Preimage of Union under Relation

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Theorem

Let $S$ and $T$ be sets.

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

Let $T_1$ and $T_2$ be subsets of $T$.


Then:

$\mathcal R^{-1} \left[{T_1 \cup T_2}\right] = \mathcal R^{-1} \left[{T_1}\right] \cup \mathcal R^{-1} \left[{T_2}\right]$


General Result

Let $S$ and $T$ be sets.

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

Let $\mathcal P \left({T}\right)$ be the power set of $T$.

Let $\mathbb T \subseteq \mathcal P \left({T}\right)$.


Then:

$\displaystyle \mathcal R^{-1} \left[{\bigcup \mathbb T}\right] = \bigcup_{X \mathop \in \mathbb T} \mathcal R^{-1} \left[{X}\right]$

where $\mathcal R^{-1} \left[{X}\right]$ denotes the preimage of $X$ under $\mathcal R$.


Family of Sets

Let $S$ and $T$ be sets.

Let $\left\langle{T_i}\right\rangle_{i \mathop \in I}$ be a family of subsets of $T$.

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


Then:

$\displaystyle \mathcal R^{-1} \left[{\bigcup_{i \mathop \in I} T_i}\right] = \bigcup_{i \mathop \in I} \mathcal R^{-1} \left[{T_i}\right]$

where:

$\displaystyle \bigcup_{i \mathop \in I} T_i$ denotes the union of $\left\langle{T_i}\right\rangle_{i \mathop \in I}$
$\mathcal R^{-1} \left[{T_i}\right]$ denotes the preimage of $T_i$ under $\mathcal R$.


Proof

We have that $\mathcal R^{-1}$ is a relation

The result follows from Image of Union under Relation.

$\blacksquare$


Also see