hcju/proofwikips huggingface dataset:
Product of Products over Overlapping Domains
:$\displaystyle \prod_{R \left({j}\right)} a_j \prod_{S \left({j}\right)} a_j = \left({\prod_{R \left({j}\right) \mathop \lor S \left({j}\right)} a_j}\right) \left({\prod_{R \left({j}\right) \mathop \land S \left({j}\right)} a_j}\right)$ where $\lor$ and $\land$ signify logical disjunction and logical conjunction respectively.
Topology is Locally Compact iff Ordered Set of Topology is Continuous
Let $T = \left({S, \tau}\right)$ be a topological space. Let $L = \left({\tau, \preceq}\right)$ be the ordered set where $\preceq$ is the inclusion relation. Then :$(1): \quad T$ is locally compact implies $L$ is continuous :$(2): \quad T$ is $T_3$ space and $L$ is continuous implies $T$ is locally compact
Sum of Sequence of Power by Index
:$\displaystyle \sum_{j \mathop = 0}^n j x^j = \frac {n x^{n + 2} - \paren {n + 1} x^{n + 1} + x} {\paren {x - 1}^2}$ for $x \ne 1$.
Ordering of Series of Ordered Sequences
Let $\sequence {a_n}$ and $\sequence {b_n}$ be two sequences. Let $\displaystyle \sum_{n \mathop = 1}^{\infty} a_n$ and $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ be convergent series. For each $n \in \N$, let $a_n < b_n$. Then: :$\displaystyle \sum_{n \mathop = 0}^\infty a_n < \displaystyle \sum_{n \mathop = 0}^\infty b_n$
Reciprocal of Real Exponential
:$\dfrac 1 {\map \exp x} = \map \exp {-x}$
Telescoping Series/Example 2
Let $\left \langle {b_n} \right \rangle$ be a sequence in $\R$. Let $\left \langle {a_n} \right \rangle$ be a sequence whose terms are defined as: :$a_k = b_k - b_{k - 1}$ Then: :$\displaystyle \sum_{k \mathop = m}^n a_k = b_n - b_{m - 1}$
Repunit Integer as Product of Base - 1 by Increasing Digit Integer
{{begin-eqn}} {{eqn | l = 9 \times 1 + 2       | r = 11 }} {{eqn | l = 9 \times 12 + 3       | r = 111 }} {{eqn | l = 9 \times 123 + 4       | r = 1111 }} {{eqn | l = 9 \times 1234 + 5       | r = 11111 }} {{eqn | l = 9 \times 12345 + 6       | r = 11111 }} {{eqn | o = \ldots }} {{end-eqn}} That is: :$\displaystyle 9 \sum_{j \mathop = 0}^n \left({n - j}\right)10^j + n + 1 = \sum_{j \mathop = 0}^n 10^j$
Repunit Integer as Product of Base - 1 by Increasing Digit Integer/General Result
:$\displaystyle \paren {b - 1} \sum_{j \mathop = 0}^n \paren {n - j}b^j + n + 1 = \sum_{j \mathop = 0}^n b^j$
Exponential of Real Number is Strictly Positive/Proof 5/Lemma
:$\forall x \in \R: \exp x \ne 0$
Product of Products
Let $R: \Z \to \set {\mathrm T, \mathrm F}$ be a propositional function on the set of integers. Let $\displaystyle \prod_{R \paren i} x_i$ denote a product over $R$. Let the fiber of truth of $R$ be finite. Then: :$\displaystyle \prod_{R \paren i} \paren {b_i c_i} = \paren {\prod_{R \paren i} b_i} \paren {\prod_{R \paren i} c_i}$
Summation of General Logarithms
Let $R: \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ be a propositional function on the set of integers. Let $\displaystyle \prod_{R \left({i}\right)} a_i$ denote a product over $R$. Let the fiber of truth of $R$ be finite. Then: :$\displaystyle \log_b \left({\prod_{R \left({i}\right)} a_i}\right) = \sum_{R \left({i}\right)} \log_b a_i$
Way Below Closure is Directed in Bounded Below Join Semilattice
Let $\left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $x \in S$. Then :$x^\ll$ is directed.
Bottom is Way Below Any Element
Let $\left({S, \preceq}\right)$ be a bounded below ordered set. Let $x \in S$. Then :$\bot \ll x$ where $\bot$ denotes the smallest element in $S$.
Equivalence of Definitions of Real Exponential Function/Proof 2
The following definitions of the exponential function are equivalent.
