hcju/proofwikips huggingface dataset:
Condition for Woset to be Isomorphic to Ordinal
Let $\struct {S, \preceq}$ be a woset. Let $\struct {S, \preceq}$ be such that $\forall a \in S$, the initial segment $S_a$ of $S$ determined by $a$ is order isomorphic to some ordinal. Then $\struct {S, \preceq}$ itself is order isomorphic to an ordinal.
Woset is Isomorphic to Unique Ordinal
Every woset is order isomorphic to a unique ordinal.
Product of the Incidence Matrix of a BIBD with its Transpose
For any BIBD with parameters $v,k,\lambda$, let $A$ be its block incidence matrix. Then: : $A^\intercal \cdot A = \left({a_{ij}}\right) = \left({r - \lambda}\right) I_v + \lambda J_v$ where: * $A$ is $v \times b$ * $A^\intercal$ is the transpose of $A$ * $J_v$ is the all $v\times{v}$ $ 1$'s matrix * $I_v$ is the $v\times{v}$ identity matrix. That is: : $A^\intercal \cdot A = \begin{bmatrix} r & \lambda & \cdots & \lambda \\ \lambda & r & \cdots & \lambda \\ \vdots & \vdots & \ddots & \vdots \\ \lambda & \lambda & \cdots & r \\ \end{bmatrix}$
Fisher's Inequality
For any BIBD $\struct {v, k, \lambda}$, the number of blocks $b$ must be greater then or equal to the number of points $v$: :$ b \ge v$
Self-Distributive Law for Conditional
The following is known as the Self-Distributive Law:
Disjunction of Conditional and Converse
: $\vdash \left({p \implies q}\right) \lor \left({q \implies p}\right)$
Characteristics of Eulerian Graph
A finite (undirected) graph is Eulerian  it is connected and each vertex is even. Note that the definition of graph here includes: * Simple graph * Loop-graph * Multigraph * Loop-multigraph but does not include directed graph.
De Morgan's Laws
'''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:
Four Color Theorem
Any planar graph $G$ can be assigned a proper vertex $k$-coloring such that $k \le 4$.
Dirac's Theorem
If a connected graph $G$ has $n \ge 3$ vertices and the degree of each vertex is at least $\dfrac n 2$, then $G$ is Hamiltonian.
Functionally Complete Logical Connectives
These sets of logical connectives are functionally complete:
Properties of NAND
Let $\uparrow$ signify the NAND operation. The following results hold:
Properties of NOR
Let $\downarrow$ signify the NOR operation. The following results hold:
NAND and NOR are Functionally Complete
The NAND and NOR operators are each functionally complete. That is, NAND and NOR are Sheffer operators.
Equivalence of Semantic Consequence and Logical Implication
Let $U = \set {\phi_1, \phi_2, \ldots, \phi_m, \ldots}$ be a countable set of propositional formulas. Let $\psi$ be a propositional formula. Then $U \models \psi$  $U \vdash \psi$. That is, semantic consequence is equivalent to provable consequence.
WFFs of PropLog of Length 1
The only WFFs of propositional logic of length $1$ are: * The letters of the formal grammar of propositional logic $\mathcal L_0$ * The tautology symbol $\top$ * The contradiction symbol $\bot$.
WFF of PropLog is Balanced
Let $\mathbf A$ be a WFF of propositional logic. Then $\mathbf A$ is a balanced string.
Initial Part of WFF of PropLog is not WFF
Let $\mathbf A$ be a WFF of propositional logic. Let $\mathbf S$ be an initial part of $\mathbf A$. Then $\mathbf S$ is not a WFF of propositional logic.
Construction of Parallel Line
Given a straight line, and a given point not on that straight line, it is possible to draw a parallel to the given straight line. {{:Euclid:Proposition/I/31}}
Sum of Angles of Triangle equals Two Right Angles
In a triangle, the sum of the three interior angles equals two right angles. {{:Euclid:Proposition/I/32}}
Equivalence of Logical Implication and Conditional
: $\left({p \implies q}\right) \dashv \vdash \left({p \vdash q}\right)$ That is, the conditional is logically equivalent to logical implication.
