query: Let S be an infinite set and A a finite subset. Prove that $|S| = |S -A|$.
positive_passages.docid: 142631
positive_passages.text: Show that A has the same cardinality as $A\setminus\{x,y,z\}$
negative_passages.docid: 120562
negative_passages.text: If A is an infinite set and B is at most countable set, prove that A and $A \cup B$ have the same cardinality
negative_passages.docid: 135694
negative_passages.text: Proving $\kappa^{\lambda} = |\{X: X \subseteq \kappa, |X|=\lambda\}|$
negative_passages.docid: 152382
negative_passages.text: Can an infinite set A be larger than an infinite set B but still have the same cardinality?
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: 762
negative_passages.text: Proving every infinite set $S$ contains a denumerable subset
negative_passages.docid: 43068
negative_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: 13415
negative_passages.text: How to show if $A \subseteq B$, then $A \cap B = A$?
negative_passages.docid: 144360
negative_passages.text: For any sets $A$ and $B$, show that $(B\smallsetminus A)\cup A=B \iff A\subseteq B$.
negative_passages.docid: 147213
negative_passages.text: Prove that finite unions of compact sets are compact
negative_passages.docid: 9021
negative_passages.text: An infinite subset of a countable set is countable
negative_passages.docid: 44974
negative_passages.text: Infinite set and countable subsets
negative_passages.docid: 132868
negative_passages.text: Bijection between an infinite set and its union of a countably infinite set
negative_passages.docid: 6478
negative_passages.text: Prove that for any infinite poset there is an infinite subset which is either linearly ordered or antichain.
negative_passages.docid: 136306
negative_passages.text: Let $A$ be a finite set and let $B$ be a subset of $A$ with $|A|=n$,$|B|=m$ and $0<m<n$. Count the subsets of $A$ containing $ B.$
negative_passages.docid: 33157
negative_passages.text: Proof that an infinite subset of $\mathbb{N}$ is countable
negative_passages.docid: 80180
negative_passages.text: Discrete and compact subset must be finite