Open main menu

Wikibooks β

Set Theory/Sets

< Set Theory

Contents

Sets and elementsEdit

A mathematical set is defined as an unordered collection of distinct elements. That is, elements of a set can be listed in any order and elements occurring more than once are equivalent to occurring only once.

We say that an element is a member of a set. An element of a set can be anything. It's easiest to begin with only numbers as elements. For that reason, most of the examples in this book will only include numbers, but this is only a technique to make the topic less abstract.

TerminologyEdit

For a set   having an element   , the following are all used synonymously:

  is a member of  
  is contained in  
  is included in  
  is an element of the set  
  contains  
  includes  

NotationEdit

We specify a set by specifying its members. The curly brace notation is used for this purpose.

 

is the set containing 1, 2, 3 as members. Or, {mother, this ipod, my school, the planet Jupiter, 12} is also a set. The curly brace notation can be extended to specify a set by specifying a rule for set membership. ("|" means "such that".)

 

is again the set containing 1, 2, and 3 as members.

 

is the set of all natural numbers. This form or representing set can be generalized as:

 

where   is a statement about the variable   . The set defined by above notation is a set of all objects   such that   is true. **** (For a concrete example, consider   . Here the property   is “ ” Thus,   is the set of all real numbers whose square is one.). EXPLANATION: [A set may be defined by a property. For instance, the set of all planets in the solar system, the set of all even integers, the set of all polynomials with real coefficients, and so on. For a property   and an element   of a set   , we write   to indicate that   has the property   . Then the notation   indicates that the set consists of all elements   of   having the property   . The vertical bar | is commonly read as “such that,” and can be also written using a colon instead. So   is an alternative notation for   . For a concrete example, consider  . Here the property   is   . Thus,   is the set of all real numbers (  of   (i.e. 1)) whose square   is one.] ***

A modified epsilon notation is used for set membership. Thus

 

means that   is a member of   . We can also say that   is not a member of   :

 

Characteristics of setsEdit

A set is uniquely identified by its members.

 

Moreover, the sets   are said to be equal if and only if every element of   is also an element of   , and every element of   is an element of   .

All the above expressions specify the same set even though the concept of an even prime is different from the concept of a positive square root. Repetition of members is inconsequential in specifying a set. The expressions

 

all specify the same set.


Sets are unordered. The expressions

 

all specify the same set.

Sets can have other sets as members. There is, for example, the set

 

Some special setsEdit

As stated above, sets are defined by their members. However some sets are given names to ease referencing them.

The set with no members is the empty or null set. The expressions

 

all specify the empty set.

A set with exactly one member is called a singleton. A set with exactly two members is called a doubleton. Thus   is a singleton and   is a doubleton.

Subsets, power sets, set operationsEdit

SubsetsEdit

A set   is a subset of set   if every member of   is a member of   . We use the horseshoe notation to indicate subsets. The expression

 

says that   is a subset of   . The empty set is a subset of every set. Every set is a subset of itself. A proper subset of   is a subset of   that is not identical with   . The expression

 

says that   is a proper subset of   .

Power setsEdit

A power set of a set is the set of all its subsets.   is used for the power set. Note that the empty set and the set itself are members of the power set.

 

UnionEdit

The union of two sets A and B, written  , is the set that contains all the members of A and all the members of B (and nothing else). That is,

 

As an example,

 

IntersectionEdit

The intersection of two sets   , written   , is the set that contains everything that is a member of both   and   (and nothing else). That is,

 

As an example,

 

Two sets are disjoint if their intersection is empty. That is, if   and   are disjoint sets,

 

Relative complementEdit

The relative complement of   , denoted   (sometimes  ), is the set containing all the members of   that are not members of   . That is,

 

As an example,

 

Absolute complementEdit

If we define a universe, or a set containing all of the elements we wish to consider, then we can discuss the absolute complement of a set. For a universe   , define the absolute complement of a subset   of   to be

 

The absolute complement of   is denoted by   (according to the ISO 31-11 standard) if   is fixed.

Some properties of set operationsEdit

Union and intersectionEdit

Based on the preceding definitions, we can derive some useful properties for the operations on sets. The proofs of these properties are left as an exercise to the reader.

The union and intersection operations are symmetric. That is, for sets  

 

Furthermore, they are associative. That is, for sets  

 

Furthermore, union distributes over intersection and intersection distributes over union. That is, for sets  

 

De Morgan's lawsEdit

Two important propositions for sets are De Morgan's laws. They state that, for sets  

 

When   is a universe to which   and   belong, De Morgan's laws can be stated more simply as,

 

Families of setsEdit

A set of sets is usually referred to as a family or collection of sets. Often, families of sets are written with either a script or Fraktur font to easily distinguish them from other sets. For a family of sets   , define the union and intersection of the family by,

 

For a family of sets, we say that it is pairwise disjoint if any two distinct sets we choose from the family are disjoint.

Introduction · Axioms