Image of Union under Mapping/Family of Sets

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Theorem

Let $S$ and $T$ be sets.

Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$.

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


Then:

$\displaystyle f \sqbrk {\bigcup_{i \mathop \in I} S_i} = \bigcup_{i \mathop \in I} f \sqbrk {S_i}$

where $\displaystyle \bigcup_{i \mathop \in I} S_i$ denotes the union of $\family {S_i}_{i \mathop \in I}$.


Proof

As $f$, being a mapping, is also a relation, we can apply Image of Union under Relation: Family of Sets:

$\displaystyle \mathcal R \sqbrk {\bigcup_{i \mathop \in I} S_i} = \bigcup_{i \mathop \in I} \mathcal R \sqbrk {S_i}$

$\blacksquare$


Sources