Product to n of Product to Index
:$\displaystyle \prod_{i \mathop = 0}^n \prod_{j \mathop = 0}^i a_i a_j = \prod_{i \mathop = 0}^n {a_i}^{n + 2}$
Bernoulli's Inequality/Corollary/General Result
For all $n \in \Z_{\ge 0}$: :$\displaystyle \prod_{j \mathop = 1}^n \left({1 - a_j}\right) \ge 1 - \sum_{j \mathop = 1}^n a^j$ where $0 < a_j < 1$ for all $j$.
Product of Sequence of 1 minus Reciprocal of Squares
For all $n \in \Z_{\ge 1}$: :$\displaystyle \prod_{j \mathop = 2}^n \paren {1 - \dfrac 1 {j^2} } = \dfrac {n + 1} {2 n}$
Product of Summations
:$\displaystyle \prod_{j \mathop = 1}^n \sum_{i \mathop = 1}^m a_{i j} = \sum_{1 \mathop \le i_1, \mathop \ldots \mathop , i_n \le m} a_{i_1 1} \cdots a_{i_n n}$
Summation of Powers over Product of Differences
:$\displaystyle \sum_{j \mathop = 1}^n \begin{pmatrix} {\dfrac { {x_j}^r} {\displaystyle \prod_{\substack {1 \mathop \le k \mathop \le n \\ k \mathop \ne j} } \left({x_j - x_k}\right)} } \end{pmatrix} = \begin{cases} 0 & : 0 \le r < n - 1 \\ 1 & : r = n - 1 \\ \displaystyle \sum_{j \mathop = 1}^n x_j & : r = n \end{cases}$
Summation by k of Product by r of x plus k minus r over Product by r less k of k minus r
:$\displaystyle \sum_{k \mathop = 1}^n \left({\dfrac {\displaystyle \prod_{\substack {1 \mathop \le r \mathop \le n \\ r \mathop \ne m} } \left({x + k - r}\right)} {\displaystyle \prod_{\substack {1 \mathop \le r \mathop \le n \\ r \mathop \ne k} } \left({k - r}\right)} }\right) = 1$ where $1 \le m \le n$ and $x$ is arbitrary.
Summation by k of Product by r of x plus k minus r over Product by r less k of k minus r/Example
:$\dfrac {x \left({x - 2}\right) \left({x - 3}\right)} {\left({-1}\right) \left({-2}\right) \left({-3}\right)} + \dfrac {\left({x + 1}\right) \left({x - 1}\right) \left({x - 2}\right)} {\left({1}\right) \left({-1}\right) \left({-2}\right)} + \dfrac {\left({x + 2}\right) x \left({x - 1}\right)} {\left({2}\right) \left({1}\right) \left({-1}\right)} + \dfrac {\left({x + 3}\right) \left({x + 1}\right) x} {\left({3}\right) \left({2}\right) \left({1}\right)} = 1$
Inverse of Cauchy Matrix
Let $C_n$ be the square Cauchy matrix of order $n$: :$C_n = \begin{bmatrix} \dfrac 1 {x_1 + y_1} & \dfrac 1 {x_1 + y_2} & \cdots & \dfrac 1 {x_1 + y_n} \\ \dfrac 1 {x_2 + y_1} & \dfrac 1 {x_2 + y_2} & \cdots & \dfrac 1 {x_2 + y_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac 1 {x_n + y_1} & \dfrac 1 {x_n + y_2} & \cdots & \dfrac 1 {x_n + y_n} \\ \end{bmatrix}$ Then its inverse $C_n^{-1} = \sqbrk b_n$ can be specified as: :$\begin{bmatrix} b_{ij} \end{bmatrix} = \begin{bmatrix} \dfrac {\displaystyle \prod_{k \mathop = 1}^n \paren {x_j + y_k} \paren {x_k + y_i} } {\displaystyle \paren {x_j + y_i} \paren {\prod_{\substack {1 \mathop \le k \mathop \le n \\ k \mathop \ne j} } \paren {x_j - x_k} } \paren {\prod_{\substack {1 \mathop \le k \mathop \le n \\ k \mathop \ne i} } \paren {y_i - y_k} } } \end{bmatrix}$
Sum of Elements in Inverse of Cauchy Matrix