Models for Propositional Logic
This page gathers together some useful results that can be used in the derivation of proofs by propositional tableau. Let $\mathcal M$ be a model for propositional logic, and let $\mathbf A$ and $\mathbf B$ be WFFs of propositional logic. Then the following results hold. The symbol $\models$ is used throughout to denote semantic consequence.
No Boolean Interpretation Models a WFF and its Negation
Let $v$ be a boolean interpretation. Let $\mathbf A$ be a WFF of propositional logic. Then $v$ can not model both $\mathbf A$ and $\neg \mathbf A$.
Extended Soundness Theorem for Propositional Tableaus and Boolean Interpretations
Tableau proofs (in terms of propositional tableaus) are a strongly sound proof system for boolean interpretations. That is, for every collection $\mathbf H$ of WFFs of propositional logic and every WFF $\mathbf A$: :$\mathbf H \vdash_{\mathrm{PT}} \mathbf A$ implies $\mathbf H \models_{\mathrm{BI}} \mathbf A$
Extended Completeness Theorem for Propositional Tableaus and Boolean Interpretations
Tableau proofs (in terms of propositional tableaus) are a strongly complete proof system for boolean interpretations. More precisely, for every countable collection $\mathbf H$ of WFFs of propositional logic and every WFF $\mathbf A$: :$\mathbf H \models_{\mathrm{BI}} \mathbf A$ implies $\mathbf H \vdash_{\mathrm{PT}} \mathbf A$
Completeness Theorem for Propositional Tableaus and Boolean Interpretations
Tableau proofs (in terms of propositional tableaus) are a complete proof system for boolean interpretations. That is, for every WFF $\mathbf A$: :$\models_{\mathrm{BI}} \mathbf A$ implies $\vdash_{\mathrm{PT}} \mathbf A$
König's Tree Lemma
Let $T$ be a rooted tree with an infinite number of nodes, each with a finite number of children. Then $T$ has a branch of infinite length.
Compactness Theorem for Boolean Interpretations
Let $\mathbf H$ be a countable set of WFFs of propositional logic. Suppose $\mathbf H$ is finitely satisfiable for boolean interpretations. That is, suppose that every finite subset $\mathbf H' \subseteq \mathbf H$ is satisfiable for boolean interpretations. Then $\mathbf H$ has a model.
De Morgan's Laws (Predicate Logic)
{{:De Morgan's Laws (Predicate Logic)/Assertion of Universality}}
Initial Part of WFF of Predicate Logic is not WFF
Let $\mathbf A$ be a WFF of predicate logic. Let $\mathbf S$ be an initial part of $\mathbf A$. Then $\mathbf S$ is not a WFF of predicate logic.
Unique Readability Theorem of Predicate Calculus
Each WFF of predicate logic which starts with a left bracket or a negation sign has exactly one main connective. {{Explain|probably, in the correct formulation of 'main connective', the quantifiers ought to be connectives as well}}
Quantifier has Unique Scope
Let $\mathbf A$ be a WFF of predicate logic. Let $Q$ be an occurrence of a quantifier in $\mathbf A$. Then there exists a unique well-formed part of $\mathbf A$ which (omitting outermost parentheses) begins with that occurrence $Q$. This unique well-formed part of $\mathbf A$ is called the scope of the occurrence of $Q$.
Restriction of Operation Distributivity
Let $\left({S, *, \circ}\right)$ be an algebraic structure. Let $T \subseteq S$. If the operation $\circ$ is distributive over $*$ in $\left({S, *, \circ}\right)$, then it is also distributive over $*$ on a restriction $\left({T, * \restriction_T, \circ \restriction_T}\right)$.
