Geometric Congruence is Equivalence Relation

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
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