APPENDIX I Some Major Algebraic Systems Semigroup GroupI ~~ Abelian group Operator group / Ring / Commutative ring R-module DoL ~ \ /0""0' Field Vector space A ring (always containing 1 -1= 0) is a set with two operations, addition and multiplication. It is an abelian group under addition, a semi group with 1 under multiplication, and the two operations are linked by the distributive laws. A commutative ring is a ring in which multiplication is commutative. 476 Appendix I. Some Major Algebraic Systems A domain (or integral domain) is a commutative ring in which ab = 0 implies a = 0 or b = 0; equivalently, the cancellation law holds: if ab = ac and a =F 0, then b = c. A division ring (or skew field) is a (not necessarily commutative) ring in which every nonzero element has a multiplicative inverse: if a =F 0, then there is b E K with ab = 1 = ba. The set of nonzero elements K X = K - {O} is thus a multiplicative group. Afield K is a commutative division ring. It is a theorem of Wedderburn (1905) that every finite division ring is a field. APPENDIX II Equivalence Relations and Equi valence Classes A relation on a set X is a subset == of X x X. One usually writes x == y instead of (x, y) E ==; for example, the relation < on the set of real numbers IR consists of all points in the plane IR x IR lying above the line with equation y = x, and one usually writes 2 < 3 instead of (2, 3) E <. A relation == on a set X is reflexive if x == x for all x EX; it is symmetric if, for all x, y E X, X == Y implies y == x; it is transitive if, for all x, y, Z E X, X == Y and y == z imply x == z. A relation == on a set X is an equivalence relation if it is reflexive, symmetric, and transitive. If == is an equivalence relation on a set X and if x E X, then the equivalence class of x is [x] = {y E X: y == x} eX. Proposition 11.1. Let == be an equivalence relation on a set X. If x, a E X, then [x] = [a] if and only if x == a. Proof. If [x] = [a], then x E [x], by reflexivity, and so x E [a] = [x]; that is, x == a. Conversely, if x == a, then a == x, by symmetry. If y E [x], then y == x. By transitivity, y == a, y E [a], and [x] c [a]. For the reverse inclusion, if z E [a], then z == a. By transitivity, z == x, so that z E [x] and [a] c [x], as desired• . A partition of a nonempty set X is a family of nonempty subsets {S;: i E I} such that X = U;EI S; and the subsets are pairwise disjoint: if i =I j, then S;nSj = 0. 478 Appendix II. Equivalence Relations and Equivalence Classes Proposition 11.2. If == is an equivalence relation on a nonempty set X, then the family of all equivalence classes is a partition of X. Proof. For each x E X, reflexivity gives x E [x]; this shows that the equiva­ lence classes are nonempty and that X = UX EX [x]. To check pairwise disjointness, assume that [x] n [y] i= 0. Therefore, there exists an element Z E [x] n [y]; that is, Z == x and Z == y. By the first proposition, [z] = [x] and [z] = [y], so that [x] = [y]. • Proposition 11.3. If {Si: i E I} is a partition of a nonempty set X, then there is an equivalence relation on X whose equivalence classes are the Si' Proof. If x, y E X, define x == y to mean that there exists Si containing both x and y. It is plain that == is reflexive and symmetric. To prove transitivity, assume that x == y and y == z; that is, x, y E Si and y, z E Sj. Since y E Si n Sj' pairwise disjointness gives Si = Sj; hence x, Z E Si and x == z. If x E X, then x E Si for some i. If y E Si' then y, x E Si and y == x; that is, Si C [x]. For the reverse inclusion, if Z E [x], then Z == x, and so z, x E Si; that is, [x] c Si' • Proposition II.l signals the importance of equivalence relations. If == is an equivalence relation on a set X and if E is the family of equivalence classes, then x == y in X if and only if [x] = [y] in E; equivalence of elements in X becomes equality of elements in E. The construction of the new set E thus identifies equivalent elements. For example, the fractions t and i are called equal if the numerators and denominators satisfy "cross multiplication." In reality, one defines a relation == on X = {(a, b) E 71. x 71.: b i= O} by (a, b) == (c, d) if ad = bc, and a straight­ forward calculation shows th,at == is an equivalence relation on X. The equiv­ alence class containing (a, b) is denoted by alb, and the set of all rational numbers q) is defined as the family of all such equivalence classes. In particu­ lar, (1, 2) and (2, 4) are identified in q), because (1, 2) == (2,4), and so t = i. APPENDIX III Functions If X and Yare sets, a relation from X to Y is a subset f of X x Y (if X = 1', one also says that f is a relation on X). A/unction from X to 1', denoted by f: X ~ Y, is a relation f from X to Y such that for each x E X, there exists a unique Y E Y with (x, y) E f. If x E X, then the unique element y in the defini­ tion is denoted by f(x), and it is called the value of f at x or the image of x under f. With this notation, the relation f consists of all (x, f(x)) E X X Y; that is, a function is what is usually called its graph. The set X is called the domain of f and the set Y is called the target of f. One defines two functions f and g to be equal if they have the same domain X, the same target 1', and f(x) = g(x) for all x E X (this says that their graphs are the same subset of X x Y). In practice, one thinks of a function f as something dynamic: f assigns a value f(x) in Y to each element x in X. For example, the squaring function f: IR ~ IR is the parabola consisting of all (x, x 2 ) E IR x IR, but one usually thinks of f as assigning x2 to x; indeed, we often use a footed arrow to denote the value of fon a typical element x in the domain: for example, f: Xf-+X2. Most elementary texts define a function as "a rule which assigns, to each x in X, a unique value f(x) in Y." The idea is correct, but not good enough. For example, consider the functions f, g: IR ~ IR defined as follows: f(x) = (x + 1)2; g(x) = x2 + 2x + 1. Are f and g different functions? They are differ­ ent "rules" in the sense that the procedures involved in computing each of them are different. However, the definition of equality given above shows thatf = g. If X is a nonempty set, then a sequence in X is a function f: IP' ~ X, where IP is the set of positive integers. Usually, one writes Xn instead of f(n) and one describes f by displaying its values: Xl' X2' X3' .... It follows that two 480 Appendix III. Functions sequences Xl' X 2 , x 3,··· and Y1' Yz, Y3"" are equal if and only if Xn = Yn for all n ~ 1. The uniqueness of values in the definition of function deserves more com­ ment: it says that a function is "single valued" or, as we prefer to say, it is well defined. For example, if X is a nonnegative real number, then f(x) = JX is usually defined so that f(x) ~ 0; with no such restriction, it would not be a function (if J4 = 2 and J4 = - 2, then a unique value has not been assigned to 4). When attempting to define a function, one must take care that it is well defined lest one define only a relation. The diagonal {(x, x) E X xX: x E X} is a function X --+ X; it is called the identity function on X, it is denoted by lx, and Ix: xt---+x for all x E X. If X is a subset of Y, then the inclusion function i: X --+ Y (often denoted by i: X ~ Y) is defined by xt---+x for all x E X. The only difference between Ix and i is that they have different targets, but if X is a proper subset of Y, this is sufficient to guarantee that Ix =f. i. If f: X --+ Y and g: Y --+ Z are functions, then their composite go f: X --+ Z is the function defined by x t---+ g(f(x)). If h: Z --+ W is a function, then the associativity formula h 0 (g 0 f) = (h 0 g) 0 f holds, for both are functions X --+ W with x t---+ h(g(f(x))) for all x E X.