query: Let $a,b,c,d$ be real numbers such that $a<b < c<d$. Express the set $[a,b] \cup [c,d]$ as the difference of two sets
positive_passages.docid: 148234
positive_passages.text: Express set $[a,b] \cup [c,d]$ as the difference of two intervals
negative_passages.docid: 80876
negative_passages.text: Proof of $A\cap(B\cup C) = (A\cap B)\cup(A\cap C)$
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: 46831
negative_passages.text: Show that $ a，b, \sqrt{a}+ \sqrt{b} \in\mathbb Q \implies \sqrt{a},\sqrt{b} \in\mathbb Q $
negative_passages.docid: 5369
negative_passages.text: Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$
negative_passages.docid: 43246
negative_passages.text: Showing that if $a<b$, then $a^{1/n} < b^{1/n}$
negative_passages.docid: 47434
negative_passages.text: Proof that $A \subseteq B \Leftrightarrow A \cup B = B$
negative_passages.docid: 83996
negative_passages.text: For any two sets $A,B$ , $|A|\leq|B|$ or $|B|\leq|A|$
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: 76826
negative_passages.text: Are there sets $A$ and $B$ such that $A \in B$ and $B \in A$?
negative_passages.docid: 13167
negative_passages.text: Prove $A = (A \setminus B) \cup (A \cap B)$
negative_passages.docid: 115606
negative_passages.text: Proofs involving $ (A\setminus B) \cup (A \cap B)$
negative_passages.docid: 77902
negative_passages.text: If $A,B,$ and $C$ are sets, then $A\times(B-C)$ = $(A \times B)$ $-$ $(A \times C)$.
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: 79165
negative_passages.text: How to show that $P(A\cup B) = P(A) + P(B) - P(A\cap B)$
negative_passages.docid: 39708
negative_passages.text: Proving $(A\le B)\vee (B\le A)$ for sets $A$ and $B$
negative_passages.docid: 39621
negative_passages.text: Let a, b, c, d be rational numbers...