Let $C_n$ be the Cauchy matrix of order $n$ given by: :$C_n = \begin{bmatrix} \dfrac 1 {x_1 + y_1} & \dfrac 1 {x_1 + y_2 } & \cdots & \dfrac 1 {x_1 + y_n} \\ \dfrac 1 {x_2 + y_1} & \dfrac 1 {x_2 + y_2 } & \cdots & \dfrac 1 {x_2 + y_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac 1 {x_m + y_1} & \dfrac 1 {x_m + y_2 } & \cdots & \dfrac 1 {x_m + y_n} \\ \end{bmatrix}$ Let $C_n^{-1}$ be its inverse, from Inverse of Cauchy Matrix: :$b_{i j} = \dfrac {\displaystyle \prod_{k \mathop = 1}^n \paren {x_j + y_k} \paren {x_k + y_i} } {\displaystyle \paren {x_j + y_i} \paren {\prod_{\substack {1 \mathop \le k \mathop \le n \\ k \mathop \ne j} } \paren {x_j - x_k} } \paren {\prod_{\substack {1 \mathop \le k \mathop \le n \\ k \mathop \ne i} } \paren {y_i - x_k} } }$ The sum of all the elements of $C_n^{-1}$ is: :$\displaystyle \sum_{1 \mathop \le i, \ j \mathop \le n} b_{i j} = \sum_{k \mathop = 1}^n x_k + \sum_{k \mathop = 1}^n y_k$
Hilbert Matrix is Cauchy Matrix
A Hilbert matrix is a special case of a Cauchy matrix.
Inverse of Hilbert Matrix
Let $H_n$ be the Hilbert matrix of order $n$: :$\begin{bmatrix} a_{i j} \end{bmatrix} = \begin{bmatrix} \dfrac 1 {i + j - 1} \end{bmatrix}$ Then its inverse $H_n^{-1} = \sqbrk b_n$ can be specified as: :$\begin{bmatrix} b_{i j} \end{bmatrix} = \begin{bmatrix} \dfrac {\paren {-1}^{i + j} \paren {i + n - 1}! \paren {j + n - 1}!} {\paren {\paren {i - 1}!}^2 \paren {\paren {j - 1}!}^2 \paren {n - j}! \paren {n - i}! \paren {i + j - 1} } \end{bmatrix}$
Sum of Elements in Inverse of Hilbert Matrix
Let $H_n$ be the Hilbert matrix of order $n$: :$\begin{bmatrix} a_{i j} \end{bmatrix} = \begin{bmatrix} \dfrac 1 {i + j - 1} \end{bmatrix}$ Consider its inverse $H_n^{-1}$. All the elements of $H_n^{-1}$ are integers. The sum of all the elements $b_{i j}$ of $H_n^{-1}$ is: :$\displaystyle \sum_{1 \mathop \le i, \ j \mathop \le n} b_{i j} = n^2$
Suprema of two Real Sets
Let $S$ and $T$ be real sets. Let $S$ and $T$ admit suprema. Then: : $\sup S \le \sup T \iff \forall \epsilon \in \R_{>0}: \forall s \in S: \exists t \in T: s < t + \epsilon$
Sequential Continuity is Equivalent to Continuity in the Reals
Let $A \subseteq \R$ be a subset of the real numbers. Let $c \in A$. Let $f : A \to \R$ be a real function. Then $f$ is continuous at $c$ : :for each sequence $\sequence {x_n}$ in $A$ that converges to $c$, the sequence $\sequence {\map f {x_n} }$ converges to $\map f c$.
Rising Factorial as Quotient of Factorials
:$x^{\overline n} = \dfrac {\paren {x + n - 1}!} {\paren {x - 1}!} = \dfrac {\map \Gamma {x + n} } {\map \Gamma x}$
Equivalent Definition for Alternating Bilinear Mapping
Let $\left({A_R, \oplus}\right)$ be an algebra over a ring $R$ with the property that $\operatorname{Char}\left({R}\right) \neq 2$. Then the following definitions for alternating bilinear maps are equivalent: * $\oplus$ is an alternating bilinear map  for all $a \in A_R$, $a \oplus a = 0$. * $\oplus$ is an alternating bilinear map  for all $a, b \in A_R$, $a \oplus b + b \oplus a = 0$.
