0
A countable set is an infinite set that can be put into a one-to one correspondence with the counting numbers. In other words, it is possible to arrange all of the elements of the set in a sequence with a first element, a second element, and so on.
Georg Cantor proved that the rational numbers are countable, but the real numbers are not.
One proof (not Cantor's) that the rationals are countable: Choose any rational number, write it out in its simplest form. Whatever number you wrote is represented by a finitely long string of either the numerals or the division slash (possibly preceded the negative sign). If you consider a base-eleven number system with the division slash as the eleventh numeral, then whatever rational number you just wrote out corresponds directly and unambiguously to one specific integer. Since there exists a mapping scheme that assigns any arbitrary rational number to a specific integer, the rational numbers are countable.
What else can I help you with?
What is the Difference between a finite set and countable set?
all finite set is countable.but,countable can be finite or infinite
What definition proves that the set of all odd natural numbers is a countable set?
It is NOT a 'countable set'. It is an infinite set. 1, 3, 5, 7, 9, 11, ... you can count to infinity and keep going.
What are the different types a of set?
A null set, a finite set, a countable infinite set and an uncountably infinite set.
Can infinity be an element of a set?
Yes.The set of {Aleph-null, Aleph-one, ...}, which is the set of the different infinities, has infinity as an element.Aleph-null is the countable infinity.
What is the difference between classical set theory and fuzzy set theory?
Classical theory is a reference to established theory. Fuzzy set theory is a reference to theories that are not widely accepted.
Can set of rational numbers forms a borel set?
Yes. the set of rational numbers is a countable set which can be generated from repeatedly taking countable union, countable intersection and countable complement, etc. Therefore, it is a Borel Set.
Is every subset of a countable set is countable?
Yes.
What is the Difference between a finite set and countable set?
all finite set is countable.but,countable can be finite or infinite
Is the set of all irrational number countable?
No, the set of all irrational numbers is not countable. Countable sets are those that can be put into a one-to-one correspondence with the natural numbers (1, 2, 3, ...). The set of irrational numbers is uncountable because it has a higher cardinality than the set of natural numbers. This was proven by Georg Cantor using his diagonalization argument.
Is counting measure indeed a measure and is this always sigma-finite?
It is a measure, but it isn't always sigma-finite. Take your space X = [0,1], and u = counting measure if u(E) < infinity, then E is a finite set, but there is no way to cover the uncountable set [0,1] by a countable collection of finite sets.
What definition proves that the set of all odd natural numbers is a countable set?
It is NOT a 'countable set'. It is an infinite set. 1, 3, 5, 7, 9, 11, ... you can count to infinity and keep going.
How can you prove that the set of all languages that are not recursively enumerable is not countable?
One way to prove that the set of all languages that are not recursively enumerable is not countable is by using a diagonalization argument. This involves assuming that the set is countable and then constructing a language that is not in the set, leading to a contradiction. This contradiction shows that the set of all languages that are not recursively enumerable is uncountable.
What are the different types a of set?
A null set, a finite set, a countable infinite set and an uncountably infinite set.
Is set R of real numbers is countable set or not?
It is uncountable, because it contains infinite amount of numbers
How prove that the set of irrational numbers are uncountable?
Proof By Contradiction:Claim: R\Q = Set of irrationals is countable.Then R = Q union (R\Q)Since Q is countable, and R\Q is countable (by claim), R is countable because the union of countable sets is countable.But this is a contradiction since R is uncountable (Cantor's Diagonal Argument).Thus, R\Q is uncountable.
Is personality a countable noun?
Yes, the noun personality is a countable noun; the plural form is personalities.
Can infinity be an element of a set?
Yes.The set of {Aleph-null, Aleph-one, ...}, which is the set of the different infinities, has infinity as an element.Aleph-null is the countable infinity.
