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Overview
Two sets are congruent if their points can be put in one-to-one correspondence in such a way that distances are preserved.
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Basic
Suppose sets P and Q are congruent. What does this imply the existence of?
Back: A bijection between P and Q that preserves distances between points.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Suppose sets P and Q are congruent and f is the corresponding bijection. What FOL proposition follows?
Back: \forall p_1, p_2 \in P, \lvert p_1 - p_2 \rvert = \lvert f(p_1) - f(p_2) \rvert
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Bibliography
- Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).