Inverse is Mapping implies Mapping is Injection and Surjection
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Theorem
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.
Proof
This is divided into two parts:
Inverse is Mapping implies Mapping is Injection
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 $f$ is an injection.
Inverse is Mapping implies Mapping is Surjection
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 $f$ is a surjection.