DEFINITION1: The set with at least two distinct elements and satisfying the following five axioms is called the set of real numbers and each element of is called a real number:
I. AXIOMS OF ADDITION:
The function defined as for each in satisfies the following properties:
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Let be a set. A set is called an “index set” of the set if and that set are equipollent. I.e., a set is called an index set of if there exists a bijective function (injective and surjective) between and . As is clear from the definition, the number of all the index sets of a set may be more than one, even infinite. Besides, we can say from the definition: there exists at least one index set of any set. Because, the function is bijective. This is a trivial example since the set is indexed by itself. If we indexed every set with itself, the indexing operation would be meaningless. We must give a reinforcement example: Let be . We will give three index sets for this set: The sets , and can be given as index sets of . (Can be given more index sets for this set). The type of index set that is widely used by the mathematicians is since it one by one counts the elements of . In general, can be chosen as an index set for a set with elements. An index set of a set is directly associated with the cardinality of that set. Since the cardinality relation on any family of sets is an equivalence relation, an index set of a set can be actually considered as the most reasonable “representation of class” of the equivalence class of a set. For example, the most reasonable index set of any countable set is naturally the set of the natural numbers. , or is widely used as an index set for a set that is equipollent with the set of the real numbers.
By the end of the 19th century, the mathematicians had called “set” a collection of any objects. For example, the set of the natural numbers, the set of integers, the set of even numbers, the set of the real numbers, the set of any sets, the set of all the sets. We can give further similar examples. All the mathematicians had no doubt about that the unique condition to be a set was to gather any objects by the time Bertrand Russell’s paradox emerged. Russell had proved that when the term "set" is defined as “a collection of any objects”, there emerges a paradox in the set theory . Now, let’s examine Russell’s paradox and its proof: Assume that a collection of any objects is a set. In that case, the collection of all the sets is a set. We denote this set by . Hence, any set is an element of the set i.e., if is a set, then . Since is also a set, then . Let’s construct a subset of :
Which proposition of the two is the true one or ?
i) Let’s assume that the proposition is the true one. In that case, since any element of is a set that is not an element of itself, the proposition is true.
ii) Let’s assume that the proposition is the true one. In that case, according to the definition of , the proposition is true. As a result, the following proposition has been proved:
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The concept of the “set” is one of the basic concepts of mathematics. In spite of this fact, there is no definition agreed on by the authority. Some mathematicians define sets as “the class of objects that have certain properties”. Although this definition is widespread, there are deficiencies.
Objects that form the set are called “elements”. The sets are represented with capital letters such as , , , , , and the elements of sets are represented with lower-case letter such as , , , , . If is an element of , this case is denoted by , if is not an element of , this case is denoted by . There are three types of representations to display the sets:
1. List method: In this representation, the elements of set are written into the curly braces, by putting the commas between the elements. In a set, an element cannot be written twice. As an example, can be given.
2. Venn diagram: In this representation, the elements of set are written inside a circle or rectangle. Let’s show the above example, , by using Venn diagram:
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