Image of Subset under Relation is Subset of Image
Theorem
Let $S$ and $T$ be sets.
Let $\RR \subseteq S \times T$ be a relation from $S$ to $T$.
Let $A, B \subseteq S$ such that $A \subseteq B$.
Then the image of $A$ is a subset of the image of $B$:
- $A \subseteq B \implies \RR \sqbrk A \subseteq \RR \sqbrk B$
In the notation of direct image mappings, this can be written:
- $A \subseteq B \implies \map {\RR^\to} A \subseteq \map {\RR^\to} B$
Corollary 1
The same applies to the preimage, as follows.
Let $C, D \subseteq T$.
Then:
- $C \subseteq D \implies \RR^{-1} \sqbrk C \subseteq \RR^{-1} \sqbrk D$
where $\RR^{-1} \sqbrk C$ is the preimage of $C$ under $\RR$.
Corollary 2
The same applies for a mapping $f: S \to T$ and its inverse $f^{-1} \subseteq T \times S$, whether $f^{-1}$ is a mapping or not.
Let $A, B \subseteq S$ such that $A \subseteq B$.
Then the image of $A$ is a subset of the image of $B$:
- $A \subseteq B \implies f \sqbrk A \subseteq f \sqbrk B$
This can be expressed in the language and notation of direct image mappings as:
- $\forall A, B \in \powerset S: A \subseteq B \implies \map {f^\to} A \subseteq \map {f^\to} B$
Corollary 3
Similarly:
Let $C, D \subseteq T$.
Then:
- $C \subseteq D \implies f^{-1} \sqbrk C \subseteq f^{-1} \sqbrk D$
This can be expressed in the language and notation of inverse image mappings as:
- $\forall C, D \in \powerset T: C \subseteq D \implies \map {f^\gets} C \subseteq \map {f^\gets} D$
Proof
Suppose $\RR \sqbrk A \nsubseteq \RR \sqbrk B$.
\(\ds \RR \sqbrk A\) | \(\nsubseteq\) | \(\ds \RR \sqbrk B\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \exists t \in \RR \sqbrk A: \exists \tuple {s, t} \in \RR: \, \) | \(\ds s\) | \(\notin\) | \(\ds B\) | Definition of Image of Subset under Relation | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \exists s \notin B: \, \) | \(\ds s\) | \(\in\) | \(\ds A\) | Definition of Ordered Pair | |||||||||
\(\ds \leadsto \ \ \) | \(\ds A\) | \(\nsubseteq\) | \(\ds B\) | Definition of Subset |
The result follows by the Rule of Transposition.
$\blacksquare$
Sources
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.12$: Set Inclusions for Image and Inverse Image Sets: Theorem $12.1$