Set theory


Set theory is a branch of mathematics (and logic) that is dedicated to studying the characteristics of sets and the operations that can be performed between them.

That is, set theory is an area of ​​study focused on sets. Therefore, it is in charge of analyzing both the attributes they possess and the relationships that can be established between them. That is, its union, intersection, complement or other.

We must remember that a set is a grouping of elements, whether they are numbers, letters, words, functions, symbols, geometric figures or others.

To determine a set, the characteristic that its elements have in common is usually defined. For example, a set A with the integers, positive and even numbers less than 20.

A = {2, 4, 6, 8, 10, 12, 14, 16, 18}

History of set theory

The history of set theory can be traced back to the work of Georg Cantor, a German mathematician of Russian origin, who is considered the father of this discipline.

Among the topics that Cantor studied, for example, that of infinite sets and numerical sets stands out.

Cantor's first work on set theory dates from 1874. In addition, it is worth mentioning that he had a frequent exchange of ideas with the mathematician Richard Dedekind, who contributed to the study of natural numbers.

Numeric sets

Numerical sets are the different groupings in which numbers are classified according to their different characteristics. It is an abstract construction that has an important application in mathematics.

Numerical sets are complex, imaginary, real, irrational, rational, integer, and natural, and can be illustrated in the following Venn diagram:

Complex numbers Imaginary numbers Real numbers Irrational numbers Rational numbers Integer numbers Natural numbers

Set algebra

The algebra of sets encompasses the relationships that can be established between them.

Thus, the following operations stand out:

  • Union of sets: The union of two or more sets contains each element that is contained in at least one of them.
  • Intersection of sets: The intersection of two or more sets includes all the elements that these sets share or have in common.
  • Difference of sets: The difference of one set with respect to another is equal to the elements of the first set minus the elements of the second.
  • Complementary sets: The complement of a set includes all the elements that are not contained in that set (but that do belong to another reference set).
  • Symmetric difference: The symmetric difference of two sets includes all elements that are in one or the other, but not both at the same time.
  • Cartesian product: It is an operation that results in a new set. It contains as elements the ordered pairs or the tuples (ordered series) of the elements that belong to two or more sets. They are ordered pairs if they are two sets and tuples if they are more than two sets.

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