Banach-Tarski Paradox/Lemma 2

Lemma for Banach-Tarski Paradox

Relation Definition

Let $\approx$ denote the relation between sets in Euclidean space of $3$ dimensions defined as follows:

$X \approx Y$

if and only if:

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 $i \in \set {1, 2, \ldots, m}$.

Let $X$ and $Y$ be disjoint unions of $X_1, X_2$ and $Y_1, Y_2$ respectively.

Let $X_i \approx Y_i$ for each $i \in \set {1, 2}$.

Then:

$X \approx Y$

Proof

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Sources

• 1973: Thomas J. Jech: The Axiom of Choice ... (previous) ... (next): $1.$ Introduction: $1.3$ A paradoxical decomposition of the sphere: Lemma $1.5$