ThmDex – An index of mathematical definitions, results, and conjectures.
Finite set union is invariant under bijective shifting of indices
Formulation 0
Let $X_1, \ldots, X_N$ each be a D11: Set.
Let $\pi : \{ 1, \ldots, N \} \to \{ 1, \ldots, N \}$ be a D353: Set automorphism.
Then \begin{equation} \bigcup_{n = 1}^N X_n = \bigcup_{n = 1}^N X_{\pi(n)} \end{equation}
Formulation 1
Let $X_1, \ldots, X_N$ each be a D11: Set.
Let $\pi : \{ 1, \ldots, N \} \to \{ 1, \ldots, N \}$ be a D353: Set automorphism.
Then \begin{equation} X_1 \cup X_2 \cup \cdots \cup X_N = X_{\pi(1)} \cup X_{\pi(2)} \cup \cdots \cup X_{\pi(N)} \end{equation}
Subresults
R2223: Binary set union is commutative
Proofs
Proof 0
Let $X_1, \ldots, X_N$ each be a D11: Set.
Let $\pi : \{ 1, \ldots, N \} \to \{ 1, \ldots, N \}$ be a D353: Set automorphism.