hcju/proofwikips huggingface dataset:
Limit Point of Sequence in Discrete Space not always Limit Point of Open Set
Let $T = \struct {S, \tau}$ be a discrete topological space. Let $U \in \tau$ be an open set of $T$. Let $\sequence {x_n}$ be a sequence in $U$. Let $x$ be the limit point of $\sequence {x_n}$. Then $x$ is not always a limit point of $U$.
Accumulation Point of Sequence of Distinct Terms is Omega-Accumulation Point of Range
Let $T = \struct {S, \tau}$ be a topological space. Let $\sequence {x_n}$ be a sequence of distinct terms of $S$. Let $\alpha$ be an accumulation point of $\sequence {x_n}$. Then $\alpha$ is also an $\omega$-accumulation point of $\set {x_n: n \in \N}$.
Limit Point of Sequence is Adherent Point of Range
Let $T = \left({S, \tau}\right)$ be a topological space. Let $\left \langle {x_n} \right \rangle$ be a sequence in $S$. Let $\alpha$ be a limit point of $\left \langle {x_n} \right \rangle$. Then $\alpha$ is an adherent point of $\left\{ {x_n: n \in \N}\right\}$.
Limit Point of Sequence of Distinct Terms is Omega-Accumulation Point of Range
Let $T = \struct {S, \tau}$ be a topological space. Let $\sequence {x_n}$ be a sequence of distinct terms of $S$. Let $\alpha$ be a limit point of $\sequence {x_n}$. Then $\alpha$ is also an $\omega$-accumulation point of $\set {x_n: n \in \N}$.
Equivalence of Definitions of Isolated Point
{{TFAE|def = Isolated Point of Subset|view = isolated point}} Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a subset of $S$.
Equivalence of Definitions of Closed Set
{{TFAE|def = Closed Set (Topology)|view = Closed Set|context = Topology (Mathematical Branch)|contextview = topology}} Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$.
Kuratowski's Closure-Complement Problem
Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq S$ be a subset of $T$. By successive applications of the operations of complement relative to $S$ and the closure, there can be as many as $14$ distinct subsets of $S$ (including $A$ itself).
Kuratowski's Closure-Complement Problem/Complement
The complement of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A'       | r = \left({\gets \,.\,.\, 0}\right]       | c = {{Defof|Unbounded Closed Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \left\{ {1} \right\}       | c = {{Defof|Singleton}} }} {{eqn | o =       | ro= \cup       | r = \left[{2 \,.\,.\, 3}\right)       | c = {{Defof|Half-Open Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \left({3 \,.\,.\, 4}\right]       | c = ... adjacent to Half-Open Real Interval }} {{eqn | o =       | ro= \cup       | r = \left({\R \setminus \Q \cap \left[{4 \,.\,.\, 5}\right]}\right)       | c = Irrational Numbers from $4$ to $5$ }} {{eqn | o =       | ro= \cup       | r = \left[{5 \,.\,.\, \to}\right)       | c = {{Defof|Unbounded Closed Real Interval}} }} {{end-eqn}} :500px
Kuratowski's Closure-Complement Problem/Interior
The interior of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^\circ       | r = \left({0 \,.\,.\, 1}\right) \cup \left({1 \,.\,.\, 2}\right)       | c = Union of Adjacent Open Intervals }} {{end-eqn}} :500px
Kuratowski's Closure-Complement Problem/Closure
The closure of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^-       | r = \closedint 0 2       | c = {{Defof|Closed Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \set 3       | c = {{Defof|Singleton}} }} {{eqn | o =       | ro= \cup       | r = \closedint 4 5       | c = {{Defof|Closed Real Interval}} }} {{end-eqn}} :500px
Kuratowski's Closure-Complement Problem/Exterior
The exterior of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^e       | r = \openint \gets 0       | c = {{Defof|Unbounded Open Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \openint 2 3 \cup \openint 3 4       | c = {{Defof|Union of Adjacent Open Intervals}} }} {{eqn | o =       | ro= \cup       | r = \openint 5 \to       | c = {{Defof|Unbounded Open Real Interval}} }} {{end-eqn}} :500px
Kuratowski's Closure-Complement Problem/Closure of Complement