Union of Intersections
:$\paren {S_1 \cap S_2} \cup \paren {T_1 \cap T_2} \subseteq S_1 \cup T_1$
Union of Symmetric Differences
Let $R, S, T$ be sets. Then: :$\left({R * S}\right) \cup \left({S * T}\right) = \left({R \cup S \cup T}\right) \setminus \left({R \cap S \cap T}\right)$ where $R * S$ denotes the symmetric difference between $R$ and $S$.
Difference of Unions is Subset of Union of Differences
Let $I$ be an indexing set. Let $S_\alpha, T_\alpha$ be sets, for all $\alpha \in I$. Then: :$\displaystyle \paren {\bigcup_{\alpha \mathop \in I} S_\alpha} \setminus \paren {\bigcup_{\alpha \mathop \in I} T_\alpha} \subseteq \bigcup_{\alpha \mathop \in I} \paren {S_\alpha \setminus T_\alpha}$ where $S_\alpha \setminus T_\alpha$ denotes set difference.
Symmetric Difference of Unions is Subset of Union of Symmetric Differences
Let $I$ be an indexing set. Let $S_\alpha, T_\alpha$ be sets, for all $\alpha \in I$. Then: :$\displaystyle \bigcup_{\alpha \mathop \in I} S_\alpha * \bigcup_{\alpha \mathop \in I} T_\alpha \subseteq \bigcup_{\alpha \mathop \in I} \paren {S_\alpha * T_\alpha}$ where $S * T$ is the symmetric difference between $S$ and $T$.
Pseudometric Defines an Equivalence Relation
Let $X$ be a set on which there is a pseudometric $d: X \times X \to \R$. For any $x, y \in X$, let $x \sim y$ denote that $d \left({x, y}\right) = 0$. Then $\sim$ is an equivalence relation, and the equivalence classes consist of sets of elements of $X$ at zero distance from each other.
First-Order Reaction
Let a substance decompose spontaneously in a '''first-order reaction'''. Let $x_0$ be a measure of the quantity of that substance at time $t = 0$. Let the quantity of the substance that remains after time $t$ be $x$. Then: :$x = x_0 e^{-k t}$ where $k$ is the rate constant.
Solution to First Order ODE
Let: :$\displaystyle \Phi = \frac {\d y} {\d x} = \map f {x, y}$ be a first order ordinary differential eqn. Then $\Phi$ has a general solution which can be expressed in terms of an indefinite integral of $\map f x$: :$\displaystyle y = \int \map f {x, y} \rd x + C$ where $C$ is an arbitrary constant.
Solution to First Order Initial Value Problem
Let $\map y x$ be a solution to the first order ordinary differential equation: :$\dfrac {\d y} {\d x} = \map f {x, y}$ which is subject to an initial condition: $\tuple {a, b}$. Then this problem is equivalent to the integral equation: :$\displaystyle y = b + \int_a^x \map f {t, \map y t} \rd t$
Condition for Lipschitz Condition to be Satisfied
Let $f$ be a real function. Then $f$ satisfies the Lipschitz condition on a closed real interval $\closedint a b$ if: :$\forall y \in \closedint a b: \exists A \in \R: \size {\map {\phi'} y} \le A$
Separation of Variables
Suppose a first order ordinary differential equation can be expressible in this form: :$\dfrac {\d y} {\d x} = \map g x \, \map h y$ Then the equation is said to '''have separable variables''', or '''be separable'''. Its general solution is found by solving the integration: :$\displaystyle \int \frac {\d y} {\map h y} = \int \map g x \rd x + C$
Quotient of Homogeneous Functions
Let $\map M {x, y}$ and $\map N {x, y}$ be homogeneous functions of the same degree. Then: :$\dfrac {\map M {x, y} } {\map N {x, y} }$ is homogeneous of zero degree.