Way Below if Between is Compact Set in Ordered Set of Topology
Let $T = \left({S, \tau}\right)$ be a topological space. Let $L = \left({\tau, \preceq}\right)$ be an ordered set where $\preceq \mathop = \subseteq\restriction_{\tau \times \tau}$ Let $x, y \in \tau$ such that :$\exists H \subseteq S: x \subseteq H \subseteq y \land H$ is compact Then :$x \ll y$
Locally Compact iff Open Neighborhood contains Compact Set
Let $T = \struct{S, \tau}$ be a topological space. Then :$T$ is locally compact  :$\forall U \in \tau, x \in U: \exists K \subseteq S: x \in K^\circ \land K \subseteq U \land K$ is compact where $K^\circ$ denotes the interior of $K$.
Cauchy-Binet Formula
Let $\mathbf A$ be an $m \times n$ matrix. Let $\mathbf B$ be an $n \times m$ matrix. Let $1 \le j_1, j_2, \ldots, j_m \le n$. Let $\mathbf A_{j_1 j_2 \ldots j_m}$ denote the $m \times m$ matrix consisting of columns $j_1, j_2, \ldots, j_m$ of $\mathbf A$. Let $\mathbf B_{j_1 j_2 \ldots j_m}$ denote the $m \times m$ matrix consisting of rows $j_1, j_2, \ldots, j_m$ of $\mathbf B$. Then: :$\displaystyle \map \det {\mathbf A \mathbf B} = \sum_{1 \mathop \le j_1 \mathop < j_2 \mathop < \cdots \mathop < j_m \le n} \map \det {\mathbf A_{j_1 j_2 \ldots j_m} } \map \det {\mathbf B_{j_1 j_2 \ldots j_m} }$ where $\det$ denotes the determinant.
Auxiliary Relation is Congruent
Let $\left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $\mathcal R$ be relation on $S$ satisfying conditions $(ii)$ and $(iii)$ of auxiliary relation. Then :$\forall x, y, z, u \in S: \left({x, z}\right) \in \mathcal R \land \left({y, u}\right) \in \mathcal R \implies \left({x \vee y, z \vee u}\right) \in \mathcal R$
Cauchy-Binet Formula/Example/m equals 1
Let $\mathbf A = \left[{a}\right]_{1 n}$ be a row matrix with $n$ columns. and $\mathbf B = \left[{b}\right]_{n 1}$ be a column matrix with $n$ rows. Let $\mathbf A \mathbf B$ be the (conventional) matrix product of $\mathbf A$ and $\mathbf B$. Then: :$\displaystyle \det \left({\mathbf A \mathbf B}\right) = \sum_{j \mathop = 1}^n a_j b_j$ where: :$a_j$ is element $a_{1 j}$ of $\mathbf A$ :$b_j$ is element $b_{j 1}$ of $\mathbf B$.
Preceding is Auxiliary Relation
Let $\left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Then :$\preceq$ is auxiliary relation.
Cauchy-Binet Formula/Example/Matrix by Transpose
Let $\mathbf A$ be an $m \times n$ matrix. Let $\mathbf A^\intercal$ be the transpose $\mathbf A$. Let $1 \le j_1, j_2, \ldots, j_m \le n$. Let $\mathbf A_{j_1 j_2 \ldots j_m}$ denote the $m \times m$ matrix consisting of columns $j_1, j_2, \ldots, j_m$ of $\mathbf A$. Let $\mathbf A^\intercal_{j_1 j_2 \ldots j_m}$ denote the $m \times m$ matrix consisting of rows $j_1, j_2, \ldots, j_m$ of $\mathbf A^\intercal$. Then: :$\displaystyle \det \left({\mathbf A \mathbf A^\intercal}\right) = \sum_{1 \mathop \le j_1 \mathop < j_2 \mathop < \cdots \mathop < j_m \le n} \left({\det \left({\mathbf A_{j_1 j_2 \ldots j_m} }\right)}\right)^2$ where $\det$ denotes the determinant.
Preceding is Top in Ordered Set of Auxiliary Relations
Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let ${\it Aux}\left({L}\right)$ be the set of all auxiliary relations on $S$. Let $P = \left({ {\it Aux}\left({L}\right), \precsim}\right)$ be an ordered set where $\precsim \mathop = \subseteq\restriction_{{\it Aux}\left({L}\right) \times {\it Aux}\left({L}\right)}$ Then :$\preceq \mathop = \top_P$ where $\top_P$ denotes the greatest element in $P$.
Bottom Relation is Auxiliary Relation
Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $B = \left\{ {\left({\bot, x}\right): x \in S}\right\}$ where $\bot$ denotes the smallest element in $L$. Then :$B$ is auxiliary relation.