The closure of the complement of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^{\prime \, -}       | r = \hointl \gets 0       | c = {{Defof|Unbounded Closed Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \set 1       | c = {{Defof|Singleton}} }} {{eqn | o =       | ro= \cup       | r = \hointr 2 \to       | c = {{Defof|Unbounded Closed Real Interval}} }} {{end-eqn}} :500px
Kuratowski's Closure-Complement Problem/Closure of Interior
The closure of the interior of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^{\circ \, -}       | r = \closedint 0 2       | c = {{Defof|Closed Real Interval}} }} {{end-eqn}} :500px
Kuratowski's Closure-Complement Problem/Interior of Closure
The interior of the closure of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^{- \, \circ}       | r = \openint 0 2       | c = {{Defof|Open Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \openint 4 5       | c = {{Defof|Open Real Interval}} }} {{end-eqn}} :500px
Kuratowski's Closure-Complement Problem/Interior of Closure of Interior
The interior of the closure of the interior of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^{\circ \, - \, \circ}       | r = \left({0 \,.\,.\, 2}\right)       | c = {{Defof|Open Real Interval}} }} {{end-eqn}} :500px
Kuratowski's Closure-Complement Problem/Interior of Complement of Interior
The interior of the complement of the interior of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^{\circ \, \prime \, \circ}       | r = \openint \gets 0       | c = {{Defof|Unbounded Open Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \openint 2 \to       | c = {{Defof|Unbounded Open Real Interval}} }} {{end-eqn}} :500px
Kuratowski's Closure-Complement Problem/Closure of Interior of Complement
The closure of the interior of the complement of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^{\prime \, \circ \, -}       | r = \hointl \gets 0       | c = {{Defof|Unbounded Closed Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \closedint 2 4       | c = {{Defof|Closed Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \hointr 5 \to       | c = {{Defof|Unbounded Closed Real Interval}} }} {{end-eqn}} :500px
Kuratowski's Closure-Complement Problem/Closure of Interior of Closure
The closure of the interior of the closure of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^{- \, \circ \, -}       | r = \left[{0 \,.\,.\, 2}\right]       | c = {{Defof|Closed Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \left[{4 \,.\,.\, 5}\right]       | c = {{Defof|Closed Real Interval}} }} {{end-eqn}} :500px
Kuratowski's Closure-Complement Problem/Interior of Complement of Interior of Closure
The interior of the complement of the interior of the closure of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^{- \, \circ \, \prime \, \circ}       | r = \left({\gets \,.\,.\, 0}\right)       | c = {{Defof|Unbounded Open Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \left({2 \,.\,.\, 4}\right)       | c = {{Defof|Open Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \left({5 \,.\,.\, \to}\right)       | c = {{Defof|Unbounded Open Real Interval}} }} {{end-eqn}} :500px
Kuratowski's Closure-Complement Problem/Complement of Interior of Closure of Interior
The complement of the interior of the closure of the interior of $A$ in $\R$ is given by: {{begin-eqn}} {{eqn | l = A^{\circ \, - \, \circ \, \prime}       | r = \left({\gets \,.\,.\, 0}\right]       | c = {{Defof|Unbounded Closed Real Interval}} }} {{eqn | o =       | ro= \cup       | r = \left[{2 \,.\,.\, \to}\right)       | c = {{Defof|Unbounded Closed Real Interval}} }} {{end-eqn}} :500px
Sum of Integrals on Adjacent Intervals for Integrable Functions
Let $f$ be a real function which is Darboux integrable on any closed interval $\mathbb I$. Let $a, b, c \in \mathbb I$. Then: :$\displaystyle \int_a^c \map f t \rd t + \int_c^b \map f t \rd t = \int_a^b \map f t \rd t$
Closure of Irrational Numbers is Real Numbers
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the same topology. Then: :$\paren {\R \setminus \Q}^- = \R$ where $\paren {\R \setminus \Q}-$ denotes the closure of $\R \setminus \Q$.