Solution to Homogeneous Differential Equation
Let: :$\map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$ be a homogeneous differential equation. It can be solved by making the substitution $z = \dfrac y x$. Its solution is: :$\displaystyle \ln x = \int \frac {\d z} {\map f {1, z} - z} + C$ where: :$\map f {x, y} = -\dfrac {\map M {x, y} } {\map N {x, y} }$
Solution to Exact Differential Equation
The first order ordinary differential equation: :$F = \map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$ is an exact differential equation : :$\dfrac {\partial M} {\partial y} = \dfrac {\partial N} {\partial x}$ The general solution of such an equation is: :$\map f {x, y} = C$ where: :$\dfrac {\partial f} {\partial x} = M$ :$\dfrac {\partial f} {\partial y} = N$
Existence of Integrating Factor
Let the first order ordinary differential equation: :$(1): \quad M \left({x, y}\right) + N \left({x, y}\right) \dfrac {\mathrm d y} {\mathrm d x} = 0$ be such that $M$ and $N$ are real functions of two variables which are ''not'' homogeneous functions of the same degree. Suppose also that: :$\dfrac {\partial M} {\partial y} \ne \dfrac {\partial N} {\partial x}$ that is, $(1)$ is not exact. Finally, suppose that $(1)$ has a general solution. Then it is always possible to find an integrating factor $\mu \left({x, y}\right)$ such that: :$\mu \left({x, y}\right) \left({M \left({x, y}\right) + N \left({x, y}\right) \dfrac {\mathrm d y} {\mathrm d x}}\right) = 0$ is an exact differential equation. Hence it is possible to find that solution by Solution to Exact Differential Equation.
Integrating Factor for First Order ODE
Let the first order ordinary differential equation: :$(1): \quad \map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$ be non-homogeneous and not exact. By Existence of Integrating Factor, if $(1)$ has a general solution, there exists an integrating factor $\map \mu {x, y}$ such that: :$\displaystyle \map \mu {x, y} \paren {\map M {x, y} + \map N {x, y} \frac {\d y} {\d x} } = 0$ is an exact differential equation. Unfortunately, there is no systematic method of finding such a $\map \mu {x, y}$ for all such equations $(1)$.<ref>"In general this is quite difficult." ::: -- {{BookReference|Differential Equations|1972|George F. Simmons}}: $\S 2.9$: Integrating Factors</ref> However, there are certain types of first order ODE for which an integrating factor ''can'' be found procedurally.
Solution to Linear First Order Ordinary Differential Equation
A linear first order ordinary differential equation in the form: :$\dfrac {\d y} {\d x} + \map P x y = \map Q x$ has the general solution: :$\ds y = e^{-\int P \rd x} \paren {\int Q e^{\int P \rd x} \rd x + C}$
Extremal Length of Composition
Let $\Gamma_1$ and $\Gamma_2$ be families of (unions of) rectifiable curves on a Riemann surface $X$. Let $\Gamma_1$ and $\Gamma_2$ be disjoint in the sense that there exist disjoint Borel sets $E_1 \subseteq X$ and $E_2 \subseteq X$ with $\bigcup \Gamma_1 \subset E_1$ and $\bigcup \Gamma_2 \subset E_2$. The extremal length of the family: :$\Gamma := \left\{{\gamma_1 \cup \gamma_2:\ \gamma_1 \in \Gamma_1 \text{ and }\gamma_2 \in \Gamma_2}\right\}$ satisfies: :$\lambda \left({\Gamma}\right) = \lambda \left({\Gamma_1}\right) + \lambda \left({\Gamma_2}\right)$
Squares with All Odd Digits
The only squares whose digits<ref>That is, when written in base 10 notation.</ref> are all odd are $1$ and $9$.
Parity of Integer equals Parity of its Square
Let $p \in \Z$ be an integer. Then $p$ is even  $p^2$ is even.
Weierstrass M-Test
Let $f_n$ be a sequence of real functions defined on a domain $D \subseteq \R$. Let $\displaystyle \sup_{x \mathop \in D} \size {\map {f_n} x} \le M_n$ for each integer $n$ and some constants $M_n$ Let $\displaystyle \sum_{i \mathop = 1}^\infty M_i < \infty$. Then $\displaystyle \sum_{i \mathop = 1}^\infty f_i$ converges uniformly on $D$.
