query: Let $A,B,C$ be sets. Prove that $(A \setminus B)\setminus C = A \setminus (B \cup C)$ for any sets $A,B,C$.
positive_passages.docid: 150738
positive_passages.text: Proof of $(A - B) - C = A - (B \cup C)$
negative_passages.docid: 13026
negative_passages.text: Prove that if AB=I_m, then rk(A)=rk(B).
negative_passages.docid: 144502
negative_passages.text: Showing that $A\cap B = A\setminus (A\setminus B)$
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: 68918
negative_passages.text: Show that $A\cap B\subseteq A$ and $A\subseteq A\cup B$
negative_passages.docid: 13253
negative_passages.text: Prove $f(S \cup T) = f(S) \cup f(T)$
negative_passages.docid: 60707
negative_passages.text: Show that if $ \ A \cup B = A$ and $ \ A\cap B = A$ then $ \ A = B$
negative_passages.docid: 134770
negative_passages.text: For sets $A,B$, $A \cap (B \setminus A) \subseteq \varnothing$
negative_passages.docid: 10909
negative_passages.text: $|A \cup B| = \mathfrak{c}$, then $A$ or $B$ has cardinality $\mathfrak{c}$.
negative_passages.docid: 80876
negative_passages.text: Proof of $A\cap(B\cup C) = (A\cap B)\cup(A\cap C)$
negative_passages.docid: 50939
negative_passages.text: If $A$ and $B$ are nonempty sets, prove that $A \times B = B \times A$ if and only if $A = B$
negative_passages.docid: 70278
negative_passages.text: $A\subseteq B\to C\setminus B\subseteq C\setminus A\,$ -- how to prove this?
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: 56399
negative_passages.text: Proof for $A \cup B = B$ if and only if $A \subset B$
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: 31196
negative_passages.text: Prove $\binom{n}{a}\binom{n-a}{b-a} = \binom{n}{b}\binom{b}{a}$
negative_passages.docid: 87327
negative_passages.text: Unions and intersections: $(A \cup B = A ∪ C) \land (A \cap B = A ∩ C) \implies B = C.$