Classical Analysis I


 Donald Nicholson
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1 Classical Analysis I 1 Sets, relations, functions A set is considered to be a collection of objects. The objects of a set A are called elements of A. If x is an element of a set A, we write x A, and if x is not an element of A, we write x A. If every element of a set A is also an element of B, then A is subset of B. This is denoted as A B. Two sets A and B are equal, denoted A = B, if A and B have the same elements. Equivalently, A = B if A B and B A. If A is a set and P is a property, then {x P(x)} is the subset of X consisting of all elements x of X such that P(x) is true. The empty set has no elements; it has the property that it is a subset of any set. Given two sets A and B we define: The union A B is the set The intersection A B is the set A B = {x x A or x B}. A B = {x x A and x B}. The relative complement A \ B is the set A \ B = {x x A and x B}. Sets A and B are called disjoint if A B =. When it is understood that all sets under considerations are subsets of a fixed set X, then the complement A c of the set A X is defined by A c = X \ A = {x X x A}. The set of all subsets of a given set X is called the power set and is denoted by P(X). 1
2 The concept of union and intersection of two sets extends to unions and intersections of arbitrary families of sets. By a family of sets we mean a nonempty set F whose elements are sets themselves. If F is a family of sets, then A = {x x A for some A F} A F A F A = {x x A for all A F}. Proposition 1.1 (de Morgan s laws). Let {A i i J} be the family of ssubsets of X. Then ( i I A ) c i = i I Ac i. ( i I A ) c i = i I Ac i. If a A and b B, then (a,b) is called an ordered pair. Two ordered pairs (a,b) and (a,b ) are equal if a = a and b = b. The (Cartesian) product A B of A and B is the set of all ordered pairs (a,b) where a A and b B. The product A B = if and only if A = or B =. In general, A B B A. The product of three sets A,B and C is defines as A B C = (A B) C. Proceeding inductively one defines the product of n by A 1 A n = (A 1 A n 1 ) A n. For a A 1 A n we write (a 1,...,a n ) instead of ( ((a 1,a 2 ),a 3 ),,a n ). We call a k the kth component of a. 1.1 Relations Let X and Y be two sets. A relation R from X to Y is a subset R X Y. If X = Y, then R is said to be a relation on X. If x X and y Y, we write xry if (x,y) R Equivalence relation A relation on X is called an equivalence relation if it satisfies the following conditions: (a) (reflexivity) For all x X, x x. 2
3 (b) (symmetry ) If x y, then y x. (c) (transitivity) If x y and y z, then x y. Example 1.2. On the set Z = N N define the relation: (m,n) (j,k) if and only if m + k = n + j. This is an equivalence relation. Indeed, it is obvious that is reflexive and symmetric. If (m,n) (j,k) and (j,k) (r,s), then m + k = n + j and j + s = k + r. Then m + k + j + s = n + j + k + r from which we conclude that m + s = n + r. This means that (m,n) (r,s). Example 1.3. If is an equivalence relation on X, then the equivalence class of x as the set [x] = {y A x y}. The quotient set of A and is the set of all equivalence classes of A with respect to, that is, the set {[x] x A}. The quotient set is denoted by A/. We have (1) Let x,y A. If x y, then [x] = [y] and, if x y, then [x] [y] = (2) x A [x] = A Functions Given two sets X and Y, a relation f from X to Y is called a function, denoted by f : X Y, if for every x X, there is exactly one y Y such that (a,b) f. In other words, if (x,y) f and (x,y ) f, then y = y. This definition of a function identifies a function with its graph. We think of f as a rule of assigning, to the element x X, the element y Y for which (x,y) f. If (x,y) f, then y is the value of f at x, and we write y = f(x). The set X is called the domain of the function and the set Y is called the codomain. the set {f(x) x X} is called the range of f, denoted by im(f), or the image of f, denoted by R(f). If f : X Y is a function, A X and B Y, then f(a) = {f(a) a A} is called the image of A under f and the set f 1 (B) = {x X f(x) C} 3
4 is called the preimage of B under f. Given two functions f : X Y and g : Y Z, we define the composition g f of f and g by g f : X Z x g(f(x)). If X Y, then f is surjective if im(f) = Y, injective if f(x) = f(y) implies x = y for all x,y X, and bijective if f is both injective and surjective. Proposition 1.4. Let f : X Y. Then f is bijective if and only if there is a function g : Y X satisfying g f = id X and f g = id Y. Proof. = Since f is bijective, for every y Y, there is x X such that f(x) = y. Since f is injective, this x is uniquely determined. This defines a function g : Y X with the desired properties. = From f g = id Y follows that f is surjective. If x,y X and f(x) = f(y), then x = g(f(x)) = g(f(y)) = y showing that f is injective. Let f : X Y bijective. Then the inverse function f 1 of f is the unique function f 1 : Y X such that f f 1 = id Y and f 1 f = id X. Proposition 1.5. Let f : X Y and g : Y Z be bijective. Then g f : X Z is bijective and (g f) 1 = f 1 g 1. Proposition 1.6. The following hold for the function f : X Y. If A B X, then f(a) f(b). If A i X for every i J, then f ( i J A ) i i J f(a i). If A i X for every i J, then f ( i J A ) i i J f(a i). If A X, then f(a c ) f(x) \ f(a). If B i Y for every i J, f 1( i I B ) i = i I f 1 (B i ). If B i Y for every i J, f 1( i I B ) i = i I f 1 (B i ). If B Y, f 1 (B c ) = [f 1 (B)] c. 4
5 1.1.3 Order relations A relation < is called a partial ordering on X if it satisfies: (1) For all x X, x x. (2) (transitivity) If x < y and y < z, then x < y. If this holds, then the pair (X,<) is called a partially ordered set. If, in addition, the partial order < satisfies: (4) trichotomy) For every x,y X, either x < y or y < x or x = y, then then is called total order on X and (X,<) is a totally ordered set. Example 1.7. Let (P(X), ) be the power set of X. For A,B P(X), define A < B if and only if A B and A B. Hence < is a proper inclusion of subsets of X. Then < is a partial order on P(X) and (P(X),<) is a partially ordered set. We also write x y to mean either x < y or x = y. Let (X,<) be a partially ordered set and A be a nonempty subset of X. An element x X is called an upper boundof A if a x for all a A, It is called a lower bound if x a for all a A. The subset A bounded above if it has an upper bound, bounded below if it has a lower bound, and bounded if it is both bounded above and below. An element m X is the maximum of A, written m = max(a), if m is an upper bound of A and m A. An element m X is the minimum of A, written m = min(a), if m is a lower bound of A and m A. Let A be a nonempty subset of a partially ordered subset of X which is bounded above. If the set of all upper bounds of A has a minimum, then this element is called the least upper bound of A or supremum of A and is written sup(a). Similarly, if A is bounded below and the set of all lower bounds has a maximum, then this element is called the greatest lower bound of A or infimum and is written inf(a). 5
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