Measurable Sets form Sigma-Algebra
Let $\mu^*$ be an outer measure on a set $X$. Then the set $\map {\mathfrak M} {\mu^*}$ of all $\mu^*$-measurable subsets of $X$ is a $\sigma$-algebra.
Markov's Inequality
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $f$ be an $A$-measurable function where $A \in \Sigma$. Then: :$\displaystyle \mu \left({\left\{ {x \in A: \left\vert{f \left({x}\right)}\right\vert \ge t}\right\} }\right) \le \frac 1 t \int_A \left\vert{f}\right\vert \mathrm d \mu$ for any positive $t \in \R$.
Limit Superior includes Limit Inferior
Let $\set {E_n : n \in \N}$ be a sequence of sets. Then: :$\ds \liminf_{n \mathop \to \infty} E_n \subseteq \limsup_{n \mathop \to \infty} E_n$
Construction of Outer Measure
{{questionable|The sense of this seems to have got lost at around the time of the edit of 25th March 2012. Presumably the intention was to extend this definition from the set of mappings on the powerset of $X$ to an arbitrary subset of them -- but the definition of $\mu^*$ seems to be inconsistent with this. Either a more rigorous explanation of exactly what all the concepts are here, or the theorem has been mis-stated.}} Let $X$ be a set. Let $\powerset X$ be the power set of $X$. Let $\AA$ be a subset of $\powerset X$ which contains the empty set. Let $\overline \R_{\ge 0}$ denote the set of positive extended real numbers. Let $\gamma: \AA \to \overline \R_{\ge 0}$ be a mapping such that $\map \gamma \O = 0$. Let $\mu^*: \AA \to \overline \R_{\ge 0}$ be the mapping defined as: :$\displaystyle \forall S \in \powerset X: \map {\mu^*} S = \inf \set {\sum_{n \mathop = 0}^\infty \map \gamma {A_n}: \forall n \in \N : A_n \in \AA, \ S \subseteq \bigcup_{n \mathop = 0}^\infty A_n}$ Then $\mu^*$ is an outer measure on $X$. The infimum of the empty set is the greatest element, $+\infty$.
Additive and Countably Subadditive Function is Countably Additive
Let $\Sigma$ be a $\sigma$-algebra over a set $X$. Let $f: \Sigma \to \overline \R_{\ge 0}$ be an additive and countably subadditive function, where $\overline \R_{\ge 0}$ denotes the set of positive extended real numbers. Then $f$ is countably additive.
Banach Fixed-Point Theorem
Let $\struct {M, d}$ be a complete metric space. Let $f: M \to M$ be a contraction. That is, there exists $q \in \hointr 0 1$ such that for all $x, y \in M$: :$\map d {\map f x, \map f y} \le q \, \map d {x, y}$ Then there exists a unique fixed point of $f$.
Measure of Empty Set is Zero
Let $\struct {X, \Sigma, \mu}$ be a measure space. Then $\map \mu \O = 0$. That is, $\O$ is a $\mu$-null set.
Convergence a.u. Implies Convergence a.e.
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $f_n: D \to \R$ be a sequence of $\Sigma$-measurable functions for $D \in \Sigma$. Let $f_n$ converge a.u. to a function $f$ on $D$. Then $f_n$ converges a.e. to $f$.
Existence of Solution to System of First Order ODEs
Consider the system of initial value problems: :$\begin{cases} \dfrac {\mathrm d y} {\mathrm d x} = f \left({x, y, z}\right) & : y \left({x_0}\right) = y_0 \\ & \\ \dfrac {\mathrm d z} {\mathrm d x} = g \left({x, y, z}\right) & : z \left({x_0}\right) = z_0 \\ \end{cases}$ where $f \left({x, y, z}\right)$ and $g \left({x, y, z}\right)$ are continuous real functions in some region of space $x y z$ that contains the point $\left({x_0, y_0, z_0}\right)$. Then this system of equations has a unique solution which exists on some interval $\left|{x - x_0}\right| \le h$.
