query: If A is independent of C and B is independent of C, is their conjuction also independent of C?
positive_passages.docid: 21097
positive_passages.text: If $A$ and $B$ are independent events, and $A$ and $C$ are independent events, how do I show that $A$ and $B\cup C$ are independent?
negative_passages.docid: 57692
negative_passages.text: If A and B are correlated with C, why are A and B not necessarily correlated?
negative_passages.docid: 11509
negative_passages.text: Relationship between $\operatorname{ord}(ab), \operatorname{ord}(a)$, and $\operatorname{ord}(b)$
negative_passages.docid: 111003
negative_passages.text: Prove that if $A\triangle B = C\triangle B$, then $A = C$
negative_passages.docid: 137626
negative_passages.text: If $a,b,c$ are positive integers,and $\cfrac{a}{b}+\cfrac{b}{c}+\cfrac{c}{a}$ and $\cfrac{b}{a}+\cfrac{c}{b}+\cfrac{a}{c}$ are integers then $a=b=c$
negative_passages.docid: 23066
negative_passages.text: If $\gcd(a, b) = 1$, then $\gcd(ab, c) = \gcd(a, c) \cdot\gcd(b, c)$
negative_passages.docid: 71717
negative_passages.text: Is this always true: $P(A|B) = 1-P(A^c|B)$?
negative_passages.docid: 35242
negative_passages.text: If $a$, $b$, $c$, $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$, and $\frac{a}{c}+\frac{b}{a}+\frac{c}{b}$ are all integers, then $|a|=|b|=|c|$
negative_passages.docid: 70311
negative_passages.text: Trying to prove (A ∩ B) − C = (A − C) ∩ (B − C)
negative_passages.docid: 11168
negative_passages.text: Proof of divisibility: if $a|b$ and $a|(b+c)$ then $a|c$
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: 21726
negative_passages.text: Show that if $(b,c)=1$, then $(a,bc)=(a,b)(a,c)$.
negative_passages.docid: 18915
negative_passages.text: "A and B both are" vs. "A and B are both" vs. "Both A and B are" vs. "Both of A and B are"?
negative_passages.docid: 40609
negative_passages.text: Prove that $f(X)$ and $g(Y)$ are independent if $X$ and $Y$ are independent
negative_passages.docid: 36114
negative_passages.text: conjunction of disjunctions and disjunction of conjunctions
negative_passages.docid: 21893
negative_passages.text: Probability that a < b < c
negative_passages.docid: 32180
negative_passages.text: Why is "$A$ unless $B$" equivalent to $A \lor B$?