# Set Theory/Sets

## Sets and elements[edit | edit source]

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.

### Terminology[edit | edit source]

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

### Notation[edit | edit source]

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 sets[edit | edit source]

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 sets[edit | edit source]

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 operations[edit | edit source]

### Subsets[edit | edit source]

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 sets[edit | edit source]

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.

### Union[edit | edit source]

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,

### Intersection[edit | edit source]

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 complement[edit | edit source]

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 complement[edit | edit source]

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 operations[edit | edit source]

### Union and intersection[edit | edit source]

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 laws[edit | edit source]

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 sets[edit | edit source]

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.