Lecture 1 Mathematical Preliminaries A. Banerji July 26, 2016 (Delhi School of Economics) Introductory Math Econ July 26, 2016 1 / 27 Outline 1 Preliminaries Sets Logic Sets and Functions Linear Spaces (Delhi School of Economics) Introductory Math Econ July 26, 2016 2 / 27 Preliminaries Sets Basic Concepts Take as understood the notion of a set. Usually use upper case letters for sets and lower case ones for their elements. Notation. a 2 A, b 2= A. A ⊆ B if every element of A also belongs to B. A = B if A ⊆ B and B ⊆ A are both true. A [ B = fxjx 2 A or x 2 Bg. The ‘or’ is inclusive of ‘both’. A \ B = fxjx 2 A and x 2 Bg. What if A and B have no common elements? To retain meaning, we invent the concept of an empty set, ;. A [; = A, A \; = ;. For the latter, note that there’s no element in common between the sets A and ; because the latter does not have any elements. ; ⊆ A, for all A. Every element of ; belongs to A is vacuously true since ; has no elements. This brings us to some logic. (Delhi School of Economics) Introductory Math Econ July 26, 2016 3 / 27 Preliminaries Logic Logic Statements or propositions must be either true or false. P ) Q. If the statement P is true, then the statement Q is true. But if P is false, Q may be true or false. For example, on real numbers, let P = x > 0 and Q = x2 > 0. If P is true, so is Q, but Q may be true even if P is not, e.g. x = −2. We club these together and say P ) Q. Let P = x2 < 0 and Q = x = 5. Then P ) Q is vacuously true. ∼ Q ⇒∼ P is the contrapositive of P ) Q. For example: If x3 ≤ 0, then x ≤ 0 is the contrapositive of “if x > 0, then x3 > 0 ". (Delhi School of Economics) Introductory Math Econ July 26, 2016 4 / 27 Preliminaries Logic Logic Claim. A statement and its contrapositive are equivalent. Proof. Suppose P ) Q is true. Suppose Q is false. Then P must be false, for if not, then via P ) Q, Q would be true, contradicting our assumption that Q is false. On the other hand, suppose P ) Q is false. It follows that P is true and Q is false (for if P is false, then P ) Q is vacuously true). But then, ∼ Q ⇒∼ P cannot be true. Definition Q ) P is called the converse of the statement P ) Q. No relationship between a statement and its converse. e.g., if x > 0 then x2 > 0 is true, but its converse is not. On the other hand, if some statement and its converse are both true, we say P if and only if Q or P , Q. (Delhi School of Economics) Introductory Math Econ July 26, 2016 5 / 27 Preliminaries Logic Quantifiers and Negation 2 logical quantifiers: ‘for all’ and ‘there exists’. P : For all a 2 A, property Π(a) holds. The negation of P (i.e. ∼ P) is the statement: There exists at least one a 2 A s.t. property Π(a) does not hold. e.g. P : For every x 2 <, x2 > 0. The negation of P : There exists x 2 < s.t. x2 ≤ 0. Q : There exists b 2 B s.t. property Θ(b) holds. ∼ Q : For all b 2 B, property Θ(b) does not hold. Note. Order of quantifiers matters. e.g. ‘for all x > 0, there exists y > 0 s.t. y 2 = x’, says that every positive real has a positive square root. This is not the same as ‘there exists y > 0 s.t. for all x > 0; y 2 = x’, which says some number y is the common square root of every positive real. (Delhi School of Economics) Introductory Math Econ July 26, 2016 6 / 27 Preliminaries Logic Logic Let Π(a; b) be a property defined on elements a and b in sets A and B respectively. Let P : For every a 2 A there exists b 2 B s.t. Π(a; b) holds. Then, ∼ P : There exists a 2 A s.t. for all b 2 B, Π(a; b) does not hold. Necessary and Sufficient Conditions Let P ) Q be true. We say Q is necessary for P. Or P is sufficient for Q. So, if P , Q, P is necessary and sufficient for Q. (Delhi School of Economics) Introductory Math Econ July 26, 2016 7 / 27 Preliminaries Sets and Functions Sets Set Difference: A − B = fxjx 2 A and x 2= Bg. Also called the complement of B relative to A. More familiar is the notion of the universal set X, and X − B = Bc. Some set-theoretic ‘laws’ Distributive Laws (i) A \ (B [ C) = (A \ B) [ (A \ C) (ii) A [ (B \ C) = (A [ B) \ (A [ C) Proof. (i) Suppose x 2 LHS. So, x 2 A and x 2 (B [ C). So x 2 A and either x 2 B or x 2 C or both. So, either x 2 (A \ B) or x 2 (A \ C) (or both). Converse? DeMorgan’s Laws (i) A − (B [ C) = (A − B) \ (A − C) or more familiarly, (B [ C)c = Bc \ Cc (ii) A − (B \ C) = (A − B) [ (A − C) or (B \ C)c = Bc [ Cc. (Delhi School of Economics) Introductory Math Econ July 26, 2016 8 / 27 Preliminaries Sets and Functions Sets Arbitrary Unions and Intersections Let A be a collection of sets. Then S A2A = fxjx 2 A for at least one A 2 Ag T A2A = fxjx 2 A for every A 2 Ag Cartesian Products We’ll say (a; b) is an ordered pair if the order of writing these 2 objects matters: i.e. if (a; b) and (b; a) are not the same thing. (Think of points on the plane). Alternatively, we can derive the notion of ordered pair from the more primitive notion of a set as follows. Define (a; b) = ffag; fa; bgg. On the right is a set of 2 sets; the first of these is the singleton that we want to be ‘first’ in the ordered pair. The 2nd is the set of both objects (obviously, fa; bg = fb; ag, so that alone cannot be sufficient to define an ordered pair). Thus (b; a) is defined to be the set ffbg; fa; bgg. Let A and B be sets. We then have (Delhi School of Economics) Introductory Math Econ July 26, 2016 9 / 27 Preliminaries Sets and Functions Functions Definition The Cartesian product A × B = f(x; y)jx 2 A and y 2 Bg. We can formally define a function using the notion of Cartesian product, as follows. Let C; D be 2 sets. A rule of assignment r is a subset of C × D s.t. elements of C appear as first coordinates of ordered pairs belonging to r at most once. The set A of elements of C appearing as first coordinates in r is called the domain of r. The set of elements of D comprising 2nd coordinates of r is called the image set of r. Then Definition A function f is a rule of assignment r along with a set B that contains the image set of r. A is called the domain of f and the image set of r is called the image set or range of f . (Delhi School of Economics) Introductory Math Econ July 26, 2016 10 / 27 Preliminaries Sets and Functions Functions We write f : A ! B and think of f as a rule carrying every element a 2 A to exactly one element b 2 B. Examples. 1. f : < ! < defined by f (x) = x2; 8x 2 <. 2. Using the notation <2 for the Cartesian product < × < (representing 2 2 the plane), let f : < ! < be defined by f (x1; x2) = x1x2; 8(x1; x2) 2 < . 2 2 2 3. g : < ! < defined by g(x1; x2) = (x1x2; x1 + x2); 8(x1; x2) 2 < . 4. Arbitrary Cartesian Products. We first formally define n-tuples in terms of functions. Let X be a set and define the function x : f1; :::; ng ! X. This function is called an n-tuple of elements of X. It’s image at i 2 f1; :::; ng, x(i), is written as xi . The n-tuple is written as n (x1; :::; xn). The order counts. We write X for the set of all n-tuples of elements of X. The leading example is <n, n-dimensional Euclidean space. (Delhi School of Economics) Introductory Math Econ July 26, 2016 11 / 27 Preliminaries Sets and Functions Functions Definition Sn Let A1; :::An be a collection of sets, and let X = 1 Ai . The cartesian n product of this collection of sets, written as A1 × ::: × An or Πi=1Ai , is the set of all n-tuples (x1; :::; xn) such that xi 2 Ai ; 8i 2 f1; :::; ng. 5. Let X be a set. A sequence or infinite sequence of elements of X is a function x : Z++ ! X.(Z++ is the set of positive integers). Sequences are written by collecting images in order. We write x = (x1; x2; ::::::). This definition generalizes the notion of infinite sequences of real numbers. (Delhi School of Economics) Introductory Math Econ July 26, 2016 12 / 27 Preliminaries Sets and Functions Functions - Images, Preimages Let f : A ! B. Definition Let A0 ⊆ A. The image of A0 under f , denoted f (A0), is the set fbjb = f (a); for some a 2 A0g. −1 Let B0 ⊆ B.