# Mapping is Injection and Surjection iff Inverse is Mapping

## Theorem

Let $S$ and $T$ be sets.

Let $f: S \to T$ be a mapping.

Then:

- $f: S \to T$ can be defined as a bijection in the sense that:
- $(1): \quad f$ is an injection
- $(2): \quad f$ is a surjection

## Proof

### Necessary Condition

Let $S$ and $T$ be sets.

Let $f: S \to T$ be a mapping such that:

- $(1): \quad f$ is an injection
- $(2): \quad f$ is a surjection.

Then the inverse $f^{-1}$ of $f$ is itself a mapping.

### Sufficient Condition

Let $S$ and $T$ be sets.

Let $f: S \to T$ be a mapping.

Let the inverse $f^{-1} \subseteq T \times S$ itself be a mapping.

Then:

- $(1): \quad f$ is an injection
- $(2): \quad f$ is a surjection.

## Sources

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: Mappings: $\S 13$ - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 9$: Inverse Functions, Extensions, and Restrictions