Union is Commutative/Family of Sets
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Theorem
Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of sets.
Let $\ds I = \bigcup_{i \mathop \in I} S_i$ denote the union of $\family {S_i}_{i \mathop \in I}$.
Let $J \subseteq I$ be a subset of $I$.
Then:
- $\ds \bigcup_{i \mathop \in I} S_i = \bigcup_{j \mathop \in J} S_j \cup \bigcup_{k \mathop \in \relcomp I J} S_k = \bigcup_{k \mathop \in \relcomp I J} S_k \cup \bigcup_{j \mathop \in J} S_j$
where $\relcomp I J$ denotes the complement of $J$ relative to $I$.
Proof
We have that both $\ds \bigcup_{j \mathop \in J} S_j$ and $\ds \bigcup_{k \mathop \in \relcomp I J} S_k$ are sets.
Hence by Union is Commutative we have:
- $\bigcup_{j \mathop \in J} S_j \cup \bigcup_{k \mathop \in \relcomp I J} S_k = \bigcup_{k \mathop \in \relcomp I J} S_k \cup \bigcup_{j \mathop \in J} S_j$
It remains to be demonstrated that $\ds \bigcup_{i \mathop \in I} S_i = \bigcup_{j \mathop \in J} S_j \cup \bigcup_{k \mathop \in \relcomp I J} S_k$.
So:
\(\ds x\) | \(\in\) | \(\ds \bigcup_{i \mathop \in I} S_i\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \exists i \in I: x\) | \(\in\) | \(\ds S_i\) | Definition of Union of Family | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \exists j \in J: x\) | \(\in\) | \(\ds S_j\) | Definition of Relative Complement | ||||||||||
\(\, \ds \lor \, \) | \(\ds \exists k \in \relcomp I J: x\) | \(\in\) | \(\ds S_k\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(\in\) | \(\ds \bigcup_{j \mathop \in J} S_j\) | Definition of Union of Family | ||||||||||
\(\, \ds \lor \, \) | \(\ds x\) | \(\in\) | \(\ds \bigcup_{k \mathop \in \relcomp I J} S_k\) | Definition of Union of Family | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(\in\) | \(\ds \bigcup_{j \mathop \in J} S_j \cup \bigcup_{k \mathop \in \relcomp I J} S_k\) | Definition of Set Union |
That is:
- $\ds x \in \bigcup_{i \mathop \in I} S_i \iff x \in \bigcup_{j \mathop \in J} S_j \cup \bigcup_{k \mathop \in \relcomp I J} S_k$
The result follows by definition of set equality.
$\blacksquare$
Also see
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 9$: Families