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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 (Hi, Ri), i = 1, 2, are flags then there is one and only one M such that
[H1] = H2 and [R1] = R2.
We will call the members of M motions
.
Definition. Congruence of sets of points. We say the sets X and Y of points are congruent and
write
X Y
if there exists 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