query: If A is finite, B is countable, and A and B are disjoint, prove that A union B is countable.
positive_passages.docid: 66398
positive_passages.text: Union of a finite set and a countably infinite set is countably infinite
negative_passages.docid: 36315
negative_passages.text: Is $[0,1]$ a countable disjoint union of closed sets?
negative_passages.docid: 137707
negative_passages.text: How to prove that $(A\cup B) \cap C \subseteq A\cup (B\cap C)$
negative_passages.docid: 102168
negative_passages.text: Prove that there exists disjoint open subsets $U,V$ of $X$ such that $x \in U$ and $B \subseteq V$.
negative_passages.docid: 60707
negative_passages.text: Show that if $ \ A \cup B = A$ and $ \ A\cap B = A$ then $ \ A = B$
negative_passages.docid: 47434
negative_passages.text: Proof that $A \subseteq B \Leftrightarrow A \cup B = B$
negative_passages.docid: 67709
negative_passages.text: Stuck with proof for $\forall A\forall B(\mathcal{P}(A)\cup\mathcal{P}(B)=\mathcal{P}(A\cup B)\rightarrow A\subseteq B \vee B\subseteq A)$
negative_passages.docid: 35692
negative_passages.text: The set of all finite subsets of the natural numbers is countable
negative_passages.docid: 21892
negative_passages.text: Any open subset of $\Bbb R$ is a countable union of disjoint open intervals
negative_passages.docid: 149317
negative_passages.text: An open subset of $\mathbb{R}$ is an at-most-countable union of disjoint open intervals
negative_passages.docid: 12472
negative_passages.text: Let $X$ and $Y$ be countable sets. Then $X\cup Y$ is countable
negative_passages.docid: 158695
negative_passages.text: Prove that if $AB$ is invertible then $B$ is invertible.
negative_passages.docid: 147213
negative_passages.text: Prove that finite unions of compact sets are compact
negative_passages.docid: 90434
negative_passages.text: Prove for every two sets $A$ and $B$ that $A-B$, ${B-A}$ and $A \cap B$ are pairwise disjoint.
negative_passages.docid: 32774
negative_passages.text: Finite Cartesian Product of Countable sets is countable?
negative_passages.docid: 136306
negative_passages.text: Let $A$ be a finite set and let $B$ be a subset of $A$ with $|A|=n$,$|B|=m$ and $0<m<n$. Count the subsets of $A$ containing $ B.$
negative_passages.docid: 136006
negative_passages.text: Prove that any countable cartesian product of countable sets is countable