Image of Union under Mapping/Family of Sets

From ProofWiki
Jump to navigation Jump to search

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:

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

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


This can be expressed in the language and notation of direct image mappings as:

$\ds \map {f^\to} {\bigcup_{i \mathop \in I} S_i} = \bigcup_{i \mathop \in I} \map {f^\to} {S_i}$


Proof

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

$\ds \RR \sqbrk {\bigcup_{i \mathop \in I} S_i} = \bigcup_{i \mathop \in I} \RR \sqbrk {S_i}$

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Image_of_Union_under_Mapping/Family_of_Sets&oldid=710356"