query: Boundary of union of two sets equals the union of their boundaries
positive_passages.docid: 30231
positive_passages.text: The boundary of union is the union of boundaries when the sets have disjoint closures
negative_passages.docid: 116527
negative_passages.text: Validating a proof for necessary and sufficient conditions on sets A & B such that $P(A) \cup P(B) = P(A\cup B)$
negative_passages.docid: 140668
negative_passages.text: Union of $2$ set $A, B$ where $ A$ is a subset of $B$
negative_passages.docid: 23442
negative_passages.text: Cardinality of the union of infinite and countable sets
negative_passages.docid: 123613
negative_passages.text: Double Complement of a set proof
negative_passages.docid: 13132
negative_passages.text: If the union of two sets is contained in the intersection, then one is contained in the other ($[A\cup B\subseteq A\cap B]\implies A \subseteq B$)
negative_passages.docid: 114599
negative_passages.text: $A \cup B$ and $A \cap B$ connected $\implies A$ and $B$ are connected
negative_passages.docid: 158850
negative_passages.text: Is it true that a boundary of a simply connected and bounded set is connected in $\mathbb{C}$?
negative_passages.docid: 86505
negative_passages.text: Let $A,B \subset \mathbb{R}$. Show that $\sup(A \cup B) = \max\{\sup A, \sup B\}$
negative_passages.docid: 28641
negative_passages.text: Suppose $Y\subset X$ and $X,Y$ are connected and $A,B$ form separation for $X-Y$ then, prove that $Y\cup A$ and $Y\cup B$ are connected
negative_passages.docid: 5132
negative_passages.text: Given $A\cap \overline{B}\neq \emptyset$, prove $A\cup B$ connected.
negative_passages.docid: 46240
negative_passages.text: Show that if $A$ and $B$ are sets, then $(A\cap B) \cup (A\cap \overline{B})=A$.
negative_passages.docid: 12404
negative_passages.text: Union of definable sets
negative_passages.docid: 64871
negative_passages.text: Closure of Union contains Union of Closures
negative_passages.docid: 106434
negative_passages.text: Prove that for a dense set $A$ and an open set $U$, $U\subset \overline{A\cap U}$
negative_passages.docid: 88982
negative_passages.text: Prove that for any two sets $A$ and $B$, $A\notin B$ or $B\notin A$
negative_passages.docid: 48797
negative_passages.text: Proving Two Sets are Equal - Infinite Sets - Example