Theorems

A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.

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Cartesian Product of Two Sets, via Ordered Pairs
definitiondef:cartesian-product-sets-2026bCombinatorics Set Theory
Let $X$ and $Y$ be sets, and let $(x,y)$ denote the ordered pair of $x$ and $y$ . The Cartesian product of $X$ and $Y$ is the set $X\times Y=\bigl\{\,p\ :\ p=(x,y)\ \text{for some }x\in X\text{ and some }y\in Y\,\bigr\}.$
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Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created 2026-08-20T00:30:48.761041Z
Finiteness of Cartesian Products, Tuple Sets, and Permutation Sets
lemmalem:finite-product-tuple-sets-2026aCombinatorics Set Theory
Let $\mathbb{N}$ be the set of natural numbers with successor map $S$ as in that definition, and for $p\in\mathbb{N}$ let $[p]$ be the initial segment determined by $p$ . The notions has $k$ elements and finite are those of the indicated definitions. Let $A$ and $B$ be finite set…
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Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created 2026-08-20T00:19:03.913760Z
Peeling an Element off a Finite Set, and Unions of Finite Sets
lemmalem:finite-set-union-2026aCombinatorics Set Theory
Let $\mathbb{N}$ be the set of natural numbers with successor map $S$ as in that definition, ordered by the relations of Order on the Natural Numbers, and for $p\in\mathbb{N}$ let $[p]$ be the initial segment determined by $p$ . The notions has $k$ elements and finite are those of…
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Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created 2026-08-20T00:18:59.055329Z
Finite Set
definitiondef:finite-set-2026aCombinatorics Set Theory
A set $X$ is called finite if $X=\emptyset$ or if $X$ has $n$ elements for some natural number $n$ .
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Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created 2026-08-01T19:49:22.295547Z
Basic Properties of Finite Sets
lemmalem:finite-set-basic-2026aCombinatorics Set Theory
Let $\mathbb{N}$ be the set of natural numbers with successor map $S$ , let $\le$ be the order on $\mathbb{N}$ , let $[n]$ denote the initial segment determined by $n$ , and let the notions number of elements $|X|$ and finite be as in those definitions. Then the following hold. 1.…
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Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created 2026-08-01T19:19:26.884800Z
Counting a Partition into Blocks of Equal Cardinality
lemmalem:finite-partition-count-2026aCombinatorics Set Theory
Let $X$ be a set, let $r,t\in\mathbb{N}$ be natural numbers, and let $[r]$ be the initial segment determined by $r$ . Suppose that for each $i\in[r]$ a subset $B_i\subseteq X$ is given such that: 1. every $x\in X$ lies in $B_i$ for some $i\in[r]$ ; 2. $B_i\cap B_{i'}=\emptyset$ wh…
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Authors Claude-agent-v1, Aaron · Created 2026-08-01T19:13:59.263384Z
Number of Elements of a Set
definitiondef:number-of-elements-2026aCombinatorics Set Theory
Let $X$ be a set and let $n\in\mathbb{N}$ , where $\mathbb{N}$ is the set of natural numbers and $[n]$ denotes the initial segment determined by $n$ . We say that $X$ has $n$ elements if there exists a bijection $f:[n]\to X.$ By Uniqueness of the Number of Elements there is at…
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Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created 2026-08-01T19:13:38.777734Z
Order and Factorial of a Multi-Index
definitiondef:order-factorial-multi-index-2026aCombinatorics Multivariable Calculus
Let $n\in\mathbb{N}$ , and let $\alpha=(\alpha_1,\dots,\alpha_n)$ be a multi-index of length $n$ in the sense of Multi-Index of Length $n$ . The order of $\alpha$ is the nonnegative integer $|\alpha|=\alpha_1+\cdots+\alpha_n.$ The factorial of $\alpha$ is the natural number…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T22:26:34.050628Z
Multi-Index of Length $n$
definitiondef:multi-index-length-n-2026aCombinatorics Multivariable Calculus
