hcju/proofwikips huggingface dataset:
Cardinality of Generator of Vector Space is not Less than Dimension
Let $V$ be a vector space over a field $F$. Let $\BB$ be a generator for $V$ containing $m$ elements. Then: :$\map {\dim_F} V \le m$ where $\map {\dim_F} V$ is the dimension of $V$.
Trichotomy Law for Real Numbers
The real numbers obey the Trichotomy Law. That is, $\forall a, b \in \R$, exactly one of the following holds: {{begin-axiom}} {{axiom | n = 1         | lc= $a$ is greater than $b$:         | m = a > b }} {{axiom | n = 2         | lc= $a$ is equal to $b$:         | m = a = b }} {{axiom | n = 3         | lc= $a$ is less than $b$:         | m = a < b }} {{end-axiom}}
Real Number Ordering is Compatible with Multiplication
=== Positive Factor === {{:Real Number Ordering is Compatible with Multiplication/Positive Factor}} === Negative Factor === {{:Real Number Ordering is Compatible with Multiplication/Negative Factor}}
Real Number Ordering is Compatible with Addition
:$\forall a, b, c \in \R: a < b \implies a + c < b + c$
Real Number Inequalities can be Added
Let $a, b, c, d \in \R$ such that $a > b$ and $c > d$. Then: :$a + c > b + d$
Positive Real Number Inequalities can be Multiplied
Let $a, b, c, d \in \R$ such that $a > b$ and $c > d$. Let $b > 0$ and $d > 0$. Then $a c > b d$. If $b < 0$ or $d < 0$ the inequality does not hold.
Real Ordering Incompatible with Subtraction
Let $a, b, c, d \in R$ be real numbers such that $a > b$ and $c > d$. Then it does not necessarily hold that: :$a - c > b - d$
Real Ordering Incompatible with Division
Let $a, b, c, d \in \R$ be real numbers such that $a > b$ and $c > d$. Then it does not necessarily hold that: :$\dfrac a c > \dfrac b d$
Positive Real Number Inequalities can be Multiplied/Disproof for Negative Parameters
Let $a, b, c, d \in \R$ such that $a > b$ and $c > d$. Let $b > 0$ and $d > 0$. From Positive Real Number Inequalities can be Multiplied, $a c > b d$ holds. However, if $b < 0$ or $d < 0$ the inequality does ''not'' hold.
Reverse Triangle Inequality/Real and Complex Fields/Corollary
Let $x$ and $y$ be elements of either the real numbers $\R$ or the complex numbers $\C$. Then: :$\size {x - y} \ge \size x - \size y$ where $\size x$ denotes either the absolute value of a real number or the complex modulus of a complex number.
Combination Theorem for Sequences/Real/Sum Rule
:$\displaystyle \lim_{n \mathop \to \infty} \paren {x_n + y_n} = l + m$
Combination Theorem for Sequences/Real/Multiple Rule
:$\displaystyle \lim_{n \mathop \to \infty} \paren {\lambda x_n} = \lambda l$
Combination Theorem for Sequences/Real/Combined Sum Rule
:$\displaystyle \lim_{n \mathop \to \infty} \paren {\lambda x_n + \mu y_n} = \lambda l + \mu m$
Combination Theorem for Sequences/Real/Product Rule
:$\displaystyle \lim_{n \mathop \to \infty} \paren {x_n y_n} = l m$
Combination Theorem for Sequences/Real/Quotient Rule
:$\displaystyle \lim_{n \mathop \to \infty} \frac {x_n} {y_n} = \frac l m$ provided that $m \ne 0$.
Squeeze Theorem/Sequences/Real Numbers
Let $\sequence {x_n}$, $\sequence {y_n}$ and $\sequence {z_n}$ be sequences in $\R$. Let $\sequence {y_n}$ and $\sequence {z_n}$ both be convergent to the following limit: :$\displaystyle \lim_{n \mathop \to \infty} y_n = l, \lim_{n \mathop \to \infty} z_n = l$ Suppose that: :$\forall n \in \N: y_n \le x_n \le z_n$ Then: :$x_n \to l$ as $n \to \infty$ that is: :$\displaystyle \lim_{n \mathop \to \infty} x_n = l$ Thus, if $\sequence {x_n}$ is always between two other sequences that both converge to the same limit, $\sequence {x_n} $ is said to be '''sandwiched''' or '''squeezed''' between those two sequences and itself must therefore converge to that same limit.
