Image is Subset of Codomain

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Theorem

Let $\mathcal R = S \times T$ be a relation.


For all subsets $A$ of the domain of $\mathcal R$, the image of $A$ is a subset of the codomain of $\mathcal R$:

$\forall A \subseteq \Dom {\mathcal R}: \mathcal R \sqbrk A \subseteq T$


In the notation of induced mappings, this can be written as:

$\forall A \in \powerset S: \map {\mathcal R^\to} A \in \powerset T$


Corollary 1

Let $\mathcal R = S \times T$ be a relation.


The image of $\mathcal R$ is a subset of the codomain of $\mathcal R$:

$\Img {\mathcal R} \subseteq T$


These results also hold for mappings:


Corollary 2

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


For all subsets $A$ of the domain $S$, the image of $A$ is a subset of the codomain of $f$:

$\forall A \subseteq S: f \sqbrk A \subseteq T$


Corollary 3

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


The image of $f$ is a subset of the codomain of $f$:

$\Img f \subseteq T$


Proof

\(\displaystyle A\) \(\subseteq\) \(\displaystyle \Dom {\mathcal R}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \mathcal R \sqbrk A\) \(\subseteq\) \(\displaystyle \Img {\mathcal R}\) Image of Subset is Subset of Image
\(\displaystyle \) \(\subseteq\) \(\displaystyle T\)

$\blacksquare$