query: If $A$ is denumerable and $x\in A$, then $A\setminus\{x\}$ is denumerable. (With my proof)
positive_passages.docid: 147136
positive_passages.text: Subset of a countable set is itself countable
negative_passages.docid: 49709
negative_passages.text: Suppose $X$ is infinite and $A$ is a finite subset of $X$. Then $X$ and $X \setminus A$ are equinumerous
negative_passages.docid: 21820
negative_passages.text: How to prove that a set is invariant?
negative_passages.docid: 52565
negative_passages.text: Prove that if $d \mid n \in \mathbb{N}$, then $\varphi(d) \mid \varphi(n)$.
negative_passages.docid: 142631
negative_passages.text: Show that A has the same cardinality as $A\setminus\{x,y,z\}$
negative_passages.docid: 66948
negative_passages.text: Proving question on sets
negative_passages.docid: 10911
negative_passages.text: $A \in M_{n}(\mathbb{R})$ has inverse if and only if it's rank is n.
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: 109070
negative_passages.text: Question about the proof of the distributive property for sets
negative_passages.docid: 77286
negative_passages.text: Proving set theory question
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: 6291
negative_passages.text: Proving something is a linearly independent subset
negative_passages.docid: 13575
negative_passages.text: Need help for proving that: $f(f^{-1}(A)) ⊆ A$
negative_passages.docid: 39455
negative_passages.text: Prove that $(X\times Y)\setminus (A\times B)$ is connected
negative_passages.docid: 42809
negative_passages.text: How can we show that $(I-A)$ is invertible?
negative_passages.docid: 52614
negative_passages.text: (exercise from Tao's analysis book) Proof of a lemma relating to power set of X
negative_passages.docid: 19856
negative_passages.text: Prove that a subset of a separable set is itself separable