Squeeze Theorem/Sequences/Complex Numbers
Let $\left \langle {a_n} \right \rangle$ be a sequence in $\R$ which is null, that is: :$a_n \to 0$ as $n \to \infty$ Let $\left \langle {z_n} \right \rangle$ be a sequence in $\C$. Suppose $\left \langle {a_n} \right \rangle$ dominates $\left \langle {z_n} \right \rangle$. That is, suppose that: : $\forall n \in \N: \left|{z_n}\right| \le a_n$ Then $\left \langle {z_n} \right \rangle$ is a null sequence.
Squeeze Theorem/Sequences
There are two versions of this result: * one for sequences in the set of complex numbers $\C$, and more generally for sequences in a metric space. * one for sequences in the set of real numbers $\R$, and more generally in linearly ordered spaces (which is stronger). {{explain|stronger in what sense?}} === Sequences of Real Numbers === {{:Squeeze Theorem/Sequences/Real Numbers}} === Sequences of Complex Numbers === {{:Squeeze Theorem/Sequences/Complex Numbers}} === Sequences in a Linearly Ordered Space === {{:Squeeze Theorem/Sequences/Linearly Ordered Space}} === Sequences in a Metric Space === {{:Squeeze Theorem/Sequences/Metric Spaces}}
Squeeze Theorem/Functions
Let $a$ be a point on an open real interval $I$. Let $f$, $g$ and $h$ be real functions defined at all points of $I$ except for possibly at point $a$. Suppose that: :$\forall x \ne a \in I: \map g x \le \map f x \le \map h x$ :$\displaystyle \lim_{x \mathop \to a} \ \map g x = \lim_{x \mathop \to a} \ \map h x = L$ Then: :$\displaystyle \lim_{x \mathop \to a} \ \map f x = L$
Weierstrass's Theorem
There exists a real function $f: \closedint 0 1 \to \closedint 0 1$ such that: :$(1): \quad f$ is continuous :$(2): \quad f$ is nowhere differentiable.
Combination Theorem for Continuous Functions
Let $X$ be one of the standard number fields $\Q, \R, \C$. Let $f$ and $g$ be functions which are continuous on an open subset $S \subseteq X$. Let $\lambda, \mu \in X$ be arbitrary numbers in $X$. Then the following results hold: === Sum Rule === {{:Combination Theorem for Continuous Functions/Sum Rule}} === Multiple Rule === {{:Combination Theorem for Continuous Functions/Multiple Rule}} === Combined Sum Rule === {{:Combination Theorem for Continuous Functions/Combined Sum Rule}} === Product Rule === {{:Combination Theorem for Continuous Functions/Product Rule}} === Quotient Rule === {{:Combination Theorem for Continuous Functions/Quotient Rule}}
Combination Theorem for Limits of Functions/Sum Rule
:$\displaystyle \lim_{x \mathop \to c} \paren {\map f x + \map g x} = l + m$
Combination Theorem for Limits of Functions/Multiple Rule
:$\displaystyle \lim_{x \mathop \to c} \ \paren {\lambda f \paren x} = \lambda l$
Combination Theorem for Limits of Functions/Combined Sum Rule
:$\displaystyle \lim_{x \mathop \to c} \left({\lambda f \left({x}\right) + \mu g \left({x}\right)}\right) = \lambda l + \mu m$
Combination Theorem for Limits of Functions/Product Rule
:$\displaystyle \lim_{x \mathop \to c} \ \paren {\map f x \map  g x} = l m$
Combination Theorem for Limits of Functions/Quotient Rule
:$\displaystyle \lim_{x \mathop \to c} \frac {\map f x} {\map g x} = \frac l m$ provided that $m \ne 0$.
Combination Theorem for Continuous Functions/Sum Rule
:$f + g$ is continuous on $S$.
