query: Prove A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C )
positive_passages.docid: 145058
positive_passages.text: Prove $ (A \cup B) \cap C$ = $(A \cap C) \cup (B \cap C) $
negative_passages.docid: 68258
negative_passages.text: Let $A,B,C$ be sets, and $B \cap C=\emptyset$. Show $|A^{B \cup C}|=|A^B \times A^C|$
negative_passages.docid: 5149
negative_passages.text: Show $\binom{a}{b}\binom{b}{c}=\binom{a}{a-c}\binom{a-c}{a-b}$
negative_passages.docid: 33524
negative_passages.text: Prove the following $A \setminus (B\setminus C) = (A\setminus B)\cap (A \cup C)$
negative_passages.docid: 134914
negative_passages.text: Let $A,B$ and $U$ be sets so that $A\subseteq U$ and $B\subseteq U$. Prove that $A\subseteq B$ iff $(U\setminus B)\subseteq(U\setminus A)$.
negative_passages.docid: 133699
negative_passages.text: Proving that there is a bijection between $A$ and $B$.
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: 41080
negative_passages.text: If A, B, and C are set and $A \subset B ,B \subset C \rightarrow A \subset C$
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: 130156
negative_passages.text: Image of the union and intersection of sets.
negative_passages.docid: 21726
negative_passages.text: Show that if $(b,c)=1$, then $(a,bc)=(a,b)(a,c)$.
negative_passages.docid: 49578
negative_passages.text: Proving $a\mathbb{Z}\cap b\mathbb{Z}=[a,b]\mathbb{Z}$
negative_passages.docid: 79165
negative_passages.text: How to show that $P(A\cup B) = P(A) + P(B) - P(A\cap B)$
negative_passages.docid: 76049
negative_passages.text: Prove that if $a|b$ and $a|c$ then $a|(sb+tc)$ for all $s, t \in \mathbb{Z}$
negative_passages.docid: 64656
negative_passages.text: how do you prove that $A\times (B\setminus C) = (A\times B) \setminus (A\times C)$?
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: 39708
negative_passages.text: Proving $(A\le B)\vee (B\le A)$ for sets $A$ and $B$