query: Let $A,B,C$ be sets such that $\mathcal{P}(A) \cup \mathcal{P}(B) = \mathcal{P}(C)$. Then either $A=C$ or $B=C$.
positive_passages.docid: 67709
positive_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: 87327
negative_passages.text: Unions and intersections: $(A \cup B = A ∪ C) \land (A \cap B = A ∩ C) \implies B = C.$
negative_passages.docid: 1962
negative_passages.text: Polynomial $P(a)=b,P(b)=c,P(c)=a$
negative_passages.docid: 120094
negative_passages.text: Prove, that for any A, B, C sets, $A\times (B\setminus C) = (A\times B)\setminus (A\times C)$ is true.
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: 70552
negative_passages.text: Prove that $\mathcal{P}(A)⊆ \mathcal{P}(B)$ if and only if $A⊆B$.
negative_passages.docid: 68918
negative_passages.text: Show that $A\cap B\subseteq A$ and $A\subseteq A\cup B$
negative_passages.docid: 76826
negative_passages.text: Are there sets $A$ and $B$ such that $A \in B$ and $B \in A$?
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: 62770
negative_passages.text: Does $a\mid bc$ imply $a\mid b$ or $a\mid c$?
negative_passages.docid: 64586
negative_passages.text: How to prove that $\neg((A\cap B)\cup (\neg A \cap C)) = (A\cap\neg B)\cup (\neg A\cap \neg C)$
negative_passages.docid: 47434
negative_passages.text: Proof that $A \subseteq B \Leftrightarrow A \cup B = B$
negative_passages.docid: 37183
negative_passages.text: Show that the set of all $x$ such that $x \in P$ and $x \notin Q$ exists
negative_passages.docid: 113892
negative_passages.text: Show $A \in \cal F$ exists, such that $\mathbb P(A)=1$ and $\mathbb Q(A)=0$.
negative_passages.docid: 63267
negative_passages.text: Under what conditions on the sets $A$ and $B$ will it be true that $A\times B= B\times A$
negative_passages.docid: 46056
negative_passages.text: If $\mathcal{E}$ are subsets of $X$ and $A \subset X$. Show that $\mathcal{A}\mathcal{(E \cap}$ A)=$\mathcal{A}\mathcal{(E)} \cap$ A
negative_passages.docid: 136835
negative_passages.text: ZF Set Theory - If $|A|^{|B|} = |B|$, then $|A| = 1 = |B|.$