Mapping is Injection and Surjection iff Inverse is Mapping

Theorem

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

if and only if:

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

Proof

Necessary Condition

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