Binary Relations 1 Are We Related ?

ly, we assume there is some function that extracts the angles, size, color, or whatever other property of elements we’re interested in. Two elements would be considered equivalent iff the function extracts the same value for each. � Course Notes, Week 4: Binary Relations 3 For example, if fc is the function mapping a triangle to the lengths its sides, then fc determines the congruence relation. If fs is the function mapping a triangle to the sizes of its angles, then fs determines the similarity relation. Definition 2.1. Given any total function, f , with domain A, define the binary relation ≡f on A by the rule: a ≡f b iff f(a) = f(b) (1) for all a, b ∈ A. A binary relation is an equivalence relation iff it equals ≡f for some f . So congruence of triangles is an equivalence relation because it is ≡fc , as is triangle similarity because it is ≡fs . Likewise representation­equivalence on number lists is an equivalence relation because it is ≡fr , where fr maps a representation to the set it represents. Quick exercise: Show that the equality relation on elements of a set, A, is actually an equivalence relation according to Definition 2.1 by describing an I : A → A such that equality is ≡I . Congruence modulo n is another equivalence that we will explore in detail when we introduce elementary number theory and its role in modern cryptography. Integers k and m are congruent modulo an integer n > 1, written m ≡ k mod n, iff m and k have the same remainder on division by n. So congruence modulo n is the equiv­ alence relation determined by the remainder­on­division­by­n function. This relation is called a congruence because adding or multiplying equivalent integers yields equivalent integers. That is, Lemma 2.2. If m1 ≡ k1 mod n and m2 ≡ k2 mod n, then m1 + m2 ≡ k1 + k2 mod n, and m1m2 ≡ k1k2 mod n. We leave the proof of Lemma 2.2 as an easy exercise for the reader.