Some_basic_mathematical_concepts

# Some_basic_mathematical_concepts - University of Waterloo...

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University of Waterloo Department of Economics Economics 301 - Microeconomic Theory 2 - Section 001 Fall 2011 Lecture note - Mathematical Review: 1. Set notation: A set is a collection of distinct objects. These objects are called the points or elements of the set. Examples: A set that contains all positive integers can be written as S = {1, 2, 3, ………} A set of all real numbers is . To indicate that a particular element x belongs to a set S , we write x S If r does not belong to S , we write r S Set are characterized by the properties of their elements. S ={ x | x has property s } Example: + is the set of all nonnegative real numbers + = { x | x + and x 0} Or, + = { x | x 0} Set A is a subset of B or A is contained in B, then relationship is written as

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A B The null set or empty set ( ) includes no elements. The union of two sets is denoted by A B , is the set of all elements contained in A , in B or in both A and B . A B ={ x | x A or x B } Example: A = {3, 5, 7} B = {2, 3, 4, 8} A B = {2, 3, 4, 5, 7, 8} The intersection of two sets is denoted by A B , is the set of all elements contained in both A and B. A B ={ x | x A or x B } Example: A = {3, 5, 7} B = {2, 3, 4, 8} A B = { 3 } If two set have no common elements, they are called disjoint sets . Set A and B are disjoint if A B = . In general, the order of the set does not matter. In sets of certain types, though, order maters. These are called ordered pairs and written as (a, b). The Cartesian product of two sets, x × y , is the set of all ordered pairs, where the first element comes from set x and the second element comes from set y . x × y = {( a, b ) | a x and b y } Example 1: Euclidean 2-space is × = 2 Example 2: x = {1,2} and y = {3,4} x × y = {(1,3)(1,4)(2,3)(2,4)} Complement of set A is . 2. Relations and Functions:
Since any ordered pair associates an x value with a y value, any collections of ordered pairs- the subset of the Cartesian product will constitute a relation between x and y . A function is a set of ordered pairs with property that any x value uniquely determines a y value. Example: The set {x, y | y =2x} is a set of ordered pairs The set {x, y | y x} is a set of ordered pairs including A function [ y = f ( x )] is also called a mapping or transformation. f represents a particular rule of mapping: f : x y In the function y = f(x), x is referred to as the argument of the function and y is called the value of the function. The set of all permissible values that x can take is called the domain of the function. The

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Some_basic_mathematical_concepts - University of Waterloo...

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