Closure of Union of Adjacent Open Intervals
Let $a, b, c \in R$ where $a < b < c$. Let $A$ be the union of the two adjacent open intervals: :$A := \left({a \,.\,.\, b}\right) \cup \left({b \,.\,.\, c}\right)$ Then: :$A^- = \left[{a \,.\,.\, c}\right]$ where: : $A^-$ is the closure of $A$.
Interior equals Complement of Closure of Complement
Let $T$ be a topological space. Let $H \subseteq T$. Let $H^-$ denote the closure of $H$ and $H^\circ$ denote the interior of $H$. Let $H^\prime$ denote the complement of $H$ in $T$: :$H^\prime = T \setminus H$ Then: :$H^\circ = H^{\prime \, - \, \prime}$
Closure of Half-Open Real Interval is Closed Real Interval
Let $\struct {\R, \tau_d}$ be the real number line under the usual (Euclidean) topology.   Let $H_1 = \hointl a b$ and $H_2 = \hointr a b$ be half-open intervals of $\R$. Then the closure of both $H_1$ and $H_2$ in $\R$ are the closed interval $\closedint a b$.
Closure of Irrational Interval is Closed Real Interval
Let $\struct {\R, \tau_d}$ be the real numbers under the usual (Euclidean) topology. Let $\struct{\R \setminus \Q, \tau_d}$ be the irrational number space under the same topology. Let $a, b \in \R$ such that $a < b$. Let $\Bbb I \subseteq \R$ be an interval of $\R$ Then the closure of the set: :$\Bbb I \cap \paren {\R \setminus \Q}$ is the closed real interval $\closedint a b$.
Closure of Rational Interval is Closed Real Interval
Let $\struct {\R, \tau_d}$ be the real numbers under the usual (Euclidean) topology. Let $\struct {\Q, \tau_d}$ be the rational number space under the same topology. Let $a, b \in \R$ such that $a < b$. Let $\Bbb I \subseteq \R$ be an interval of $\R$ Then the closure of the set : :$\Bbb I \cap \Q$ is the closed real interval $\closedint a b$.
Interior of Singleton in Real Number Line is Empty
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $a \in \R$ be a real number. Then: :$\set a^\circ = \O$ where $\set a^\circ$ denotes the interior of $\set a$ in $\R$.
Kuratowski's Closure-Complement Problem/Proof of Maximum
Let $T = \left({S, \tau}\right)$ be a topological space. Let $A \subseteq S$ be a subset of $T$. By successive applications of the operations of complement relative to $S$ and the closure, there can be no more than $14$ distinct subsets of $S$ (including $A$ itself).
Closure of Complement of Closure is Regular Closed
Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq S$ be a subset of $T$. Let $A^-$ denote the closure of $A$ in $T$. Let $A'$ denote the complement of $A$ in $S$: $A' = S \setminus A$. Then $A^{- ' -}$ is regular closed.
Topology induced by Usual Metric on Positive Integers is Discrete
Let $\Z_{>0}$ be the set of (strictly) positive integers. Let $d: \Z_{>0} \times \Z_{>0} \to \R$ be the usual (Euclidean) metric on $\Z_{>0}$. Then the metric topology for $d$ is a discrete topology.
Topology induced by Scaled Euclidean Metric on Positive Integers is Discrete
Let $\Z_{>0}$ be the set of (strictly) positive integers. Let $\delta: \Z_{>0} \times \Z_{>0} \to \R$ be the metric on $\Z_{>0}$ defined as: :$\forall x, y \in \Z_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$ Then the metric topology for $\delta$ is a discrete topology.
