1 0 ABSTRACT ALGEBRA We continue this background section with a brief discussion of the concept of a relation on a set Z), with particular attention to equivalence relations. Definition. A relation i?ona set D is any subset of D x D. If (a,b) e R, we say that a is in the relation Rtob. R is said to be reflexive if R contains the diagonal set A = {(d,d) :deD}. R is symmetric if the following holds: If (a,b) e R, then (b, a) € R. R is transitive if the following holds: If(a, b) e R and (b, c) e R, then (a, c) e R. Finally we call R an equivalence relation if R is reflexive, symmetric, and transitive. If E is an equivalence relation on D and if a e D, then the set Ea = {d e D : (a,d) e E] is called the (E)-equivalence class containing a. Obviously every element of D lies in exactly one E-equivalence class. Thus the set of E-equivalence classes forms a partition of D into disjoint subsets. Conversely any partition P of D into disjoint subsets defines an equivalence relation Ep via (a,b) e Ep if and only if a and b lie in the same subset of the partition P. Equivalence relations are central to mathematics. The quintessential equivalence relation is equality, denoted by =. It is the finest of equivalence relations: An object is equal only to itself. All other equivalence relations may be thought of as "filters" through which we look at mathematical objects, ignoring certain differences and allowing only the similarities to show through. Thus in Euclidean geometry, the congruence filter ignores the position of objects and observes only their size and shape, while the even darker similarity filter also ignores size and observes only shape. Equivalence relations are crucial to our ability to make generalizations, to prove general theorems. At a deeper level, an equivalence relation on one mathematical entity may be used to define a new mathematical entity whose members are the equivalence classes themselves. We shall see some examples of this type of construction, which appeared first in the work of Gauss around 1800 and was one of the key constituents of the newly emerging abstract algebra. We shall repeatedly be considering sets with "operations" in the following sense. Definition. A set D is said to be closed under a binary operation x (sometimes denoted + or o, etc.) if x is a function: x : D x D -+ D. Usually we denote x (a, b) by a x b. We say that x is an associative operation if a x (b x c) = (a x b) x c

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