# Tagged Questions

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### Help me find out the minimum of n(B) [duplicate]

About natural numbers a 1 ,a 2 ,…,a 20 , define set A={a i +a j |1≤i≤j≤20} . n(A)=201 , then about set B={|a i −a j ||1≤i≤j≤20} . What's the minimum of n(B) ? Last time I posted this question ...
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### Number of functions on finite set

If $A$ has $n$ elements, how many functions are there from $A \rightarrow A$? How many bijective functions are there from $A$ to $A$? My thinking was that there are $n$ possibilities for $f(a_1)$, ...
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### What is the number of ways to represent the $n$ element set as a union of distinct non-empty subsets

edit: I do not mean the number of partitions $B_n$ here. The title says it all. The n element set is $[n]=\{1,2,\dots,n\}$. One representation (the one using the most sets) for example is the union ...
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### Generalization of principle of inclusion and exclusion (PIE)

The PIE can be stated as $$|\cup_{i=1}^n Y_i| = \sum_{J\subset[n], J\neq \emptyset} (-1)^{|J|-1} |Y_J|$$ where $[n]=\{1,2,...,n\}$ and $Y_J=\cap_{i \in J} Y_i$. Problems using it are usually reduced ...
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### Find the cardinal of the set of all infinite sequences of $0,1,-1$ such that each sequence contains each digit at least once - Check my answer

As the title says, we are asked to find the cardinal of the set of all infinite sequences made from the digits $0,1,-1$ such that each sequence contains each digit at least once. My answer I solved ...
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### Unique combination of sets

We start with a finite number of $N$ sets, $\boldsymbol{X}_1,\ldots,\boldsymbol{X}_N$, each containing a finite number of integers. The sets do not in general have the same number of elements. The ...
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### Mean Element of a Finite Set

Given a finite set $S = \{A_1,A_2,A_3...\}$ containing an arbitrary number of finite sets such that for any $A_i{}\in{}S$ and $A_j{}\in{}S$, $| A_i{} | = | A_j{} |$, and given that for every ...
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### how many pairs differ in one value from another

Assume I have a family of sets $X=\{X_1,X_2,...,X_m\}$ each set $X_i\in X$ has $n$ elements $\{x^i_1,x^i_2,...,x^i_n\}$. Let $Z$ be the cartesian product of $X$. Let $z^{\downarrow V}$ be the ...
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### Language to describe a number smaller than, but related to Bell number

I understand that the Bell number $B_n$ is the number of partitions of a set of size $n$. Despite my incredible ineptitude at combinatorics, I also understand most of how the binomial coefficient ...
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