Set Equivalence behaves like Equivalence Relation/Transitive
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Theorem
Set equivalence behaves like a transitive relation:
- $S_1 \sim S_2 \land S_2 \sim S_3 \implies S_1 \sim S_3$
Proof
| \(\ds \) | \(\) | \(\ds S_1 \sim S_2 \land S_2 \sim S_3\) | ||||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds \exists \phi_1: S_1 \to S_2 \land \exists \phi_2: S_2 \to S_3\) | Definition of Set Equivalence: $\phi_1$ and $\phi_2$ are bijections | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds \exists \phi_2 \circ \phi_1: S_1 \to S_3\) | Composite of Bijections is Bijection: $\phi_2 \circ \phi_1$ is a bijection | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds S_1 \sim S_3\) | Definition of Set Equivalence |
$\blacksquare$
Sources
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $2$. Set Theoretical Equivalence and Denumerability
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 3.7$. Similar sets
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 17$: Finite Sets: Theorem $17.1$
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $3$: Cardinality: Exercise $1 \ \text{(c)}$
- 1999: András Hajnal and Peter Hamburger: Set Theory ... (previous) ... (next): $2$. Definition of Equivalence. The Concept of Cardinality. The Axiom of Choice: Theorem $2.1$