Combination Theorem for Continuous Functions/Product Rule
:$f g$ is continuous on $S$
Combination Theorem for Continuous Functions/Multiple Rule
:$\lambda f$ is continuous on $S$.
Combination Theorem for Continuous Functions/Combined Sum Rule
:$\lambda f + \mu g$ is continuous on $S$.
Combination Theorem for Continuous Functions/Quotient Rule
:$\dfrac f g$ is continuous on $S \setminus \set {x \in S: \map g x = 0}$ that is, on all the points $x$ of $S$ where $\map g x \ne 0$.
Brouwer's Fixed Point Theorem/One-Dimensional Version
Let $f: \closedint a b \to \closedint a b$ be a real function which is continuous on the closed interval $\closedint a b$. Then: :$\exists \xi \in \closedint a b: \map f \xi = \xi$ That is, a continuous real function from a closed real interval to itself fixes some point of that interval.
Brouwer's Fixed Point Theorem/Smooth Mapping
A smooth mapping $f$ of the closed unit ball $B^n \subset \R^n$ into itself has a fixed point: :$\forall f \in \map {C^\infty} {B^n \to B^n}: \exists x \in B^n: \map f x = x$
Power Rule for Derivatives/Natural Number Index
Let $n \in \N$. Let $f: \R \to \R$ be the real function defined as $f \left({x}\right) = x^n$. Then: :$f' \left({x}\right) = n x^{n-1}$ everywhere that $f \left({x}\right) = x^n$ is defined. When $x = 0$ and $n = 0$, $f^{\prime} \left({x}\right)$ is undefined.
Power Rule for Derivatives/Integer Index
Let $n \in \Z$. Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$. Then: :$\map {f'} x = n x^{n - 1}$ everywhere that $\map f x = x^n$ is defined. When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined.
Power Rule for Derivatives/Fractional Index
Let $n \in \N_{>0}$. Let $f: \R \to \R$ be the real function defined as $\map f x = x^{1 / n}$. Then: :$\map {f'} x = n x^{n - 1}$ everywhere that $\map f x = x^n$ is defined. When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined.
Power Rule for Derivatives/Rational Index
Let $n \in \Q$. Let $f: \R \to \R$ be the real function defined as $f \left({x}\right) = x^n$. Then: :$\map {f'} x = n x^{n-1}$ everywhere that $\map f x = x^n$ is defined. When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined.
Power Rule for Derivatives/Real Number Index
Let $n \in \R$. Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$. Then: :$\map {f'} x = n x^{n-1}$ everywhere that $\map f x = x^n$ is defined. When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined.
Well-Ordering Minimal Elements are Unique
Let $\left({S,\preceq}\right)$ be a well-ordered set. Then every non-empty subset of $S$ has a unique minimal element.
Fundamental Theorem of Calculus/First Part
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$. Let $F$ be a real function which is defined on $\closedint a b$ by: :$\displaystyle \map F x = \int_a^x \map f t \rd t$ Then $F$ is a primitive of $f$ on $\closedint a b$.
Fundamental Theorem of Calculus/Second Part
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$. Then: :$(1): \quad f$ has a primitive on $\closedint a b$ :$(2): \quad$ If $F$ is any primitive of $f$ on $\closedint a b$, then: :::$\displaystyle \int_a^b \map f t \rd t = \map F b - \map F a = \bigintlimits {\map F t} a b$
Natural Logarithm of 1 is 0
:$\ln 1 = 0$
Derivative of Natural Logarithm Function
Let $\ln x$ be the natural logarithm function. Then: :$\map {\dfrac \d {\d x} } {\ln x} = \dfrac 1 x$
Derivative of General Logarithm Function
Let $a \in \R_{>0}$ such that $a \ne 1$ Let $\log_a x$ be the logarithm function to base $a$. Then: :$\map {\dfrac \d {\d x} } {\log_a x} = \dfrac {\log_a e} x$
Logarithm is Strictly Increasing
:$\ln x: x > 0$ is strictly increasing.
