query: Show that $A \subset B \implies A \cap B = A$
positive_passages.docid: 49143
positive_passages.text: Showing $A \subset B \iff A\cap B=A$
negative_passages.docid: 144266
negative_passages.text: set theory proof of $A\cap B = A \setminus(A\setminus B)$
negative_passages.docid: 79697
negative_passages.text: If $A$ is a subset of $B$, then the closure of $A$ is contained in the closure of $B$.
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: 25710
negative_passages.text: $A ⊂ B$ if and only if $A − B = ∅$
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: 48523
negative_passages.text: Let $A$ be any uncountable set, and let $B$ be a countable subset of $A$. Prove that the cardinality of $A = A - B $
negative_passages.docid: 134770
negative_passages.text: For sets $A,B$, $A \cap (B \setminus A) \subseteq \varnothing$
negative_passages.docid: 37398
negative_passages.text: Proof that$ (a+A)\cap A=\varnothing$ if $A$ is countable
negative_passages.docid: 155929
negative_passages.text: Let $A, B$ be sets. Show that $\mathcal P(A ∩ B) = \mathcal P(A) ∩ \mathcal P(B)$.
negative_passages.docid: 80255
negative_passages.text: How to prove that $|a-b|<\epsilon$ implies $|b|-\epsilon<|a|<|b|+\epsilon$?
negative_passages.docid: 151852
negative_passages.text: How to prove $A \subset B \iff A \cup B = B$
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: 36631
negative_passages.text: Is $[0,1] \cap \Bbb Q$ a compact subset of $\Bbb Q$?
negative_passages.docid: 131983
negative_passages.text: Prove that $A \subset B$ if and only if $A \setminus B = \emptyset$
negative_passages.docid: 10477
negative_passages.text: Finding the closure of A=$\mathbb Q \cap [0,1]$.
negative_passages.docid: 26302
negative_passages.text: Prove that $(X \cap Y) \cup Z = (X \cup Z) \cap (Y \cup Z)$