MATH135 Lecture Notes - Equivalence Class, Congruence (Geometry), Additive Inverse

• School: University of Waterloo
• Department: Mathematics
• Course: MATH135
• Professor: Wentang Kuo

Document Summary

Let m be a xed m > 0 integer. If a, b z, we say that a is congruent to b modulo m. symbolically, Example i a b (mod m) a (cid:54) b (mod m) m = 10. We see that picking two numbers of a and b, 2 8 (mod 10), since 10 | ( 2 8) = Then: a a (mod m) re exivity, a b (mod m) b a (mod m) symmetry, a b (mod m) b c (mod m) a c (mod m) transitivity. In general, a relation satis es these three conditions is called a equivalence relation. Proof: a z, (a a) = 0 = 0(m) Thus, m | (a a) and a a (mod m: let m | (a b), m | (b c). By dic, m | 1(a b) + 1(b c) = (a c)