query: Cardinalities of the two sets and the condition of the function in between
positive_passages.docid: 20303
positive_passages.text: A function $f\colon E\to F$ between finite sets is injective iff it is surjective iff it is bijective
negative_passages.docid: 147627
negative_passages.text: Relation between condition numbers $\kappa(A^T A)$ and $\kappa(A)$
negative_passages.docid: 20085
negative_passages.text: What's the cardinality of all sequences with coefficients in an infinite set?
negative_passages.docid: 13273
negative_passages.text: Cardinality of the set of all real functions of real variable
negative_passages.docid: 46698
negative_passages.text: Showing a set is closed under inverses
negative_passages.docid: 122947
negative_passages.text: What is the cardinality of the set of infinite cardinalities?
negative_passages.docid: 144787
negative_passages.text: Show that the set of all infinite subsets of $\mathbb{N}$ has cardinality greater than $\mathbb{N}$
negative_passages.docid: 155999
negative_passages.text: How to prove the following properties of infimum and supremum involving the union and intersection of the sets $A_k$
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: 127123
negative_passages.text: Cardinality of power set and binary sequence
negative_passages.docid: 92639
negative_passages.text: Prove that the family of open sets in $\mathbb{R}$ has cardinality equal to $2^{\aleph_0}$
negative_passages.docid: 99927
negative_passages.text: The closure of $\{ z:|f(z)|<c\}$ is the set $\{z:|f(z)|\leq c\}$.
negative_passages.docid: 130156
negative_passages.text: Image of the union and intersection of sets.
negative_passages.docid: 45218
negative_passages.text: Comparing the two cardinals $\aleph_0^{\aleph_0}$ and $2^{\aleph_0}$
negative_passages.docid: 132868
negative_passages.text: Bijection between an infinite set and its union of a countably infinite set
negative_passages.docid: 95373
negative_passages.text: Additive set function on a semiring of sets
negative_passages.docid: 39880
negative_passages.text: Cardinality of Orderings of $\mathbb{R}$