functions preserve input distinction in the outputs.

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Non-Example

Suppose:

f(x) = x * x
Domain: {-2, -1, 0, 1, 2}

Then:

-2 -> 4
2 -> 4
-1 -> 1
1 -> 1
0 -> 0

Different inputs can produce the same output.

So the function is not injective.

A function is not injective when distinct inputs collapse into the same output.

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Injective vs Function

Every injective mapping is still a function, so each input has exactly one output.

Injective adds an extra requirement:

different inputs must not share an output

Function requirement:

one input -> one output

Injective requirement:

different inputs -> different outputs

Injective is stricter than being a function.

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Why Injectivity Matters

Injectivity tells us whether output still identifies the input.

If a function is injective, seeing the output is enough to know which input produced it.

If it is not injective, information may be lost.

Example:

square output: 4
possible inputs: -2 or 2

Injectivity is about preserving distinctions.

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Final Summary

• An injective function maps different inputs to different outputs.
• No output is reused by two different inputs.
• f(x) = x + 1 is injective over integers.
• f(x) = x * x is not injective over integers.
• Injectivity preserves distinctions between inputs.