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Congruence. The Congruence Axiom.(Cong) There is a group
 

Summary: Congruence.
The Congruence Axiom.(Cong) There is a group
M
of permutations of the set of points such that
M  B
and such that if (H i ; R i ); i = 1; 2; are ags then there is one and only one  2 M such that
 [H 1 ] = H 2 and  [R 1 ] = R 2 :
We will call the members of M motions
.
De nition. Congruence of sets of points. We say the sets X and Y of points are congruent and
write
X ' Y
if there exists  2 M such that  [X ] = Y in which case we say X can be moved to Y (by the motion
).
Theorem. We have
(i) If X is a set of points then X ' X .
(ii) If X and Y are sets of points and X ' Y then Y ' X .
(iii) If X;Y and Z are sets of points, X ' Y and Y ' Z then X ' Z.
Remark. Equivalently, congruence is an equivalence relation on the family of sets of points.
Proof. This follows directly from the fact that the set of motions is a group.

  

Source: Allard, William K. - Department of Mathematics, Duke University

 

Collections: Mathematics