Çınlar Erhan - Real and Convex Analysis

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Erhan nlar and Robert J. Vanderbei Undergraduate Texts in Mathematics Real and Convex Analysis 2013 10.1007/978-1-4614-5257-7_1

Springer Science+Business Media New York 2013

1. Sets and Functions

Erhan nlar 1 and Robert J. Vanderbei 1

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

Abstract

This introductory chapter is devoted to general notions regarding sets, functions, sequences, and series. We aim to introduce and review the basic notation, terminology, conventions, and elementary facts.

This introductory chapter is devoted to general notions regarding sets, functions, sequences, and series. We aim to introduce and review the basic notation, terminology, conventions, and elementary facts.

A. Sets

A set is a collection of some objects. Given a set, the objects that form it are called its elements . Given a set A , we write x A to mean that x is an element of A . To say that x A , we also say x is in A , or x is a member of A , or x belongs to A , or A includes x .

One can specify a set by listing its elements inside curly braces, but doing so is not feasible in most cases. More often we specify a set by precisely describing its elements. For example, a,b,c } is the set whose elements are a,b , and c , and is the set of all numbers exceeding 2.7. The following are some special sets:

• : The empty set . It has no elements.
• : The set of natural numbers .
• : The set of strictly positive integers .
• : The set of integers .
• : The set of rationals .
• : The set of reals .
• : The set of positive reals .
• : The closed interval with endpoints a and b .
• , defined for numbers a and b with a < b : The open interval with endpoints a and b .

We assume that the reader is familiar with these sets. For example, we take it for granted that real numbers are limits of rationals.

Subsets

A set A is said to be a subset of a set B if every element of A is an element of B . We write A B or B A to indicate this and say A is contained in B , or B contains A , to the same effect. The sets A and B are the same, if and only if A B and A B , and then we write A = B . For the contrary situations, we write A B when A and B are not the same. The set A is called a proper subset of B if A is a subset of B , and A and B are not the same.

The empty set is a subset of every set. Let A be a set. The claim is that A , that is, that every element of is also an element of A , or equivalently, there is no element of that does not belong to A . But the last phrase is true simply because has no elements.

Set Operations

Let A and B be sets. Their union , denoted by A B , is the set consisting of all elements that belong to either A or B (or both). Their intersection , denoted by A B , is the set of all elements that belong to both A and B . The complement of A in B , denoted by B A , is the set of all elements of B that are not in A . Sometimes, when B is understood from context, B A is also called the complement of A and is denoted by A c . Regarding these operations, the following statements hold:

• Commutative laws:

  
• Associative laws:

  
• Distributive laws:

  

The associative laws show that A B C and A B C have unambiguous meanings.

Definitions of unions and intersections can be extended to arbitrary collections of sets. Let I be a set. For each i in I , let A i be a set. The union of the sets A i , i I , is the set A such that x A if and only if x A i for some i in I . The intersection of them is the set A such that x A if and only if x A i for every i in I . The following notations are used to denote the union and intersection respectively:

When , it is customary to write

All of these notations follow the conventions for sums of numbers. For instance,

Disjoint Sets

Two sets are said to be disjoint if their intersection is empty; that is, if they have no elements in common. A collection of sets is said to be disjointed if A i and A j are disjoint for every i and j in I with i j .

Products of Sets

Let A and B be sets. Their product , denoted by A B , is the set of all pairs ( x , y ) with x in A and y in B . It is also called the rectangle with sides A and B .

If are sets, then their product is the set of all n -tuples where . This product is called, variously, a rectangle, or a box, or an n -dimensional box. If

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