Little Bézout Theorem
Let $\map {P_n} x$ be a polynomial of degree $n$ in $x$. Let $a$ be a constant. Then the remainder of $\map {P_n} x$ when divided by $x - a$ is equal to $\map {P_n} a$.
Thales' Theorem
Let $A$ and $B$ be two points on opposite ends of the diameter of a circle. Let $C$ be another point on the circle such that $C \ne A, B$. Then the lines $AC$ and $BC$ are perpendicular to each other. :400px
Edge of Polyhedron has no Curvature
The edge of a polyhedron has zero curvature.
Properties of Dot Product
Let $\mathbf u, \mathbf v, \mathbf w$ be vectors in the vector space $\R^n$. Let $c$ be a real scalar. The dot product has the following properties:
Dot Product is Inner Product
The dot product is an inner product.
Factor Matrix in the Inner Product
:$\left \langle {A \mathbf u,\mathbf v} \right \rangle = \left \langle {\mathbf u, A^T\mathbf v} \right \rangle$ where $\mathbf u$ and $\mathbf v$ are both $1 \times n$ column vectors.
Empty Set Disjoint with Itself
The empty set is disjoint with itself: :$\varnothing \cap \varnothing = \varnothing$
Pi is Irrational
Pi ($\pi$) is irrational.
Properties of Hadamard Product
Let $\map {\MM_S} {m, n}$ be a $m \times n$ matrix space over $S$ over an algebraic structure $\struct {S, \cdot}$. Let $\mathbf A, \mathbf B \in \map {\MM_S} {m, n}$. Let $\mathbf A \circ \mathbf B$ be defined as the Hadamard product of $\mathbf A$ and $\mathbf B$. The operation $\circ$ of Hadamard product satisfies the following properties: :$\circ$ is closed on $\map {\MM_S} {m, n}$  $\cdot$ is closed on $\struct {S, \cdot}$ :$\circ$ is associative on $\map {\MM_S} {m, n}$  $\cdot$ is associative on $\struct {S, \cdot}$ :$\circ$ is commutative on $\map {\MM_S} {m, n}$  $\cdot$ is commutative on $\struct {S, \cdot}$.
Graph Isomorphism is Equivalence Relation
Graph isomorphism is an equivalence relation.
Relation Isomorphism is Equivalence Relation
Relation isomorphism is an equivalence relation.
Vertex Condition for Isomorphic Graphs
Let $G_1$ and $G_2$ be isomorphic graphs. Then the degrees of the vertices of $G_1$ are exactly the same as the degrees of the vertices of $G_2$.
Pigeonhole Principle
Let $S$ be a finite set whose cardinality is $n$. Let $S_1, S_2, \ldots, S_k$ be a partition of $S$ into $k$ subsets. Then: :at least one subset $S_i$ of $S$ contains at least $\ceiling {\dfrac n k}$ elements where $\ceiling {\, \cdot \,}$ denotes the ceiling function.
Graph Connectedness is Equivalence Relation
Let $G = \struct {V, E}$ be a graph. Let $\to$ denote the relation '''is connected to''' on the set $V$. Then $\to$ is an equivalence relation.
Condition for Edge to be Bridge
Let $G = \struct {V, E}$ be a connected graph. Let $e \in E$ be an edge of $G$. Then $e$ is a bridge  $e$ does not lie on any circuit of $G$.
Characteristics of Traversable Graph
A finite loop-multigraph is traversable  it is connected and no more than two vertices are odd. Any Eulerian trail which is not an Eulerian circuit must start and end at an odd vertex.
Number of Edges of Regular Graph
An $r$-regular graph of order $n$ is of size $\dfrac {n r} 2$.