Logarithm Tends to Infinity
:$\ln x \to +\infty$ as $x \to +\infty$
Logarithm Tends to Negative Infinity
:$\ln x \to -\infty$ as $x \to 0^+$
Second Derivative of Natural Logarithm Function
Let $\ln x$ be the natural logarithm function. Then: :$\map {\dfrac {\d^2} {\d x^2} } {\ln x} = -\dfrac 1 {x^2}$
Exponential of Zero and One
=== Exponential of Zero === {{:Exponential of Zero}} === Exponential of One === {{:Exponential of One}}
Exponential is Strictly Increasing
:The function $\map f x = \exp x$ is strictly increasing.
Exponential Tends to Zero and Infinity
:$\exp x \to +\infty$ as $x \to +\infty$ :$\exp x \to 0$ as $x \to -\infty$ Thus the exponential function has domain $\R$ and image $\openint 0 \to$.
Exponential of Natural Logarithm
: $\forall x > 0: \exp \left({\ln x}\right) = x$ : $\forall x \in \R: \ln \left({\exp x}\right) = x$
Exponent Combination Laws/Product of Powers
:$a^x a^y = a^{x + y}$
Exponent Combination Laws/Power of Product
:$\paren {a b}^x = a^x b^x$
Exponent Combination Laws/Negative Power
: $a^{-x} = \dfrac 1 {a^x}$
Exponent Combination Laws/Power of Power
Let $x, y \in \R$ be real numbers. Let $a^x$ be defined as $a$ to the power of $x$. Then: :$\paren {a^x}^y = a^{x y}$
Exponent Combination Laws/Quotient of Powers
:$\dfrac{a^x} {a^y} = a^{x - y}$
Exponent Combination Laws/Power of Quotient
:$\paren {\dfrac a b}^x = \dfrac {a^x} {b^x}$
Exponent Combination Laws/Negative Power of Quotient
: $\left({\dfrac a b}\right)^{-x} = \left({\dfrac b a}\right)^x$
Binomial Theorem/General Binomial Theorem
Let $\alpha \in \R$ be a real number. Let $x \in \R$ be a real number such that $\size x < 1$. Then: {{begin-eqn}} {{eqn | l = \paren {1 + x}^\alpha       | r = \sum_{n \mathop = 0}^\infty \frac {\alpha^{\underline n} } {n!} x^n       | c =  }} {{eqn | r = \sum_{n \mathop = 0}^\infty \dbinom \alpha n x^n       | c =  }} {{eqn | r = \sum_{n \mathop = 0}^\infty \frac 1 {n!} \paren {\prod_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } x^n       | c =  }} {{eqn | r = 1 + \alpha x + \dfrac {\alpha \paren {\alpha - 1} } {2!} x^2 + \dfrac {\alpha \paren {\alpha - 1} \paren {\alpha - 2} } {3!} x^3 + \cdots       | c =  }} {{end-eqn}} where: :$\alpha^{\underline n}$ denotes the falling factorial :$\dbinom \alpha n$ denotes a binomial coefficient.
Real Sine Function is Continuous
:$\sin x$ is continuous on $\R$.
Sine Function is Absolutely Convergent
:$\sin x$ is absolutely convergent for all $x \in \R$.
Sine of Zero is Zero
:$\sin 0 = 0$
Sine Function is Odd
:$\map \sin {-z} = -\sin z$ That is, the sine function is odd.
Cosine Function is Continuous
:$\cos x$ is continuous on $\R$.
Cosine Function is Absolutely Convergent
:$\cos x$ is absolutely convergent for all $x \in \R$.
Cosine of Zero is One
:$\cos 0 = 1$
Cosine of Integer Multiple of Pi
:$\forall n \in \Z: \cos n \pi = \paren {-1}^n$
Cosine Function is Even
:$\map \cos {-z} = \cos z$ That is, the cosine function is even.
Image of Interval by Derivative
Let $f$ be a real function that is everywhere differentiable. Let $I \subseteq \Dom f$ be a real interval. Then: :$f' \sqbrk I$ is a real interval where $f'$ denotes the derivative of $f$
Intermediate Value Theorem for Derivatives
Let $I$ be an open interval. Let $f : I \to \R$ be everywhere differentiable. Then $f'$ satisfies the Intermediate Value Property.
