CHAPTER 1-A SET THEORY AND THE REAL LINE 1. Ordered Sets Let A be a set • x 2 A means \x is a member of A" • x2 = A means \x is not a member of A" • the set with no elements is called the empty set, denoted ; • let B be another set: A ⊆ B means \every element of A is in B". We say that A is a subset of B, and that B is a superset of A; when A is a proper subset of B we will write A ⊂ B and means that 8x 2 A, x 2 B and 9y 2 B, y2 = A; note that A ⊆ B and B ⊆ A if an only if A = B. A * B means that A is not a subset of B • A [ B is the set of elements that belong to A or belong to B • A \ B is the set of elements that belong to both A and B • A − B is the set of elements of A that are not members of B • Ac is the set of elements that are not in A; it is called the complement of A • The Cartesian product of A and B is A × B = f(x; y): x 2 A; y 2 Bg: • the set formed by all subsets of A including the empty set is denoted 2A. In what follows, all the sets are suppose to be nonempty. Definition 1.1. Given sets A and B, a (binary) relation R from A to B is a subset of A × B. Notation 1.2. We will write most often xRy for (x; y) 2 R; x R y means (x; y) 2= R. Definition 1.3. Given sets A and B and the binary relation R from A to B, the inverse relation of R is R−1 = f(y; x):(x; y) 2 Rg: Definition 1.4. Let R be a binary relation defined on a set A. We say that R is (1) Reflexive, if 8x 2 A, xRx. (2) Irreflexive, if 8x 2 A, x R x. (3) Symmetric, if xRy implies yRx. (4) Antisymmetric, if xRy and yRx imply x = y. 1 2 (5) Transitive, if xRy and yRz imply xRz. (6) Total, if 8x; y 2 A, xRy or yRx or both. Definition 1.5. Let R be a relation defined on a set A. The relation R is an equivalence relation if it is reflexive, symmetric and transitive. Remark 1.6. An equivalence relation R partitions A into disjoint equivalence classes such that x and y are in the same class if and only if xRy. For, writing [x] = fz : zRxg, we have x 2 [x] , so [fx : x 2 Ag = A. Also, for any x; y 2 A, either [x] = [y] or [x]\[y] = ;. Definition 1.7. Let R be a relation defined on a set A. The relation R is a partial order it is reflexive, antisymmetric and transitive. It is an order if it is total, reflexive, antisymmetric and transitive. We will use ≤ (or ≥) instead of R to denote an order relation. When x ≤ y and x 6= y, we will write x < y. Definition 1.8. An ordered set (partially ordered set, or poset) is a set in which an order (a partial order) is defined. Definition 1.9. Let (A; ≤) be a poset. Let S ⊆ A be nonempty. An element b 2 A is a maximal element of S if for any x 2 S, b ≤ x implies x = b. Definition 1.10. Let (A; ≤) be a poset. Let S ⊆ A be nonempty. An element b 2 A is a minimal element of S if for any x 2 S, x ≤ c implies x = c. If the order is only partial, then for a maximal (minimal) element b of A, it could be neither x ≤ b nor b ≤ x. 2 2 Example 1.11. Let (R ; ≤) where for x = (x1; x2), y = (y1; y2) of R , x ≤ y if and only if x1 ≤ y1 and y1 ≤ y2. This relation is a partial order (the Pareto order). Given 2 S ⊆ R , a maximal element of S is called a Pareto optimal point of S. For instance, in 2 R , (1; 1) is Pareto optimal in (−∞; 1] × (−∞; 1], but not in (−∞; 1] × (−∞; 2]. Consider 2 now S = f(x; y) 2 R : x ≥ 0; y ≥ 0; x + y ≤ 1g. Every point of the segment joining (0; 1) and (1; 0) is Pareto optimal (efficient frontier of S). Let T = f(x; y): y ≤ −x2g. Then f(x; −x2) : 0 ≤ xg is the set of maximals points of A and there are no minimal elements. Example 1.12. Let R be the set of real numbers, and define the relation ≤ by x ≤ y if and only if y − x ≥ 0. It is an order relation. Example 1.13. Let AR be the set of functions f : A ! R. Define f ≤ g in AR if and only if f(x) ≤ g(x) for all x 2 A. It is a partial order. 