Scaled Euclidean Metric is Metric
Let $\R_{>0}$ be the set of (strictly) positive integers. Let $\delta: \R_{>0} \times \R_{>0} \to \R$ be the metric on $\R_{>0}$ defined as: :$\forall x, y \in \R_{>0}: \delta \left({x, y}\right) = \dfrac {\left\lvert{x - y}\right\rvert} {x y}$ Then $\delta$ is a metric.
Topologies induced by Usual Metric and Scaled Euclidean Metric on Positive Integers are Homeomorphic
Let $\Z_{>0}$ be the set of (strictly) positive integers. Let $d: \Z_{>0} \times \Z_{>0} \to \R$ be the usual (Euclidean) metric on $\Z_{>0}$. Let $\delta: \Z_{>0} \times \Z_{>0} \to \R$ be the metric on $\Z_{>0}$ defined as: :$\forall x, y \in \Z_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$ Let $\tau_d$ denote the metric topology for $d$. Let $\tau_\delta$ denote the metric topology for $\delta$. Then $\struct {\Z_{>0}, \tau_d}$ and $\struct {\Z_{>0}, \tau_\delta}$ are homeomorphic.
Cauchy Sequence in Positive Integers under Usual Metric is eventually Constant
Let $\Z_{>0}$ be the set of (strictly) positive integers. Let $d: \Z_{>0} \times \Z_{>0} \to \R$ be the usual (Euclidean) metric on $\Z_{>0}$. Let $\sequence {x_n}$ be a Cauchy sequence in $\struct {\Z_{>0}, d}$. Then: :$\exists m, n \in \Z_{>0}: \forall r > n: x_r = m$ That is, $\sequence {x_n}$ is eventually constant.
Positive Integers under Usual Metric is Complete Metric Space
Let $\Z_{>0}$ be the set of (strictly) positive integers. Let $d: \Z_{>0} \times \Z_{>0} \to \R$ be the usual (Euclidean) metric on $\Z_{>0}$. Then $\struct {\Z_{>0}, d}$ is a complete metric space.
Cauchy Sequence in Positive Integers under Scaled Euclidean Metric
Let $\Z_{>0}$ be the set of (strictly) positive integers. Let $\delta: \Z_{>0} \times \Z_{>0} \to \R$ be the scaled Euclidean metric on $\Z_{>0}$ defined as: :$\forall x, y \in \Z_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$ The sequence $\sequence {x_n}$ in $\Z_{>0}$ defined as: :$\forall n \in \N: x_n = n$ is a Cauchy sequence in $\struct {\Z_{>0}, \delta}$.
Positive Integers under Scaled Euclidean Metric is not Complete Metric Space
Let $\Z_{>0}$ be the set of (strictly) positive integers. Let $\delta: \Z_{>0} \times \Z_{>0} \to \R$ be the scaled Euclidean metric on $\Z_{>0}$ defined as: :$\forall x, y \in \Z_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$ Then $\struct {\Z_{>0}, \delta}$ is not a complete metric space.
Subset of Euclidean Plane whose Product of Coordinates are Greater Than or Equal to 1 is Closed
Let $\struct {\R^2, \tau_d}$ be the real number plane with the usual (Euclidean) topology. Let $A \subseteq R^2$ be the set of all points defined as: :$A := \set {\tuple {x, y} \in \R^2: x y \ge 1}$ Then $A$ is a closed set in $\struct {\R^2, d}$.
Projection on Real Euclidean Plane is Open Mapping
Let $\struct {\R^2, d}$ be the real number plane with the usual (Euclidean) topology. Let $\rho: \R^2 \to \R$ be the first projection on $\R^2$ defined as: :$\forall \tuple{x, y} \in \R^2: \map \rho {x, y} = x$ Then $\rho$ is an open mapping. The same applies with the second projection on $\R^2$.