Sine of Half-Integer Multiple of Pi
:$\forall n \in \Z: \map \sin {n + \dfrac 1 2} \pi = \paren {-1}^n$
Zeroes of Sine and Cosine
:$(1): \quad \forall n \in \Z: x = \paren {n + \dfrac 1 2} \pi \implies \cos x = 0$ :$(2): \quad \forall n \in \Z: x = n \pi \implies \sin x = 0$
Sine of Integer Multiple of Pi
:$\forall n \in \Z: \sin n \pi = 0$
Cosine of Half-Integer Multiple of Pi
:$\forall n \in \Z: \map \cos {n + \dfrac 1 2} \pi = 0$
Rational Square Root of Integer is Integer
Let $n$ be an integer. Suppose that $\sqrt n$ is a rational number. Then $\sqrt n$ is an integer.
Definite Integral of Even Function
Let $f$ be an even function with a primitive on the closedinterval $\closedint {-a} a$, where $a > 0$. Then: :$\displaystyle \int_{-a}^a \map f x \rd x = 2 \int_0^a \map f x \rd x$
Definite Integral of Odd Function
Let $f$ be an odd function with a primitive on the open interval $\closedint {-a} a$, where $a > 0$. Then: :$\displaystyle \int_{-a}^a \map f x \rd x = 0$
Union of Equivalence Classes is Whole Set
Let $\RR \subseteq S \times S$ be an equivalence on a set $S$. Then the set of $\RR$-classes constitutes the whole of $S$.
Ratio of Consecutive Fibonacci Numbers
For $n \in \N$, let $f_n$ be the $n$th Fibonacci number. Then: :$\displaystyle \lim_{n \mathop \to \infty} \frac {f_{n + 1} } {f_n} = \phi$ where $\phi = \dfrac {1 + \sqrt 5} 2$ is the golden mean.
Cardinality of Image of Injection
Let $f: S \rightarrowtail T$ be an injection. Let $A \subseteq S$ be a finite subset of $S$. Then: :$\card {f \paren A} = \card A$ where $\card A$ denotes the cardinality of $A$.
Composition of Mappings is not Commutative
The composition of mappings is '''not''' in general a commutative binary operation: :$f_2 \circ f_1 \ne f_1 \circ f_2$
Intersection of Injective Image with Relative Complement
Let $f: S \to T$ be a mapping. Then $f$ is an injection : :$\forall A \subseteq S: f \sqbrk A \cap f \sqbrk {\relcomp S A} = \O$
Injection to Image is Bijection
Let $f: S \rightarrowtail T$ be an injection. Let $X \subseteq T$ be the image of $f$. Then the restriction $f {\restriction_{S \times X}}: S \to X$ of $f$ to the image of $f$ is a bijection of $S$ onto $X$.
Cardinality of Set of Induced Equivalence Classes of Surjection
Let $f: S \to T$ be a mapping. Let $\mathcal R_f \subseteq S \times S$ be the relation induced by $f$: :$\tuple {s_1, s_2} \in \mathcal R_f \iff \map f {s_1} = \map f {s_2}$ Let $f$ be a surjection. Then there are $\card T$ different $\mathcal R_f$-classes.
Power of Elements is Subgroup
Let $G$ be an abelian group whose identity is $e$. Then for any $n \in \Z$, the set $G^n = \set {x^n: x \in G}$ is a subgroup of $G$. Moreover, if: :$(1): \quad G$ is finite and: :$(2): \quad n \ne 1$ is a divisor of the order of $G$ then $G^n$ is a proper subgroup of $G$.
Index Laws/Sum of Indices/Semigroup
Let $\struct {S, \circ}$ be a semigroup. For $a \in S$, let $\circ^n a = a^n$ be defined as the $n$th power of $a$: :$a^n = \begin{cases} a & : n = 1 \\ a^x \circ a & : n = x + 1 \end{cases}$ That is: : $a^n = \underbrace {a \circ a \circ \cdots \circ a}_{n \text{ copies of } a} = \circ^n \paren a$ Then: :$\forall m, n \in \N_{>0}: a^{n + m} = a^n \circ a^m$
Index Laws/Product of Indices/Semigroup
Let $\struct {S, \circ}$ be a semigroup. For $a \in S$, let $\circ^n a = a^n$ be the $n$th power of $a$. Then: :$\forall m, n \in \N_{>0}: a^{n m} = \paren {a^n}^m = \paren {a^m}^n$
Semigroup is Subsemigroup of Itself
Let $\struct {S, \circ}$ be a semigroup. Then $\struct {S, \circ}$ is a subsemigroup of itself.
