query: Prove that the set of positive integers can be expressed as a countably infinite union of disjoint countably infinite sets.
positive_passages.docid: 24685
positive_passages.text: Expressing a Countable Union of Sets as the Countable Union of Disjoint Sets (e.g. Eliminating Duplicate Intersections)
negative_passages.docid: 102113
negative_passages.text: Proof that the number of proofs is countably infinite
negative_passages.docid: 35692
negative_passages.text: The set of all finite subsets of the natural numbers is countable
negative_passages.docid: 94575
negative_passages.text: Inductive proof that there are infinitely many pairs of consecutive powerful integers?
negative_passages.docid: 33810
negative_passages.text: Proving there are an infinite number of pairs of positive integers $(m,n)$ such that $\frac{m+1}{n}+\frac{n+1}{m}$ is a positive integer
negative_passages.docid: 134071
negative_passages.text: If A is a finite set and B is an infinite set, then B\A is an infinite set
negative_passages.docid: 30231
negative_passages.text: The boundary of union is the union of boundaries when the sets have disjoint closures
negative_passages.docid: 118595
negative_passages.text: Proving that a set is countable by finding a bijection
negative_passages.docid: 138768
negative_passages.text: Showing that the family of Borel sets is countably generated
negative_passages.docid: 148507
negative_passages.text: Why an open set in $R^n$ might not be written as countable union of disjoint open intervals?
negative_passages.docid: 145428
negative_passages.text: Set $A$ is countably infinite if and only if there exists a bijection $f: \mathbb{N} \rightarrow A$
negative_passages.docid: 2403
negative_passages.text: Can a countable set contain uncountably many infinite subsets such that the intersection of any two such distinct subsets is finite?
negative_passages.docid: 32965
negative_passages.text: Proving that for infinite $\kappa$, $|[\kappa]^\lambda|=\kappa^\lambda$
negative_passages.docid: 129163
negative_passages.text: Showing that every positive integer can be represented in this form
negative_passages.docid: 88904
negative_passages.text: Help with a proof. Countable sets.
negative_passages.docid: 84936
negative_passages.text: The set of all finite sequences of members of a countable set is also countable
negative_passages.docid: 119549
negative_passages.text: Prove the even integers are countable (by explicitly giving a bijection $\mathbb{N} \to E$)