Graph is Bipartite iff No Odd Cycles
Let $G$ be an undirected graph. Then $G$ is bipartite  it has no odd cycles.
Ore's Theorem
Let $G = \struct {V, E}$ be a simple graph of order $n \ge 3$. Let $G$ be an Ore graph, that is: :For each pair of non-adjacent vertices $u, v \in V$: :: $(1): \quad \deg u + \deg v \ge n$ Then $G$ is Hamiltonian.
Path in Tree is Unique
Let $T$ be a graph. Then $T$ is a tree  there is exactly one path between any two vertices.
Size of Tree is One Less than Order
Let $T$ be a connected simple graph of order $n$. Then $T$ is a tree  the size of $T$ is $n-1$.
Connected Subgraph of Tree is Tree
Let $T$ be a tree. Let $S$ be a subgraph of $T$ such that $S$ is connected. Then $S$ is also a tree.
Finite Tree has Leaf Nodes
Every non-edgeless finite tree has at least two leaf nodes.
Size of Cycle Graph equals Order
The size of a cycle graph equals its order.
Path Graph from Cycle Graph
Removing one edge from a cycle graph leaves a path graph.
Cycle in Balanced Signed Graph
Let $G$ be a balanced signed graph. Let $C$ be a cycle in $G$. Then $C$ has an even number of negative edges.
Graph with Even Vertices Partitions into Cycles
Let $G = \struct {V, E}$ be a graph whose vertices are all even. Then its edge set $E$ can be partitioned into cycles, no two of which share an edge. The converse also holds: a graph which can be partitioned into cycles must have all its vertices even.
Condition for Bipartite Graph to be Hamiltonian
Let $G = \struct {A \mid B, E}$ be a bipartite graph. Let $G$ be Hamiltonian. Then $\card A = \card B$. That is, there is the same number of vertices in $A$ as there are in $B$.
Complete Hamiltonian Bipartite Graph
Let $K_{m, n}$ be a complete bipartite graph. Then $K_{m, n}$ is Hamiltonian iff $m = n > 1$.
Condition for Complete Bipartite Graph to be Semi-Hamiltonian
Let $K_{m, n}$ be a complete bipartite graph. Then $K_{m, n}$ is semi-Hamiltonian  either: :$m = n = 1$ or: :$m = n + 1$ (or $n = m + 1$).
Equivalent Definitions for Finite Tree
Let $T$ be a finite tree of order $n$. The following statements are equivalent: : $(1): \quad T$ is connected and has no circuits. : $(2): \quad T$ has $n-1$ edges and has no circuits. : $(3): \quad T$ is connected and has $n-1$ edges. : $(4): \quad T$ is connected, and the removal of any one edge renders $T$ disconnected. : $(5): \quad$ Any two vertices of $T$ are connected by exactly one path. : $(6): \quad T$ has no circuits, but adding one edge creates a cycle.
Number of Edges in Forest
Let $F = \struct {V, E}$ be a forest with $n$ nodes and $m$ components. Then $F$ contains $n - m$ edges.
Cayley's Formula
The number of distinct labeled trees with $n$ nodes is $n^{n - 2}$.
Bijection between Prüfer Sequences and Labeled Trees
There is a one-to-one correspondence between Prüfer sequences and labeled trees. That is, every labeled tree has a unique Prüfer sequence that defines it, and every Prüfer sequence defines just one labeled tree.
Tree has Center or Bicenter
Every tree has either: : $(1): \quad$ Exactly one center or: : $(2): \quad$ Exactly one bicenter, but never both. That is, every tree is either central or bicentral.
Prim's Algorithm produces Minimum Spanning Tree
Prim's Algorithm always produces a minimum spanning tree.
Number of Hamilton Cycles in Complete Graph
For all $n \ge 3$, the number of distinct Hamilton cycles in the complete graph $K_n$ is $\dfrac {\left({n-1}\right)!} 2$.