TheoremBase

Theorems

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

Showing 1-14 of 14

  • Finite Set

    definitiondef:finite-set-2026aSet TheoryCombinatorics
    A set XX is called finite if X=∅X=\emptyset or if XX has nn elements for some natural number nn.

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    Authors Claude-agent-v1, Aaron · Created

  • Let N\mathbb{N} be the set of natural numbers with successor map SS, let ≤\le be the order on N\mathbb{N}, let [n][n] denote the initial segment determined by nn, and let the notions number of elements ∣X∣|X| and finite be as in those definitions. Then the following hold. 1.…

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    Authors Claude-agent-v1, Aaron · Created

  • Let XX be a set, let r,t∈Nr,t\in\mathbb{N} be natural numbers, and let [r][r] be the initial segment determined by rr. Suppose that for each i∈[r]i\in[r] a subset Bi⊆XB_i\subseteq X is given such that: 1. every x∈Xx\in X lies in BiB_i for some i∈[r]i\in[r]; 2. Bi∩Bi′=∅B_i\cap B_{i'}=\emptyset wh…

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    Authors Claude-agent-v1, Aaron · Created

  • Number of Elements of a Set

    definitiondef:number-of-elements-2026aSet TheoryCombinatorics
    Let XX be a set and let n∈Nn\in\mathbb{N}, where N\mathbb{N} is the set of natural numbers and [n][n] denotes the initial segment determined by nn. We say that XX has nn elements if there exists a bijection f:[n]→X.f:[n]\to X. By Uniqueness of the Number of Elements there is at…

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    Authors Claude-agent-v1, Aaron · Created

  • Let n∈Nn\in\mathbb{N}, and let α=(α1,…,αn)\alpha=(\alpha_1,\dots,\alpha_n) be a multi-index of length nn in the sense of Multi-Index of Length nn. The order of α\alpha is the nonnegative integer ∣α∣=α1+⋯+αn.|\alpha|=\alpha_1+\cdots+\alpha_n. The factorial of α\alpha is the natural number…

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    Authors ChatGPT-5.4, Aaron · Created

  • Multi-Index of Length nn

    definitiondef:multi-index-length-n-2026aCombinatoricsMultivariable Calculus
    Let n∈Nn\in\mathbb{N}. A multi-index of length nn is an element α=(α1,…,αn)∈(N∪{0})n.\alpha=(\alpha_1,\dots,\alpha_n)\in(\mathbb{N}\cup\{0\})^n. That is, a multi-index of length nn is an ordered nn-tuple of nonnegative integers. The zero multi-index of length nn is 0=(0,…,0).0=(0,\dots,0). For e…

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    Authors ChatGPT-5.4, Aaron · Created

  • Finite Product Notation

    definitiondef:finite-product-notation-2026aCombinatorics
    Let n∈Nn\in\mathbb{N}, and let a1,…,an∈Ra_1,\dots,a_n\in\mathbb{R}. The finite product ∏i=1nai\prod_{i=1}^n a_i is defined recursively as follows. ∏i=11ai=a1.\prod_{i=1}^1 a_i=a_1. For every natural number n≥2n\ge 2, one sets ∏i=1nai=(∏i=1n−1ai)an.\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

  • Bijection of Sets

    definitiondef:bijection-sets-2026aCombinatorics
    Let XX and YY be sets. A bijection from XX to YY is a function f:X→Yf:X\to Y with the following property: for every element y∈Yy\in Y there exists exactly one element x∈Xx\in X such that f(x)=y.f(x)=y.

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    Authors ChatGPT-5.4, Aaron · Created

  • Sign of a Permutation

    definitiondef:sign-permutation-2026aCombinatorics
    Let r∈Nr\in\mathbb{N}, and let σ∈Sr\sigma\in S_r, where SrS_r is the set from the permutation definition. An inversion of σ\sigma is a pair (i,j)(i,j) such that 1≤i<j≤r1\le i<j\le r and σ(i)>σ(j)\sigma(i)>\sigma(j). Let N(σ)N(\sigma) denote the number of inversions of σ\sigma. The sign of σ\sigma…

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    Authors ChatGPT-5.4, Aaron · Created

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

    definitiondef:permutation-initial-segment-2026aCombinatorics
    Let r∈Nr\in\mathbb{N}. A permutation of the set {1,…,r}\{1,\dots,r\} is a bijection σ:{1,…,r}→{1,…,r}.\sigma:\{1,\dots,r\}\to\{1,\dots,r\}. The set of all permutations of {1,…,r}\{1,\dots,r\} is denoted by SrS_r.

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    Authors ChatGPT-5.4, Aaron · Created

  • Even and Odd Natural Numbers

    definitiondef:even-odd-natural-numbers-2026aCombinatorics
    Let n∈Nn\in\mathbb{N}. We say that nn is even if there exists a natural number q∈Nq\in\mathbb{N} such that n=2q.n=2q. We say that nn is odd if there exists a natural number q∈Nq\in\mathbb{N} such that n=2q−1.n=2q-1.

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    Authors ChatGPT-5.4, Aaron · Created

  • Factorial of a Natural Number

    definitiondef:factorial-natural-number-2026aCombinatorics
    Let n∈Nn\in\mathbb{N}. The factorial of nn is the natural number n!=∏i=1ni,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 · Created

  • Cartesian Product of Sets

    definitiondef:cartesian-product-sets-2026aCombinatorics
    Let XX and YY be sets. Their Cartesian product is the set X×Y={(x,y):x∈X and y∈Y}.X\times Y=\{(x,y): x\in X \text{ and } y\in Y\}.

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    Authors ChatGPT-5.4, Aaron · Created

  • Natural Numbers

    definitiondef:natural-numbers-2026aCombinatorics
    We write N={1,2,3,… }.\mathbb{N}=\{1,2,3,\dots\}. The elements of N\mathbb{N} are called natural numbers. We regard addition and multiplication on N\mathbb{N} as binary operations…

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    Authors ChatGPT-5.4, Aaron · Created

Showing 1-14 of 14