What is the countable and uncountable set?

What is the countable and uncountable set?

A set S is countable if there is a bijection f:N→S. An infinite set for which there is no such bijection is called uncountable. Every infinite set S contains a countable subset.

Which of the following is a countable set?

The sets N, Z, the set of all odd natural numbers, and the set of all even natural numbers are examples of sets that are countable and countably infinite.

What is countable sets with example?

The basic examples of (finite) countable sets are sets given by a list of their elements: The set of even prime numbers that contains only one element: {2}. The set of prime numbers less than 10: {2,3,5,7}. The set of diagonals in a regular pentagon ABCDE: {AC,AD,BD,BE,CE}.

How do you show a countable set?

Countable set

1. In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.
2. By definition, a set S is countable if there exists an injective function f : S → N from S to the natural numbers N = {0, 1, 2, 3.}.

How do you prove Q is countable?

By Cartesian Product of Natural Numbers with Itself is Countable, N×N is countable. Hence Q+ is countable, by Domain of Injection to Countable Set is Countable. The map −:q↦−q provides a bijection from Q− to Q+, hence Q− is also countable.

Is power set of Z countable?

{1,2,3,4},N,Z,Q are all countable. R is not countable. The power set P(A) is defined as a set of all possible subsets of A, including the empty set and the whole set.

What is the example of Singleton set?

A singleton set is a set containing exactly one element. For example, {a}, {∅}, and { {a} } are all singleton sets (the lone member of { {a} } is {a}). The cardinality or size of a set is the number of elements it contains. We write the cardinality of set S as |S|.

How do you prove infinitely countable?

A set X is countably infinite if there exists a bijection between X and Z. To prove a set is countably infinite, you only need to show that this definition is satisfied, i.e. you need to show there is a bijection between X and Z.

Is set of rationals countable?

The set of all rationals in [0, 1] is countable. Clearly, we can define a bijection from Q ∩ [0, 1] → N where each rational number is mapped to its index in the above set. Thus the set of all rational numbers in [0, 1] is countably infinite and thus countable.

Why is the power set not countable?

There is no bijection from a set to its power set. From Injection from Set to Power Set, we have that there exists an injection f:N→P(N). From the Cantor-Bernstein-Schröder Theorem, there can be no injection g:P(N)→N. So, by definition, P(N) is not countable.