Numerable set

Numerable Set : Fundamentally in Set Theory, a set X is called numerable in the event that there is a bijection between the set N of natural numbers and the set X. Being N = {0, 1,2,3, …}. ^[one]

Summary

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• 1 Definition
  • 1 Examples
• 2 Propositions
  • 1 Axiom of choice
• 3 Accounting set
  • 1 Equipotent sets
  • 2 Schroeder-Bernstein proposition
• 4 References
• 5 Related Topics
• 6 Sources

Definition

We will say that X is a countable set if there is a bijective function h of N (natural) in X.

Examples

• The set P of all even natural numbers.
• The set of all positive integers t such that t ²≥ 25.
• The set Z itself of all integers, both positive, negative, and zero.
• The set Q of all rational numbers.
• The set A of all algebraic numbers; these are the complex numbersthat are roots of an algebraic equation .
• The set of all m / n fractions that are equivalent to fraction 3/7.
• The set of all decimal numbers in the development of the square root of 8.
• The set of all rational numbers that are in the open interval <0, 1>.

Propositions

1. Principle of the integer minimum. Every non-empty set of natural numbers has a smaller element. It is considered, in some contexts as an axiom, but in Peano’s axiomatic it is a theorem.
2. If the set K, a subset of the set N of all natural numbers, is infinite, then the set K is countable.
3. Let Y be a countable set and K, a subset of Y and an infinite subset. In such a case, K is countable.

Axiom of choice

Given any non-empty set M, there exists a function f that corresponds to every non-empty subset A of M a certain element f (A) of this subset.

Motto .

Let T and S be non-empty sets. If f a function of T in S is surjective, then there exists an injective function g of S in T such that f (g) (x) = id _{S (x)} .

1. Proposition: If there is a surjective function f of N in S, then S is a finite set or a countable set.
2. Proposition The set N × N is a countable set.
3. Proposition: If K is a countable set then K × K is a countable set.
4. Proposición.Sea S _na succession of countable sets, then the union of all such sets from 1 to ∞ is countable.
5. C being a finite or countable set, then the family of all finite successions of elements of C is also a finite or countable family.

Accounting set

A set C is called an accounting set if C is a finite set or a countable set.

Warning

If h is a surjective function of N on S, then the set S is countable.

Equipotent sets

Two sets G and H are called equipotent if there is a bijection q between sets G and H.

Schroeder-Bernstein proposition

Let there be two sets G, H: Then G are equipotent if and only if there are two injective functions: f of G in H and k of H in G.

Uncountable infinity

The interval [0, 1] of real numbers is not countable.

References

1. go back up↑AG Tsipkin: Mathematics Manual for Secondary Education, Mir, Moscow 1985

Linked Topics

• Infinite set