Sets and applications

Set operations

We now use connectives to define the set operations[1], these allow is to build new set from given ones. Let and be subsets of the set .

DefinitionThe union

The union of and is is the set of all elements that are in or in .

DefinitionThe intersection

The intersection of and is is the set of all elements that are in both and .

Example

If and then and

Method

Let and be finite sets, then we have .

Definition

The sets and disjoint when .

DefinitionThe set difference

The set difference of and , or relative complement of with respect to , written and read “ minus ” or “the complement of with respect to ,” is the set of all elements in that are not in . .

Example

  1. , and then ,

  2. : irrational numbers.

DefinitionThe complement

The complement of the set , written and read “the complement of ,” is the set of all elements of that are not in . That is, .

Some properties

From the proprieties of the logical operations we drive the following,

Method

Cartesian products

The cartesian product of two or more sets is the set of all ordered pairs/n-tuples of the sets. It is most commonly implemented in set theory. Cartesian product is the product of any two sets, but this product is actually ordered i.e, the resultant set contains all possible and ordered pairs such that the first element of the pair belongs to the first set and the second element belongs to the second set. Since their order of appearance is important, we call them first and second elements, respectively.

Definition

The Cartesian product of and is

Example

  1. is called the Cartesian plane.

  2. If and then .

Note

and .

Sub sets

Subsets are a core concept in the study of Set Theory[2], similar to Sets.

Definition

A set is a subset of another set if all elements of the set are elements of the set . We write . If is not a subset of , we write .

Example

.

Method

Let be sets, then

  1. .

  2. .

  3. If and .

Families of sets

The elements of a set may themselves be sets. For example, the power set of a set , is the set of all subsets of . The phrase, “a set of sets” sounds confusing, and so we often use the terms collection and family when we wish to emphasize that the elements of a given set are themselves sets. We would then say that the power set of is the family (or collection) of sets that are subsets of .

Definition

For any set , .

Example

The set then

  1. 4

    M. Mignotte et J. Nervi, Algèbre : licences sciences 1ère année, Ellipses, Paris, 2004

  2. 2

    J. Franchini et J. C. Jacquens, Algèbre : cours, exercices corrigés, travaux dirigés, Ellipses, Paris, 1996.

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