Projection on Real Euclidean Plane is not Closed Mapping
Let $\struct {\R^2, d}$ be the real number plane with the usual (Euclidean) topology. Let $\rho: \R^2 \to \R$ be the first projection on $\R^2$ defined as: :$\forall \tuple {x, y} \in \R^2: \map \rho {x, y} = x$ Then $\rho$ is not a closed mapping. The same applies with the second projection on $\R^2$.
Complement of Set of Rational Pairs in Real Euclidean Plane is Arc-Connected
Let $\struct {\R^2, d}$ be the real number plane with the usual (Euclidean) topology. Let $S \subseteq \R^2$ be the subset of $\R^2$ defined as: :$\forall x, y \in \R^2: \tuple {x, y} \in S \iff x, y \in \Q$ Hence let $A := \R^2 \setminus S$: :$\tuple {x, y} \in A$  either $x$ or $y$ or both is irrational. Then $A$ is arc-connected.
Empty Set is Compact Space
Let $T = \left({S, \tau}\right)$ be a topological space. Then the empty set $\varnothing$ is a compact subspace of $T$.
Alexandroff Extension is Topology
Let $T = \struct {S, \tau}$ be a non-empty topological space. Let $p$ be a new element not in $S$. Let $S^* := S \cup \set p$. Let $T^* = \struct {S^*, \tau^*}$ be the Alexandroff extension on $S$. Then $\tau^*$ is a topology on $S^*$.
Intersection of Closed Set with Compact Subspace is Compact
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be closed in $T$. Let $K \subseteq S$ be compact in $T$. Then $H \cap K$ is compact in $T$.
Finite Union of Compact Sets is Compact
Let $T = \left({S, \tau}\right)$ be a topological space. Let $n \in \N$ be a natural number. Let $\left\langle{U_i}\right\rangle_{1 \mathop \le i \mathop \le n}$ be a finite sequence of compact subsets of $T$. Let $\mathcal U_n := \displaystyle \bigcup_{i \mathop = 1}^n U_i$ be the union of $\left\langle{U_i}\right\rangle$. Then $\mathcal U_n$ is compact in $T$.
Principle of Mathematical Induction
Let $\map P n$ be a propositional function depending on $n \in \Z$. Let $n_0 \in \Z$ be given. Suppose that: :$(1): \quad \map P {n_0}$ is true :$(2): \quad \forall k \in \Z: k \ge n_0 : \map P k \implies \map P {k + 1}$ Then: :$\map P n$ is true for all $n \in \Z$ such that $n \ge n_0$.
Principle of Finite Induction
Let $S \subseteq \Z$ be a subset of the integers. Let $n_0 \in \Z$ be given. Suppose that: :$(1): \quad n_0 \in S$ :$(2): \quad \forall n \ge n_0: n \in S \implies n + 1 \in S$ Then: :$\forall n \ge n_0: n \in S$ That is: :$S = \set {n \in \Z: n \ge n_0}$
Principle of Mathematical Induction/Peano Structure
Let $\struct {P, s, 0}$ be a Peano structure. Let $\map Q n$ be a propositional function depending on $n \in P$. Suppose that: :$(1): \quad \map Q 0$ is true :$(2): \quad \forall n \in P: \map Q n \implies \map Q {\map s n}$ Then: :$\forall n \in P: \map Q n$
Principle of Mathematical Induction/Naturally Ordered Semigroup
Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup. Let $T \subseteq S$ such that $0 \in T$ and $n \in T \implies n \circ 1 \in T$. Then $T = S$.
Consecutive Fibonacci Numbers are Coprime
Let $F_k$ be the $k$th Fibonacci number. Then: :$\forall n \ge 2: \gcd \set {F_n, F_{n + 1} } = 1$ where $\gcd \set {a, b}$ denotes the greatest common divisor of $a$ and $b$. That is, a Fibonacci number and the one next to it are coprime.
Divisibility of Fibonacci Number
:$\forall m, n \in \Z_{> 2} : m \divides n \iff F_m \divides F_n$ where $\divides$ denotes divisibility.