Bottom Relation is Bottom in Ordered Set of Auxiliary Relations
Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let ${\it Aux}\left({L}\right)$ be the set of all auxiliary relations on $S$. Let $P = \left({ {\it Aux}\left({L}\right), \precsim}\right)$ be an ordered set where $\precsim \mathop = \subseteq\restriction_{{\it Aux}\left({L}\right) \times {\it Aux}\left({L}\right)}$ Let $B = \left\{ {\left({\bot_L, x}\right): x \in S}\right\}$ where $\bot_L$ denotes smallest element in $L$. Then :$B \mathop = \bot_P$
Inverse of Conditional is Converse of Contrapositive
Let $p \implies q$ be a conditional. Then the inverse of $p \implies q$ is the converse of its contrapositive.
Inverse of Conditional is Contrapositive of Converse
Let $p \implies q$ be a conditional. Then the inverse of $p \implies q$ is the contrapositive of its converse.
Converse of Conditional is Contrapositive of Inverse
Let $p \implies q$ be a conditional. Then the converse of $p \implies q$ is the contrapositive of its inverse.
Converse of Conditional is Inverse of Contrapositive
Let $p \implies q$ be a conditional. Then the converse of $p \implies q$ is the inverse of its contrapositive.
Intersection of Auxiliary Relations is Auxiliary Relation
Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $\mathcal F$ be a non-empty set of auxiliary relations on $S$. Then :$\bigcap \mathcal F$ is auxiliary relation.
Cauchy-Binet Formula/Example/m greater than n
Let $\mathbf A$ be an $m \times n$ matrix. Let $\mathbf B$ be an $n \times m$ matrix. Let $m > n$. Then: :$\displaystyle \det \left({\mathbf A \mathbf B}\right) = 0$
Krattenthaler's Identity
:$\begin{vmatrix} \left({x + q_2}\right) \left({x + q_3}\right) & \left({x + p_1}\right) \left({x + q_3}\right) & \left({x + p_1}\right) \left({x + p_2}\right) \\ \left({y + q_2}\right) \left({y + q_3}\right) & \left({y + p_1}\right) \left({y + q_3}\right) & \left({y + p_1}\right) \left({y + p_2}\right) \\ \left({z + q_2}\right) \left({z + q_3}\right) & \left({z + p_1}\right) \left({z + q_3}\right) & \left({z + p_1}\right) \left({z + p_2}\right) \end{vmatrix} = \left({x - y}\right) \left({x - z}\right) \left({y - z}\right) \left({p_1 - q_2}\right) \left({p_1 - q_3}\right) \left({p_2 - q_3}\right)$ where $\left\vert{\, \cdot \,}\right\vert$ denotes determinant.
Infima of two Real Sets
Let $S$ and $T$ be sets of real numbers. Let $S$ and $T$ admit infima. Then: : $\inf S \ge \inf T \iff \forall \epsilon \in \R_{>0}: \forall s \in S: \exists t \in T: s + \epsilon > t$
Floor Function is Integer
Let $x$ be a real number. Then the floor function of $x$ is an integer: :$\floor x \in \Z$
Floor of Number plus Integer
:$\forall n \in \Z: \floor x + n = \floor {x + n}$
Floor is between Number and One Less
:$x - 1 < \left\lfloor{x}\right\rfloor \le x$ where $\left\lfloor{x}\right\rfloor$ is the floor of $x$.
Equivalence of Definitions of Floor Function
Let $x$ be a real number. {{TFAE|def = Floor Function}}
Properties of Floor Function
This page gathers together some basic propeties of the floor function.
Properties of Ceiling Function
This page gathers together some basic propeties of the ceiling function.
Ceiling Function is Integer
Let $x$ be a real number. Then the ceiling function of $x$ is an integer: :$\ceiling x \in \Z$
Real Number is Integer iff equals Ceiling
:$x = \ceiling x \iff x \in \Z$
Ceiling Function is Idempotent
:$\ceiling {\ceiling x} = \ceiling x$
Way Below Relation is Auxiliary Relation
Lrt $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Then :$\ll$ is auxiliary relation where $\ll$ denotes the way below relation.
Number is between Ceiling and One Less
:$\ceiling x - 1 < x \le \ceiling x$
Ceiling is between Number and One More
:$x \le \ceiling x < x + 1$ where $\ceiling x$ is the ceiling of $x$.
