All Questions
Tagged with cardinals combinatorics
52
questions
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Cardinal of a set of integers with symmetry relations
Context
In computational chemistry, there are two-electron integrals noted $(ij|kl)$ for integers (i,j,k,l) between 1 and K. The explicit expression of $(ij|kl)=\int dx_1dx_2 \chi_i(x_1)\chi_j(x_1)\...
- 13
4
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1
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82
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Induction does not preserve ordering between cardinality of sets?
Consider building a binary tree and consider it as a collection of points and edges. Here is one with five levels, numbered level $1$ at the top with $1$ node to level $5$ at the bottom with $16$ ...
- 6,360
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1
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Group action & cardinality of a set.
You can find here more details and explanation on this question.
Question:
Let $n$ be a non-negative integer. For any family $ (i_1, \ldots, i_r) $ of non-negative integers such that $ i_1 + \ldots + ...
- 719
1
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Constructing a bijection between $\mathbb{N}^2$ and $\{(x, y) \in \mathbb{N}^2: x \geq y\}$
Bijection between $\mathbb{N}^2$ and $\{(x, y) \in \mathbb{N}^2: x \geq y\}$.
My idea is to walk diagonally on $\mathbb{N}^2$, starting from $(0, 0)$, then to $(1, 0)$, $(0, 1)$ etc. Hopefully I can ...
3
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2
answers
136
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what is the Cardinality of subsets of Z of size 3
I need to find out the cardinality of the subsets of Z of size 3,
|N| = א0
|R| = א, thats how we defined it in class.
My idea is to build a ZxZ matrix, we can see that the subsets of size 2 are all ...
- 31
4
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1
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110
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How can I prove $|A_1 \cup A_2 \cup ... \cup A_n|=|A_1|+|A_2|+...+|A_n|$ using induction?
For pairwise disjoint sets $A_1,A_2,...,A_n$ how can I prove that: $|A_1 \cup A_2 \cup ... \cup A_n|=|A_1|+|A_2|+...+|A_n|$ using induction and the 2-set addition rule?
2-set addition rule: $|A_i \...
0
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0
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30
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Cardinality of a subset of euclidean space
Let $0<h, r\in \mathbb{R}$ be given and let $V\subseteq \mathcal{R}^k$ be the point $u, v\in\mathbb{R}^k$ such that $2h<||u-v||\leq r$. In my reseach, I need to find cardinality of $B(u, s)\cap ...
- 1,297
0
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0
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Number of elements in $Z^n$ with norm 2 less than some positive B [duplicate]
Is there any result or tight bound on the cardinal of : $\{\textbf{z}\in\mathbb{Z}^n / \lVert\textbf{z}\rVert_2 \leq B\}$ for some positive $B$.
Did not find any topic on this, sorry if it is a dupe.....
- 1
0
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2
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Prove that a field of a set has $2^r$ elements if it has finite cardinality.
Prove that a field of a set $A$ has $2^r$ elements if it has finite cardinality.
The definition of algebra is given here "https://en.wikipedia.org/wiki/Field_of_sets"
My try: I was trying to ...
- 9,098
1
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1
answer
65
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Is there bijection from set of multisets (of same cardinality) to the set of their ordered pairs?
Problem: Let $ A $ be a set of multisets of the same cardinality. Does there exist a bijection between $ A $ and the set of ordered pairs from all multisets in $ A $?
Examples:
$ A = \{ [1,1] , [1,2] ,...
- 1,709
0
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1
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186
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Stuck with a proof regarding cardinality
Problem: For any set $A$, finite or infinite, let $B^{A}$ be the set of all functions mapping $A$ into the set $B=\{0,1\}$. Show that the cardinality of $B^{A}$ is the same as the cardinality of the ...
-1
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1
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84
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Counting question and cardinality
A football league contains $6$ teams. During the season each team plays two matches against each other team. The result of each match is a draw or a win for one or other team. How many matches are ...
user846554
-1
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1
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96
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Choose whether or not the given sets are equinumerous to $\mathbb{N}^\mathbb{N}$.
Which of the following sets are equinumerous with $\mathbb{N}^\mathbb{N}$?
(i) $\mathbb{N} $
(ii) $\mathbb{R}$
(iii) $2^\mathbb{R}$
(iv) $\mathbb{N} \times \mathbb{R}$
(v) $\mathbb{R}^\mathbb{R}$
(vi) ...
- 15
2
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2
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127
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How should I interpret this diagram showing the bijection from $(a,b)$ to $\mathbb{R}$
In Chapter 1 of Pugh's Real Mathematical Analysis, Pugh gives the following picture:
I'm aware of other proofs to this like this one: bijection from (a,b) to R
but I'm interested in understanding how ...
- 363
5
votes
1
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317
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Almost disjoint families on uncountable sets
Suppose that $\Gamma$ is an infinite set. Let us say that a family $\mathscr A$ of subsets of $\Gamma$ is almost disjoint, whenever for any two distinct sets $A_1, A_2\in \mathscr{A}$ the intersection ...
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