Fibonacci Number in terms of Smaller Fibonacci Numbers
:$\forall m, n \in \Z_{>0}: F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$
GCD of Fibonacci Numbers
:$\forall m, n \in \Z_{> 2}: \gcd \set {F_m, F_n} = F_{\gcd \set {m, n} }$ where $\gcd \set {a, b}$ denotes the greatest common divisor of $a$ and $b$.
Catalan's Identity
:${F_n}^2 - F_{n - r} F_{n + r} = \left({-1}\right)^{n - r} {F_r}^2$
Fibonacci Number with Negative Index
:$\forall n \in \Z_{> 0} : F_{-n} = \left({-1}\right)^{n + 1} F_n$
Fibonacci Number in terms of Larger Fibonacci Numbers
:$\forall m, n \in \Z_{>0} : F_{m - n} = \left({-1}\right)^{n + 1} F_{m - 1} F_n + \left({-1}\right)^n F_m F_{n - 1}$
Vajda's Identity
==== Formulation 1 ==== {{:Vajda's Identity/Formulation 1}} ==== Formulation 2 ==== {{:Vajda's Identity/Formulation 2}}
Determinant of Matrix Product/General Case
Let $\mathbf A_1, \mathbf A_2, \cdots, \mathbf A_n$ be square matrices of order $n$, where $n > 1$. Then: :$\map \det {\mathbf A_1 \mathbf A_2 \cdots \mathbf A_n} = \map \det {\mathbf A_1} \map \det {\mathbf A_2} \cdots \map \det {\mathbf A_n}$
Opposite Sides Equal implies Parallelogram
Let $ABCD$ be a convex quadrilateral with $AB = CD$ and $BC = AD$. Then $ABCD$ is a parallelogram.
Properties of Fibonacci Numbers
Let $F_n$ denote the $n$th Fibonacci number: {{:Definition:Fibonacci Number}}
Principle of Mathematical Induction/Naturally Ordered Semigroup/General Result
Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup. Let $p \in S$. Let $T \subseteq S$ such that: :$x \in T \implies p \preceq x \land \paren {x \in T \implies x \circ 1 \in T}$ Then: :$S \setminus S_p \subseteq T$ where: :$\setminus$ denotes set difference :$S_p$ denotes the set of all elements of $S$ preceding $p$.
Principle of Mathematical Induction for Minimal Infinite Successor Set
Let $\omega$ be the minimal infinite successor set. Let $S \subseteq \omega$. Suppose that: :$(1): \quad \varnothing \in S$ :$(2): \quad \forall x: x \in S \implies x^+ \in S$ where $x^+$ is the successor set of $x$. Then: :$S = \omega$
Principle of Mathematical Induction for Natural Numbers in Real Numbers
Let $\struct {\R, +, \times, \le}$ be the field of real numbers. Let $\N$ be the natural numbers in $\R$. Suppose that $A \subseteq \N$ is an inductive set. Then $A = \N$.