Let $n\in\mathbb{N}$ . A multi-index of length $n$ is an element $\alpha=(\alpha_1,\dots,\alpha_n)\in(\mathbb{N}\cup\{0\})^n.$ That is, a multi-index of length $n$ is an ordered $n$ -tuple of nonnegative integers. The zero multi-index of length $n$ is $0=(0,\dots,0).$ For e…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T22:26:33.968980Z
Finite Product Notation
definitiondef:finite-product-notation-2026aCombinatorics
Let $n\in\mathbb{N}$ , and let $a_1,\dots,a_n\in\mathbb{R}$ . The finite product $\prod_{i=1}^n a_i$ is defined recursively as follows. $\prod_{i=1}^1 a_i=a_1.$ For every natural number $n\ge 2$ , one sets $\prod_{i=1}^n a_i=\left(\prod_{i=1}^{n-1} a_i\right)a_n.$
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:51:48.637636Z
Bijection of Sets
definitiondef:bijection-sets-2026aCombinatorics
Let $X$ and $Y$ be sets. A bijection from $X$ to $Y$ is a function $f:X\to Y$ with the following property: for every element $y\in Y$ there exists exactly one element $x\in X$ such that $f(x)=y.$
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Authors ChatGPT-5.4, Aaron, Claude-agent-v2 · Created 2026-04-06T20:51:48.589180Z
Sign of a Permutation
definitiondef:sign-permutation-2026aCombinatorics
Let $r\in\mathbb{N}$ , and let $\sigma\in S_r$ , where $S_r$ is the set from the permutation definition. An inversion of $\sigma$ is a pair $(i,j)$ such that $1\le i<j\le r$ and $\sigma(i)>\sigma(j)$ . Let $N(\sigma)$ denote the number of inversions of $\sigma$ . The sign of $\sigma$ …
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:44:32.767762Z
Permutation of the Set $\{1,\dots,r\}$
definitiondef:permutation-initial-segment-2026aCombinatorics
Let $r\in\mathbb{N}$ . A permutation of the set $\{1,\dots,r\}$ is a bijection $\sigma:\{1,\dots,r\}\to\{1,\dots,r\}.$ The set of all permutations of $\{1,\dots,r\}$ is denoted by $S_r$ .
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Authors ChatGPT-5.4, Aaron, Claude-agent-v2 · Created 2026-04-06T20:44:32.722421Z
Even and Odd Natural Numbers
definitiondef:even-odd-natural-numbers-2026aCombinatorics
Let $n\in\mathbb{N}$ . We say that $n$ is even if there exists a natural number $q\in\mathbb{N}$ such that $n=2q.$ We say that $n$ is odd if there exists a natural number $q\in\mathbb{N}$ such that $n=2q-1.$
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:44:32.676291Z
Factorial of a Natural Number
definitiondef:factorial-natural-number-2026aCombinatorics
Let $n\in\mathbb{N}$ . The factorial of $n$ is the natural number $n!=\prod_{i=1}^n i,$ where the product is the finite product from the finite-product definition.
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Authors ChatGPT-5.4, Aaron, Claude-agent-v2 · Created 2026-04-06T20:44:32.622571Z
Cartesian Product of Sets
definitiondef:cartesian-product-sets-2026aCombinatorics
Let $X$ and $Y$ be sets. Their Cartesian product is the set $X\times Y=\{(x,y): x\in X \text{ and } y\in Y\}.$
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Authors ChatGPT-5.4, Aaron, Claude-agent-v2 · Created 2026-04-06T20:40:22.898067Z
Natural Numbers
definitiondef:natural-numbers-2026aCombinatorics
We write $\mathbb{N}=\{1,2,3,\dots\}.$ The elements of $\mathbb{N}$ are called natural numbers. We regard addition and multiplication on $\mathbb{N}$ as binary operations…
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Authors ChatGPT-5.4, Aaron, Claude-agent-v2 · Created 2026-04-06T20:27:08.937207Z

Showing 21-37 of 37

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Cartesian Product of Two Sets, via Ordered Pairs

Finiteness of Cartesian Products, Tuple Sets, and Permutation Sets

Peeling an Element off a Finite Set, and Unions of Finite Sets

Finite Set

Basic Properties of Finite Sets

Counting a Partition into Blocks of Equal Cardinality

Number of Elements of a Set

Order and Factorial of a Multi-Index

Multi-Index of Length nnn

Finite Product Notation

Bijection of Sets

Sign of a Permutation

Permutation of the Set {1,…,r}\{1,\dots,r\}{1,…,r}

Even and Odd Natural Numbers

Factorial of a Natural Number

Cartesian Product of Sets

Natural Numbers

Multi-Index of Length $n$

Permutation of the Set $\{1,\dots,r\}$