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The basics of set theory

Set Theory:

The concept:

Set theory is the mathematical theory of well-defined collections of objects. Each of those objects is called a member, or an element, of the set.
The elements or members of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on.
There are, finite, infinite or even empty sets as we’ll see later.
In principle, specifying a finite set can be done by an explicit list of its members, but specifying infinite sets requires a rule or pattern to indicate membership; for example, the ellipsis in {0, 1, 2, 3, 4, 5, 6, 7, …} indicates that the list of natural numbers ℕ goes on forever.
The empty (or void, or null) set, symbolized by {} or Ø, contains no elements at all. Nonetheless, it has the status of being a set (this is one the axioms of set theory).

More on sets:

The notion of a set is so simple that it is usually introduced informally, and treated as self-evident. In set theory, however, as is usual in mathematics, sets are given axiomatically so their existence and basic properties are postulated by the appropriate formal axioms. The axioms of set theory imply the existence of a set-theoretic universe so rich that all mathematical objects can be interpreted as sets. Also, the formal language of pure set theory allows one to formalize all mathematical notions and arguments. This is why set theory has become the standard foundation for mathematics, as every mathematical object can be viewed as a set.

Notation :

In mathematics, sets are usually described using ”{}” and inside these curly brackets is a list of the elements or a description of the elements of the set. An element of a set is described as follows, for example: If a is an element of a set A, we use the notation a ∈ A and often say ”a in A” instead of ”a an element of A”. The notation a ∉ A indicates that a is not an element of A and is often read as ”a is not in A”.

some short history:

In the late 19th century, a german mathematician named Georg Cantor (1845–1918) developed the theory of sets. This theory emerged from his proof of an important theorem in real analysis. In this proof, Cantor introduced a process for forming sets of real numbers that involved an infinite iteration of the limit operation. Cantor’s novel proof led him to a deeper investigation of sets of real numbers and to his theory of abstract sets. Cantor’s creation now spread through all of mathematics and offers a versatile tool for exploring concepts that previously were considered to be inexpressible, such as infinity and infinite sets.

Between the years 1874 and 1897,Cantor fiercely published articles as his abstract set theory gradually evolved, and became a mathematical discipline. However he was met with broad criticism coming from the mathematicians of his time. Once applications to analysis began to be found, however, attitudes began to change, and by the 1890s Cantor’s ideas and results were gaining acceptance. By 1900, set theory was recognized as a distinct branch of mathematics.

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