query: How many non-empty subsets are there of the set $\{1, 2, \ldots, n\}$? Give your answer in closed form.
positive_passages.docid: 146098
positive_passages.text: Number of nonempty subsets of a set with $n$ elements
negative_passages.docid: 568
negative_passages.text: Consider a set $A$ of $10$ integers, $A = \{a_1,a_2, \ldots ,a_{10}\}$. Prove that there is at least one subset of $A$ whose sum is divisible by $10$.
negative_passages.docid: 114262
negative_passages.text: Find the cardinality of the following set $S_n=\{(x,y)|x^2+y^2=n, \text{where }x,y,n \in \mathbb Z\text{ and }n\geq0 \}$
negative_passages.docid: 153072
negative_passages.text: Guessing a subset of $\{1,...,N\}$
negative_passages.docid: 105886
negative_passages.text: A question on the proof of 14 distinct sets can be formed by complementation and closure
negative_passages.docid: 3072
negative_passages.text: If the empty set is a subset of every set, why isn't $\{\emptyset,\{a\}\}=\{\{a\}\}$?
negative_passages.docid: 79568
negative_passages.text: Cardinality of the set $D$
negative_passages.docid: 6204
negative_passages.text: How many permutations of $\{1, \ldots, n\}$ exist such that none of them contain $(i, i+1)$ (as a sequence) for $i \in {1,...,(n-1)}$?
negative_passages.docid: 117658
negative_passages.text: A set with n elements has 2^n subsets
negative_passages.docid: 73350
negative_passages.text: Geometric solution to find a Set of totally ordered Subsets of $ \mathbb{N} $ is uncountable
negative_passages.docid: 64519
negative_passages.text: Subsets of a list
negative_passages.docid: 66948
negative_passages.text: Proving question on sets
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: 858
negative_passages.text: Is $\{\emptyset\}$ a subset of $\{\{\emptyset\}\}$?
negative_passages.docid: 72750
negative_passages.text: Uncountable number of mutually disjoint countable sets in $\mathbb R$
negative_passages.docid: 78117
negative_passages.text: Closed property of nonempty finite set
negative_passages.docid: 45216
negative_passages.text: Cardinality of $[0,1]$