query: $X$ and $Y$ are sets. Prove $X = \emptyset$ iff $Y = (X \cap Y^c) \cup (X^c \cap Y)$
positive_passages.docid: 155233
positive_passages.text: $A=\emptyset $ if and only if $B = A \bigtriangleup B$
negative_passages.docid: 66948
negative_passages.text: Proving question on sets
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: 88904
negative_passages.text: Help with a proof. Countable sets.
negative_passages.docid: 142993
negative_passages.text: Proof of A ⊆ C and B ⊆ C ⇔ A ∪ B ⊆ C
negative_passages.docid: 3072
negative_passages.text: If the empty set is a subset of every set, why isn't $\{\emptyset,\{a\}\}=\{\{a\}\}$?
negative_passages.docid: 106564
negative_passages.text: Suppose $A,B,C$ are sets. Prove that $A\mathbin\triangle B\subseteq C \iff A\cup C=B\cup C$.
negative_passages.docid: 71653
negative_passages.text: Why is the Cartesian product of a set $A$ and empty set an empty set?
negative_passages.docid: 13421
negative_passages.text: Let $X,Y,Z$ be subspaces of $V$ so that $X$ is a subspace of $Y$. Prove that $Y\cap (X+Z)=X+(Y\cap Z)$
negative_passages.docid: 123613
negative_passages.text: Double Complement of a set proof
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: 10909
negative_passages.text: $|A \cup B| = \mathfrak{c}$, then $A$ or $B$ has cardinality $\mathfrak{c}$.
negative_passages.docid: 145873
negative_passages.text: Let $X$ and $Y$ be finite sets. Then the set $Y^X$ is finite and $\#(Y^X) = (\#Y)^{\#X}$.
negative_passages.docid: 134120
negative_passages.text: Prove statement with a counter-example: For all sets S, T and V, V ∪ (S ∩ T) = (V ∪ S) ∩ T is false
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: 63535
negative_passages.text: Prove that if $A – C = B – C$ and $A\cap C = B\cap C$ then $A = B$
negative_passages.docid: 93771
negative_passages.text: Prove that for any set $A$ there is some $x \notin A$