Geometric Congruence is Equivalence Relation
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Theorem
Let $S$ be the set of geometric figures.
For $F_1, F_2 \in S$, let $F_1 \cong F_2$ denote that $F_1$ is congruent to $F_2$.
Then $\cong$ is an equivalence relation on $S$.
Proof
Checking in turn each of the criteria for equivalence:
Reflexivity
Geometric Congruence is Equivalence Relation/Reflexivity $\Box$
Symmetry
Geometric Congruence is Equivalence Relation/Symmetry $\Box$
Transitivity
Geometric Congruence is Equivalence Relation/Transitivity $\Box$
$\cong$ has been shown to be reflexive, symmetric and transitive.
Hence by definition it is an equivalence relation.
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): equivalence relation
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): equivalence relation