Ceiling of Number plus Integer
:$\forall n \in \Z: \left \lceil {x} \right \rceil + n = \left \lceil {x + n} \right \rceil$
Relation Segment of Auxiliary Relation is Ideal
Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $R$ be auxiliary relation on $S$. Let $x \in S$. Then :$x^R$ is ideal in $L$ where $x^R$ denotes the $R$-segment of $x$.
Quotient of Modulo Operation with Modulus
Let $x, y \in \R$ be real numbers. Let $x \bmod y$ denote the modulo operation: :$x \bmod y := \begin{cases} x - y \left \lfloor {\dfrac x y}\right \rfloor & : y \ne 0 \\ x & : y = 0 \end{cases}$ where $\left \lfloor {\dfrac x y}\right \rfloor$ denotes the floor of $\dfrac x y$. Let $y \ne 0$. Then: :$0 \le \dfrac x y - \left \lfloor {\dfrac x y}\right \rfloor = \dfrac {x \bmod y} y < 1$
Range of Modulo Operation for Positive Modulus
Let $x, y \in \R$ be real numbers. Let $x \bmod y$ denote the modulo operation: :$x \bmod y := \begin{cases} x - y \floor {\dfrac x y} & : y \ne 0 \\ x & : y = 0 \end{cases}$ where $\floor {\dfrac x y}$ denotes the floor of $\dfrac x y$. Let $y > 0$. Then: :$0 \le x \bmod y < y$
Range of Modulo Operation for Negative Modulus
Let $x, y \in \R$ be real numbers. Let $x \bmod y$ denote the modulo operation: :$x \bmod y := \begin{cases} x - y \left \lfloor {\dfrac x y}\right \rfloor & : y \ne 0 \\ x & : y = 0 \end{cases}$ where $\left \lfloor {\dfrac x y}\right \rfloor$ denotes the floor of $\dfrac x y$. Let $y < 0$. Then: :$0 \ge x \bmod y > y$
Zero is Integer Multiple of Zero
Zero is an integer multiple of zero.
Modulo Operation/Examples/18 mod 3
:$18 \bmod 3 = 0$
Number is Divisor iff Modulo is Zero
Let $x, y \in \R$ be real numbers. Let $x \bmod y$ denote the modulo operation: :$x \bmod y := \begin {cases} x - y \floor {\dfrac x y} & : y \ne 0 \\ x & : y = 0 \end {cases}$ where $\floor {\dfrac x y}$ denotes the floor of $\dfrac x y$. Then $x \bmod y = 0$  $x$ is an integer multiple of $y$.
Number less than Integer iff Floor less than Integer
:$\left \lfloor{x}\right \rfloor < n \iff x < n$
Number not less than Integer iff Floor not less than Integer
:$x \ge n \iff \left \lfloor{x}\right \rfloor \ge n$
Number not greater than Integer iff Ceiling not greater than Integer
:$\left \lceil{x}\right \rceil \le n \iff x \le n$
Number greater than Integer iff Ceiling greater than Integer
:$\left \lceil{x}\right \rceil > n \iff x > n$
Integer equals Floor iff between Number and One Less
:$\floor x = n \iff x - 1 < n \le x$
Integer equals Floor iff Number between Integer and One More
:$\floor x = n \iff n \le x < n + 1$
Integer equals Ceiling iff between Number and One More
:$\left \lceil{x}\right \rceil = n \iff x \le n < x + 1$
Integer equals Ceiling iff Number between Integer and One Less
:$\ceiling x = n \iff n - 1 < x \le n$
Floor of Root of Floor equals Floor of Root
:$\displaystyle \floor {\sqrt {\floor x} } = \floor {\sqrt x}$
Ceiling of Root of Ceiling equals Ceiling of Root
:$\displaystyle \ceiling {\sqrt {\ceiling x} } = \ceiling {\sqrt x}$
Congruence by Divisor of Modulus/Integer Modulus
Let $r, s \in \Z$ be integers. Let $a, b \in \Z$ such that $a$ is congruent modulo $r s$ to $b$, that is: :$a \equiv b \pmod {r s}$ Then: :$a \equiv b \pmod r$ and: :$a \equiv b \pmod s$
Modulo Addition is Well-Defined/Real Modulus
Let $z \in \R$ be a real number. Let: :$a \equiv b \pmod z$ and: :$x \equiv y \pmod z$ where $a, b, x, y \in \R$. Then: : $a + x \equiv b + y \pmod z$
Congruence by Product of Moduli/Real Modulus
Let $a, b, z \in \R$. Let $a \equiv b \pmod z$ denote that $a$ is congruent to $b$ modulo $z$. Then $\forall y \in \R, y \ne 0$: :$a \equiv b \pmod z \iff y a \equiv y b \pmod {y z}$
Segment of Auxiliary Relation is Subset of Lower Closure
Let $\left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $R$ be auxiliary relation on $S$. Let $x \in S$. Then :$x^R \subseteq x^\preceq$ where :$x^R$ denotes the $R$-segment of $x$, :$x^\preceq$ denotes the lower closure of $x$.
