query: Proof by cases for sets
positive_passages.docid: 144360
positive_passages.text: For any sets $A$ and $B$, show that $(B\smallsetminus A)\cup A=B \iff A\subseteq B$.
negative_passages.docid: 45792
negative_passages.text: Rigorous proof that surjectivity implies injectivity for finite sets
negative_passages.docid: 49143
negative_passages.text: Showing $A \subset B \iff A\cap B=A$
negative_passages.docid: 67066
negative_passages.text: Proof there is a 1-1 correspondence between an uncountable set and itself minus a countable part of it
negative_passages.docid: 61747
negative_passages.text: Proof of exchange principle in set theory
negative_passages.docid: 115606
negative_passages.text: Proofs involving $ (A\setminus B) \cup (A \cap B)$
negative_passages.docid: 37183
negative_passages.text: Show that the set of all $x$ such that $x \in P$ and $x \notin Q$ exists
negative_passages.docid: 83518
negative_passages.text: Proving Two sets have same cardinality
negative_passages.docid: 136755
negative_passages.text: Direct proof of empty set being subset of every set
negative_passages.docid: 72391
negative_passages.text: linear dependence proof using subsets
negative_passages.docid: 130771
negative_passages.text: Sets and the principle of inclusion and exclusion
negative_passages.docid: 58186
negative_passages.text: Set theory proof, $\exists g(g\colon A\twoheadrightarrow C)$ or $\exists h(h\colon C\hookrightarrow B)$
negative_passages.docid: 126015
negative_passages.text: Elementary number theory in sets
negative_passages.docid: 56370
negative_passages.text: Associative operation on a set $S$
negative_passages.docid: 83996
negative_passages.text: For any two sets $A,B$ , $|A|\leq|B|$ or $|B|\leq|A|$
negative_passages.docid: 9158
negative_passages.text: Proving that a propositional theory of any cardinality has an independent set of axioms
negative_passages.docid: 37331
negative_passages.text: Prove that every set with more than one element has a permutation without fixed points