Chinese Remainder Theorem (Commutative Algebra)
Let $A$ be a commutative ring. {{explain|Presumably a Definition:Commutative and Unitary Ring is actually required here?}} Let $I_1, \ldots, I_k$ for some $k \ge 1$ be pairwise coprime ideals in $A$, that is: : $\forall i \ne j: I_i + I_j = A$ Then there is an isomorphism of rings: :$A / \left({I_1 \cap \ldots \cap I_k}\right) \to A / I_1 \times \cdots \times A / I_k$ which is induced by the ring homomorphism $\phi: A \to A / I_1 \times \cdots \times A / I_k$ defined as: :$\phi \left({x}\right) = \left({x + I_1, \ldots, x + I_k}\right)$ which passes through the quotient. {{explain|what is meant by "passes through the quotient"?}}
Sine of i
:$\sin i = \paren {\dfrac e 2 - \dfrac 1 {2 e} } i$
Cosine of i
:$\cos i = \dfrac e 2 + \dfrac 1 {2 e}$
Tangent of i
:$\tan i = \left({\dfrac {e^2 - 1} {e^2 + 1} }\right) i$
Cosecant of i
:$\csc i = \left({\dfrac {2 e} {1 - e^2} }\right) i$
Secant of i
:$\sec i = \dfrac {2 e} {e^2 + 1}$
Cotangent of i
:$\cot i = \left({\dfrac {1 + e^2} {1 - e^2} }\right) i$
Sine in terms of Cosine
{{begin-eqn}} {{eqn | l = \sin x       | r = +\sqrt {1 - \cos ^2 x}       | c = if there exists an integer $n$ such that $2 n \pi < x < \paren {2 n + 1} \pi$ }} {{eqn | l = \sin x       | r = -\sqrt {1 - \cos ^2 x}       | c = if there exists an integer $n$ such that $\paren {2 n + 1} \pi < x < \paren {2 n + 2} \pi$ }} {{end-eqn}}
Sine is Reciprocal of Cosecant
:$\sin \theta = \dfrac 1 {\csc \theta}$
Tangent is Reciprocal of Cotangent
:$\tan \theta = \dfrac 1 {\cot \theta}$
Sine of Complex Number
Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: :$\sin \paren {a + b i} = \sin a \cosh b + i \cos a \sinh b$ where: :$\sin$ denotes the sine function (real and complex) :$\cos$ denotes the real cosine function :$\sinh$ denotes the hyperbolic sine function :$\cosh$ denotes the hyperbolic cosine function.
Cosine of Complex Number
Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: :$\cos \left({a + b i}\right) = \cos a \cosh b - i \sin a \sinh b$ where: :$\cos$ denotes the cosine function (real and complex) :$\sin$ denotes the real sine function :$\sinh$ denotes the hyperbolic sine function :$\cosh$ denotes the hyperbolic cosine function
Tangent of Complex Number
Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then:
Sine in terms of Secant
{{begin-eqn}} {{eqn | l = \sin x       | r = + \frac {\sqrt{\sec ^2 x - 1} } {\sec x}       | c = if there exists an integer $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$ }} {{eqn | l = \sin x       | r = - \frac {\sqrt{\sec ^2 x - 1} } {\sec x}       | c = if there exists an integer $n$ such that $\paren {n + \dfrac 1 2} \pi < x < \paren {n + 1} \pi$ }} {{end-eqn}}
Sign of Sine
Let $x$ be a real number. {{begin-eqn}} {{eqn | l = \sin x       | o = >       | r = 0       | c = if there exists an integer $n$ such that $2 n \pi < x < \paren {2 n + 1} \pi$ }} {{eqn | l = \sin x       | o = <       | r = 0       | c = if there exists an integer $n$ such that $\paren {2 n + 1} \pi < x < \paren {2 n + 2} \pi$ }} {{end-eqn}} where $\sin$ is the real sine function.
Second Principle of Mathematical Induction
Let $\map P n$ be a propositional function depending on $n \in \Z$. Let $n_0 \in \Z$ be given. Suppose that: :$(1): \quad \map P {n_0}$ is true :$(2): \quad \forall k \in \Z: k \ge n_0: \map P {n_0} \land \map P {n_0 + 1} \land \ldots \land \map P {k - 1} \land \map P k \implies \map P {k + 1}$ Then: :$\map P n$ is true for all $n \ge n_0$. This process is called '''proof by (mathematical) induction'''.
Second Principle of Finite Induction
Let $S \subseteq \Z$ be a subset of the integers. Let $n_0 \in \Z$ be given. Suppose that: :$(1): \quad n_0 \in S$ :$(2): \quad \forall n \ge n_0: \paren {\forall k: n_0 \le k \le n \implies k \in S} \implies n + 1 \in S$ Then: :$\forall n \ge n_0: n \in S$