Preceding implies Inclusion of Segments of Auxiliary Relation
Let $\left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $R$ be an auxiliary relation on $S$. Let $x, y \in S$ such that :$x \preceq y$ Then :$x^R \subseteq y^R$ where $x^R$ denotes the $R$-segment of $x$.
Characteristic Function of Square-Free Integers is Multiplicative
Let $S \subseteq \Z$ be the set of positive integers defined as: :$S := \set {n \in \Z: \forall k \in \Z_{>1}: k^2 \nmid n}$ That is, let $S$ be the set of all square-free positive integers. Let $\chi_S: \N \to \Z$ denote the characteristic function of $S$: :$\forall n \in \Z: \map {\chi_S} n = \sqbrk {n \in S}$ where $\sqbrk {n \in S}$ is Iverson's convention. Then $\chi_S$ is multiplicative.
Segment of Auxiliary Relation Mapping is Increasing
Let $R = \left({S, \preceq}\right)$ be an ordered set. Let ${\it Ids}\left({R}\right)$ be the set of all ideals in $R$. Let $L = \left({ {\it Ids}\left({R}\right), \precsim}\right)$ be an ordered set where $\precsim \mathop = \subseteq\restriction_{ {\it Ids}\left({R}\right) \times {\it Ids}\left({R}\right)}$ Let $r$ be an auxiliary relation on $S$. Let $f: S \to {\it Ids}\left({R}\right)$ be a mapping such that :$\forall x \in S: f\left({x}\right) = x^r$ where $x^r$ denotes the $r$-segment of $x$. Then :$f$ is increasing mapping.
Product of Multiplicative Functions is Multiplicative
Let $f: \N \to \C$ and $g: \N \to \C$ be multiplicative functions. Then their pointwise product: :$f \times g: \Z \to \Z: \forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$ is also multiplicative.
Segment of Auxiliary Relation Mapping is Element of Increasing Mappings Satisfying Inclusion in Lower Closure
Let $R = \left({S, \preceq}\right)$ be an ordered set. Let $\mathit{Ids}\left({R}\right)$ be the set of all ideals in $R$. Let $L = \left({ \mathit{Ids}\left({R}\right), \precsim}\right)$ be an ordered set where $\precsim \mathop = \subseteq\restriction_{\mathit{Ids}\left({R}\right) \times \mathit{Ids}\left({R}\right)}$. Let $r$ be an auxiliary relation on $S$. Let $M = \left({F, \preccurlyeq}\right)$ be the ordered set of increasing mappings $g$ satisfying $\forall x \in S: g\left({x}\right) \subseteq x^\preceq$. Let $f: S \to \mathit{Ids}\left({R}\right)$ be a mapping such that: :$\forall x \in S: f\left({x}\right) = x^r$ where $x^r$ denotes the $r$-segment of $x$. Then: :$(1): \quad f \in F$ Let $h: S \to \mathit{Ids}\left({R}\right): x \mapsto x^\preceq$ Then: :$(2): \quad h \in F \land f \preccurlyeq h$ Let $k: S \to \mathit{Ids}\left({R}\right): x \mapsto \left\{ {\bot}\right\}$ where $\bot$ denotes the smallest element in $L$. Then: :$(3): \quad k \in F$ and $k \preccurlyeq f$
Singleton of Bottom is Ideal
Let $\struct {S, \preceq}$ be a bounded below ordered set. Then :$\set \bot$ is an ideal in $\struct {S, \preceq}$ where $\bot$ denotes the smallest element in $S$.
Rising Factorial in terms of Falling Factorial
:$x^{\overline n} = \left({x + n - 1}\right)^{\underline n}$
Rising Factorial in terms of Falling Factorial of Negative
:$x^{\overline k} = \paren {-1}^k \paren {-x}^{\underline k}$