Identity of Monoid is Unique
Let $\struct {S, \circ}$ be a monoid. Then $S$ has a unique identity.
Identity of Group is Unique
Let $\struct {G, \circ}$ be a group which has an identity element $e \in G$. Then $e$ is unique.
Unity of Integral Domain is Unique
Let $\struct {D, +, \times}$ be an integral domain. Then the unity of $\struct {D, +, \times}$ is unique.
Zero of Integral Domain is Unique
Let $\struct {D, +, \times}$ be an integral domain. Then the zero of $\struct {D, +, \times}$ is unique.
Negative in Integral Domain is Unique
Let $\struct {D, +, \times}$ be an integral domain. Let $a \in R$. Then the negative $-a$ of $a$ is unique.
Inverse in Group is Unique
Let $\struct {G, \circ}$ be a group. Then every element $x \in G$ has exactly one inverse: :$\forall x \in G: \exists_1 x^{-1} \in G: x \circ x^{-1} = e^{-1} = x \circ x$ where $e$ is the identity element of $\struct {G, \circ}$.
Product Inverse in Ring is Unique
Let $\struct {R, +, \circ}$ be a ring with unity. Let $x \in R$ be a unit of $R$. Then the product inverse $x^{-1}$ of $x$ is unique.
Units of Gaussian Integers form Group
Let $U_\C$ be the set of units of the Gaussian integers: :$U_\C = \set {1, i, -1, -i}$ where $i$ is the imaginary unit: $i = \sqrt {-1}$. Let $\struct {U_\C, \times}$ be the algebraic structure formed by $U_\C$ under the operation of complex multiplication. Then $\struct {U_\C, \times}$ forms a cyclic group under complex multiplication.
Permutation Group is Subgroup of Symmetric Group
Let $S$ be a set. Let $\struct {\map \Gamma S, \circ}$ be the symmetric group on $S$, where $\circ$ denotes the composition operation. Let $\struct {H, \circ}$ be a set of permutations of $S$ which forms a group under $\circ$. Then $\struct {H, \circ}$ is a subgroup of $\struct {\map \Gamma S, \circ}$.
Units of Field of Complex Numbers form Group
:$\C^\times = \C \setminus \set 0$ where $\C^\times$ denotes the group of units of $\C$.
Determinant of Block Diagonal Matrix
Let $\mathbf A$ be a block diagonal matrix of order $n$. Let $\mathbf A_1,\ldots,\mathbf{A}_k$ be the square matrices on the diagonal, i.e.: :$\displaystyle \mathbf A = \begin{bmatrix} \mathbf A_1 & 0 & \cdots & 0 \\ 0 & \mathbf A_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \mathbf A_k  \end{bmatrix}$ Then the determinant $\det \left({\mathbf A}\right)$ of $\mathbf A$ satisfies: :$\displaystyle \det \left({\mathbf A}\right) = \prod_{i \mathop = 1}^k \det \left({\mathbf A_i}\right)$
Coset Space forms Partition
=== Left Coset Space forms Partition === {{:Left Coset Space forms Partition}} === Right Coset Space forms Partition === {{:Right Coset Space forms Partition}}
Element of Group is in Unique Coset of Subgroup
Let $G$ be a group. Let $H$ be a subgroup of $G$. Let $x \in G$.
General Linear Group to Determinant is Homomorphism
Let $\GL {n, \R}$ be the general linear group over the field of real numbers. Let $\struct {\R_{\ne 0}, \times}$ denote the multiplicative group of real numbers. Let $\det: \GL {n, \R} \to \struct {\R_{\ne 0}, \times}$ be the mapping: :$\mathbf A \mapsto \map \det {\mathbf A}$ where $\map \det {\mathbf A}$ is the determinant of $\mathbf A$. Then $\det$ is a group homomorphism.