3 Definition 1.14. Let (A; ≤) be an ordered space. Let S ⊆ A be nonempty. • An element b 2 A is an upper bound of S if and only if for all x 2 S, x ≤ b. • An element c 2 A is a lower bound of S if and only if for all x 2 S, c ≤ x. • S is upper bounded if and only if S has upper bounds. • S is lower bounded if and only if S has lower bounds. • S is bounded if and only if it is both upper and lower bounded. • An element b 2 A is the supremum of S, b = sup S, if and only if it is the least of the upper bounds of S: (1) 8x 2 S, x ≤ b and (2) 8x 2 S, x ≤ y, implies b ≤ y. • An element c 2 A is the infimum of S, c = inf S, if and only if it is the greatest of the lower bounds of S: (1) 8x 2 S, c ≤ x and (2) 8x 2 S, y ≤ x implies y ≤ c. • If sup S 2 S, then the supremum is a maximum. • If inf S 2 S, then the infimum is a minimum. Note that to be bounded in an arbitrary ordered set is not enough to have supremum and/or infimum. See problem 3 in Problem List 1. Definition 1.15. An order has the least upper bound property if every upper bounded set admits a supremum. An order has the greatest lower bound property if every lower bounded set admits an infimum. It can be proved that an order has the least upper bound property iff it has the greatest lower bound property. Definition 1.16. Given sets A and B, a relation f from A to B, written f : A −! B, is a function provided (x; y) 2 f and (x; z) 2 f imply y = z. The domain of the function, domf = fx :(x; y) 2 fg, is the set of all possible first elements of the ordered pairs in f. The image of the function, im f = fy :(x; y) 2 fg, is the set of all possible second elements of the ordered pairs in f. Notation 1.17. f(x) = y means (x; y) 2 f. Given a subset E of A f(E) = ff(x): x 2 Eg is the image of E by f. Given a subset E of B f −1(B) = fx : f(x) 2 Bg is the inverse image of E by f. The set of all functions between A and B is denoted BA. Definition 1.18. The function f : A −! B is onto iff f(A) = B. Definition 1.19. The function f : A −! B is one-to-one iff (x1; y) 2 f and (x2; y) 2 f imply x1 = x2 4 Definition 1.20. The function f : A −! B is bijective iff it is both onto and one-to-one. Given a function f, when is the inverse relation f −1 a function? Proposition 1.21. Given the function f : A −! B, the inverse relation f −1 is a function iff f is bijective. f −1 : B −! A is called the inverse of f. 2. Finite, Countable and Uncountable Sets Definition 2.1. We say that two sets A and B have the same cardinal number and write A ∼ B, iff there is a bijective mapping between A and B. Definition 2.2. Let n 2 N and let Jn = f1; : : : ; ng. We say that the set A is (1) Finite, if A ∼ Jn for some n. (2) Infinite, if A is not finite. (3) Countable, if A ∼ N. (4) Uncountable, if A is neither finite nor countable. (5) At most countable, if A is finite or countable. n Example 2.3. Z is countable: consider f : N −! Z be defined as follows: f(n) = − 2 if n n+1 even, f(n) = 2 if n odd. It is a bijection. Example 2.4. N × N is countable: consider f : N × N −! N be defined as follows: f(n; m) = 2n−1(2m − 1). It is a bijection. This can be also proved by the following geometric argument. Arrange N × N as an infinite matrix (1; 1) (1; 2) (1; 3) ::: (2; 1) (2; 2) (2; 3) ::: (3; 1) (3; 2) (3; 3) ::: . so that (n; m) is in the nth row and mth column. We can enumerate N × N without repetitions as (1; 1); (2; 1); (1; 2); (3; 1); (2; 2); (1; 3);::: The next result means that the cardinality of A is always smaller than the cardinality of the power set of A (The set formed by all subsets of A, including the empty set).