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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.
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What is the Difference between a finite set and countable set?
all finite set is countable.but,countable can be finite or infinite
What is difference between finite and countable sets?
A finite set is one that contains a specific, limited number of elements, while a countable set can be either finite or infinite but can be put into a one-to-one correspondence with the natural numbers. In other words, a countable set has the same size as some subset of the natural numbers, meaning it can be enumerated. For example, the set of all integers is countable, even though it is infinite, whereas the set of all even integers is also countable.
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.
What is fenite set?
A finite set is a set that contains a limited or countable number of elements. For example, the set of natural numbers from 1 to 10 is a finite set because it has exactly ten elements. In contrast, an infinite set has no bounds and contains an uncountable number of elements, such as the set of all natural numbers. Finite sets can be characterized by their cardinality, which is a measure of the number of elements in the set.
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
What does it mean by countable plate?
A countable plate refers to a type of mathematical object in set theory, where a set is considered countable if its elements can be put into a one-to-one correspondence with the natural numbers. This means that even if the set is infinite, it can still be "counted" in the sense that its elements can be listed sequentially. Countable sets include finite sets and countably infinite sets, such as the set of integers or rational numbers. In some contexts, "countable plate" might also refer to a specific type of surface or geometric object, but the term is less commonly used in that sense.
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.
What is difference between finite and countable sets?
A finite set is one that contains a specific, limited number of elements, while a countable set can be either finite or infinite but can be put into a one-to-one correspondence with the natural numbers. In other words, a countable set has the same size as some subset of the natural numbers, meaning it can be enumerated. For example, the set of all integers is countable, even though it is infinite, whereas the set of all even integers is also countable.
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.
