query: How to prove that $A\subseteq B,$ given that $A\cup C\subseteq B\cup C$ and $A\cap C\subseteq B\cap C$ for all sets $C.$
positive_passages.docid: 43068
positive_passages.text: Let A and B be sets. Show that A is a subset of B if and only if for any set C, one has A union C is a subset of B union 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: 118239
negative_passages.text: If $A, B, C$ are sets prove that $(A\setminus B)\setminus C\subseteq A\setminus(B\setminus C)$
negative_passages.docid: 56399
negative_passages.text: Proof for $A \cup B = B$ if and only if $A \subset B$
negative_passages.docid: 102917
negative_passages.text: Proving statement - $(A \setminus B) \cup (A \setminus C) = B\Leftrightarrow A=B , C\cap B=\varnothing$
negative_passages.docid: 30770
negative_passages.text: Prove that $C\left(\bigcup_{\alpha\in I} A_\alpha\right)=\bigcap_{\alpha\in I}C\left(A_\alpha\right)$
negative_passages.docid: 143735
negative_passages.text: Prove/Disprove: For any sets $X$ and $Y$, $\overline{X\cap Y} = \bar{X}\cup\bar{Y}$
negative_passages.docid: 134120
negative_passages.text: Prove statement with a counter-example: For all sets S, T and V, V ∪ (S ∩ T) = (V ∪ S) ∩ T is false
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: 13167
negative_passages.text: Prove $A = (A \setminus B) \cup (A \cap B)$
negative_passages.docid: 5132
negative_passages.text: Given $A\cap \overline{B}\neq \emptyset$, prove $A\cup B$ connected.
negative_passages.docid: 13415
negative_passages.text: How to show if $A \subseteq B$, then $A \cap B = A$?
negative_passages.docid: 77859
negative_passages.text: Prove or find a counter example to the claim that for all sets A,B,C if A ∩ B = B ∩ C = A ∩ C = Ø then A∩B∩C ≠ Ø
negative_passages.docid: 115606
negative_passages.text: Proofs involving $ (A\setminus B) \cup (A \cap B)$
negative_passages.docid: 151852
negative_passages.text: How to prove $A \subset B \iff A \cup B = B$
negative_passages.docid: 66563
negative_passages.text: Is $ (A∪B)×(C∩D)=(A∪C)×(B∪D)$ true for all sets A,B,C and D?
negative_passages.docid: 138069
negative_passages.text: Proof that $\bigcap_{a\in A} \bigg(\, \bigcup_{b\in B} F_{a,b} \, \bigg) = \bigcup_{f\in ^AB} \bigg(\, \bigcap_{a \in A} F_{a,f(a)}\,\bigg) $