Banach-Tarski Paradox/Lemmata/Relation Definition
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Lemmata for Banach-Tarski Paradox: Definition of Relation
Let $\approx$ denote the relation between sets in Euclidean space of $3$ dimensions defined as follows:
- $X \approx Y$
- there exists a partition of $X$ into disjoint sets:
- $X = X_1 \cup X_2 \cup \cdots \cup X_m$
- and a partition of $Y$ into the same number of disjoint sets:
- $Y = Y_1 \cup Y_2 \cup \cdots \cup Y_m$
such that $X_i$ is congruent to $Y_i$ for each $i \in \set {1, 2, \ldots, m}$.
Sources
- 1973: Thomas J. Jech: The Axiom of Choice ... (previous) ... (next): $1.$ Introduction: $1.3$ A paradoxical decomposition of the sphere: Lemma $1.5$