Logarithm on Positive Real Numbers is Group Isomorphism
Let $\struct {\R_{>0}, \times}$ be the multiplicative group of positive real numbers. Let $\struct {\R, +}$ be the additive group of real numbers. Let $b$ be any real number such that $b > 1$. Let $\log_b: \struct {\R_{>0}, \times} \to \struct {\R, +}$ be the mapping: :$x \mapsto \map {\log_b} x$ where $\log_b$ is the logarithm to base $b$. Then $\log_b$ is a group isomorphism.
Kernel is Trivial iff Monomorphism/Group
Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a group homomorphism. Let $\map \ker \phi$ be the kernel of $\phi$. Then $\phi$ is a group monomorphism  $\map \ker \phi$ is trivial.
Kernel is Trivial iff Monomorphism/Ring
Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a ring homomorphism. Let $\map \ker \phi$ be the kernel of $\phi$. Then $\phi$ is a ring monomorphism  $\map \ker \phi = 0_{R_1}$.
First Isomorphism Theorem/Groups
Let $\phi: G_1 \to G_2$ be a group homomorphism. Let $\map \ker \phi$ be the kernel of $\phi$. Then: :$\Img \phi \cong G_1 / \map \ker \phi$ where $\cong$ denotes group isomorphism.
First Isomorphism Theorem/Rings
Let $\phi: R \to S$ be a ring homomorphism. Let $\map \ker \phi$ be the kernel of $\phi$. Then: :$\Img \phi \cong R / \map \ker \phi$ where $\cong$ denotes ring isomorphism.
Second Isomorphism Theorem/Groups
Let $G$ be a group, and let: :$(1): \quad H$ be a subgroup of $G$ :$(2): \quad N$ be a normal subgroup of $G$. Then: :$\displaystyle \frac H {H \cap N} \cong \frac {H N} N$ where $\cong$ denotes group isomorphism.
Second Isomorphism Theorem/Rings
Let $R$ be a ring, and let: :$S$ be a subring of $R$ :$J$ be an ideal of $R$. Then: :$(1): \quad S + J$ is a subring of $R$ :$(2): \quad J$ is an ideal of $S + J$ :$(3): \quad S \cap J$ is an ideal of $S$ :$(4): \quad \dfrac S {S \cap J} \cong \dfrac {S + J} J$ where $\cong$ denotes group isomorphism.
Third Isomorphism Theorem/Groups
Let $G$ be a group, and let: : $H, N$ be normal subgroups of $G$ : $N$ be a subset of $H$. Then: :$(1): \quad H / N$ is a normal subgroup of $G / N$ :::where $H / N$ denotes the quotient group of $H$ by $N$ :$(2): \quad \dfrac {G / N} {H / N} \cong \dfrac G H$ :::where $\cong$ denotes group isomorphism.
Third Isomorphism Theorem/Rings
Let $R$ be a ring, and let: :$J, K$ be ideals of $R$ :$J$ be a subset of $K$. Then: :$(1): \quad K / J$ is an ideal of $R / J$ :where $K / J$ denotes the quotient ring of $K$ by $J$ :$(2): \quad \dfrac {R / J} {K / J} \cong \dfrac R K$ :where $\cong$ denotes ring isomorphism.
Continuous Mapping on Union of Open Sets
Let $T = \left({X, \tau}\right)$ and $S = \left({Y, \sigma}\right)$ be topological spaces. Let $I$ be an indexing set. For all $i \in I$, let $C_i$ be open in $T$. Let $f: X \to Y$ be a mapping such that the restriction $f \restriction_{C_i}$ is continuous for all $i \in I$. Then $f$ is continuous on $C = \displaystyle \bigcup_{i \mathop \in I} C_i$, i.e., $f \restriction_C$ is continuous.
Inertia Principle
Let $\left \langle {a_n}\right \rangle$ be a sequence in $\R$. Let $a_n \to l$ as $n \to \infty$, and let $c < l$ where $c \in \R$. Then $\exists N \in \N$ such that: : $\forall n \in \N, n \ge N: c < a_n$