hcju/proofwikips huggingface dataset:
Definition:Abacus
An '''abacus''' (plural: '''abacuses''' or '''abaci''') is a tool for performing arithmetical calculations. It consists of: : a series of lines (for example: grooves in sand, or wires on a frame), upon which are: : a number of items (for example: pebbles in the grooves, or beads on the wires), which are manipulated by hand so as to represent numbers. As such, it is the earliest known machine for mathematics, and can be regarded as the earliest ancestor of the electronic computer.
Definition:Abacism
'''Abacism''' means '''the process of doing arithmetic using an abacus'''. === Abacist === {{:Definition:Abacism/Abacist}}
Definition:Arabic Numerals
The '''Arabic numerals''' are: {{begin-eqn}} {{eqn | l = 1:       | o = \)'''one'''\(:       | r = \circ       | c =  }} {{eqn | l = 2:       | o = \)'''two'''\(:       | r = \circ \, \circ       | c =  }} {{eqn | l = 3:       | o = \)'''three'''\(:       | r = \circ \circ \circ       | c =  }} {{eqn | l = 4:       | o = \)'''four'''\(:       | r = \circ \circ \circ \, \circ       | c =  }} {{eqn | l = 5:       | o = \)'''five'''\(:       | r = \circ \circ \circ \circ \circ       | c =  }} {{eqn | l = 6:       | o = \)'''six'''\(:       | r = \circ \circ \circ \circ \circ \, \circ       | c =  }} {{eqn | l = 7:       | o = \)'''seven'''\(:       | r = \circ \circ \circ \circ \circ \circ \circ       | c =  }} {{eqn | l = 8:       | o = \)'''eight'''\(:       | r = \circ \circ \circ \circ \circ \circ \circ \, \circ       | c =  }} {{eqn | l = 9:       | o = \)'''nine'''\(:       | r = \circ \circ \circ \circ \circ \circ \circ \circ \circ       | c =  }} {{eqn | l = 0:       | o = \)'''zero'''\(:       | r = \)or '''nought''', '''naught''', '''nil''', '''nothing''', '''cypher''', '''zilch''', '''kabutnik''', '''sweet Fanny Adams''', etc.\(       | c =  }} {{end-eqn}} They are used in conjunction with the positional decimal system of numeric representation: see Basis Representation Theorem.
Definition:Roman Numerals
The '''Roman numerals''' are: {{begin-eqn}} {{eqn | l = \mathrm I:       | o = \)'''unus'''\(:       | r = 1 }} {{eqn | l = \mathrm V:       | o = \)'''quinque'''\(:       | r = 5 }} {{eqn | l = \mathrm X:       | o = \)'''decem'''\(:       | r = 10 }} {{eqn | l = \mathrm L:       | o = \)'''quinquaginta'''\(:       | r = 50 }} {{eqn | l = \mathrm C:       | o = \)'''centum'''\(:       | r = 100 }} {{eqn | l = \mathrm D:       | o = \)'''quingenti'''\(:       | r = 500 }} {{eqn | l = \mathrm M:       | o = \)'''mille'''\(:       | r = 1000 }} {{end-eqn}}
Definition:URM Computability
Let $P$ be a URM program, and let $k$ be any positive integer. === Program === $P$ is said to '''compute the function $f: \N^k \to \N$''' : :for all ordered $k$-tuples $\tuple {n_1, n_2, \ldots, n_k} \in \N^k$, the computation of a URM using the program $P$ with input $\tuple {n_1, n_2, \ldots, n_k}$ produces the output $\map f {n_1, n_2, \ldots, n_k}$. If there are any inputs such that either of the following happens: :the output fails to equal $\map f {n_1, n_2, \ldots, n_k}$ :the program will never terminate, then the program does ''not'' compute the function $f: \N^k \to \N$. === Function === The function $f: \N^k \to \N$ is said to be '''URM computable''' if there exists a URM program which computes it. ==== Partial Function ==== $P$ is said to '''compute the partial function $f: \N^k \to \N$''' : :For all ordered $k$-tuples $\tuple {n_1, n_2, \ldots, n_k} \in \N^k$: ::If the computation of $P$ with input $\tuple {n_1, n_2, \ldots, n_k}$ '''halts''', it produces the output $\map f {n_1, n_2, \ldots, n_k}$. ::If the computation of $P$ with input $\tuple {n_1, n_2, \ldots, n_k}$ '''does not halt''', $\map f {n_1, n_2, \ldots, n_k}$ is undefined. The partial function $f: \N^k \to \N$ is said to be '''URM computable''' if there exists a URM program which computes it. Note that a URM program can be used with any number of input variables. For any positive integer $k$, the input consists of the state of the registers $R_1, R_2, \ldots, R_k$. Thus a given URM program $P$ computes a partial function $f: \N^k \to \N$ for ''each'' positive integer $k$. In this context, it is convenient to use the notation $f^k_P$ to denote the partial function of $k$ variables computed by $P$. === Set === Let $A \subseteq \N$. Then $A$ is a '''URM computable set'''  its characteristic function $\chi_A$ is a URM computable function. === Relation === Let $\mathcal R \subseteq \N^k$ be an $n$-ary relation on $\N^k$. Then $\mathcal R$ is a '''URM computable relation'''  its characteristic function $\chi_\mathcal R$ is a URM computable function.
Definition:Indeterminate Variable
An '''indeterminate variable''' is a variable whose  value has not been specified. Informally, it "could be anything".
Definition:Turing Machine
A '''Turing machine''' is an idealization of a computing machine. The idea goes as follows. A '''Turing machine''' works by manipulating symbols on an imaginary piece of paper by means of a specific set of algorithmic rules.
Definition:Substitution (Mathematical Logic)
=== Mapping === Let $S$ be a set. Let $f: S^t \to S$ be a mapping. Let $\left\{{g_1: S^k \to S, g_2: S^k \to S, \ldots, g_t: S^k \to S}\right\}$ be a set of mappings. Let the mapping $h: S^k \to S$ be defined as: :$h \left({s_1, s_2, \ldots, s_k}\right) = f \left({g_1 \left({s_1, s_2, \ldots, s_k}\right), g_2 \left({s_1, s_2, \ldots, s_k}\right), \ldots, g_t \left({s_1, s_2, \ldots, s_k}\right)}\right)$ Then $h$ is said to be '''obtained from $f, g_1, g_2, \ldots, g_k$ by substitution'''. The definition can be generalized in the following ways: * It can apply to mappings which operate on variously different sets. * Each of $g_1, g_2, \ldots, g_t$ may have different arities. If $g$ is a mapping of $m$ variables where $m > k$, we can always consider it a mapping of $k$ variables in which the additional variables play no part. So if $g_i$ is a mapping of $k_i$ variables, we can take $k = \max \left\{{k_i: i = 1, 2, \ldots, t}\right\}$ and then each $g_i$ is then a mapping of $k$ variables. === Partial Function === Let $f: \N^t \to \N$ be a partial function. Let $\left\{{g_1: \N^k \to \N, g_2: \N^k \to \N, \ldots, g_t: \N^k \to \N}\right\}$ be a set of partial functions. Let the partial function $h: \N^k \to \N$ be defined as: :$h \left({n_1, n_2, \ldots, n_k}\right) \approx f \left({g_1 \left({n_1, n_2, \ldots, n_k}\right), g_2 \left({n_1, n_2, \ldots, n_k}\right), \ldots, g_t \left({n_1, n_2, \ldots, n_k}\right)}\right)$ where $\approx$ is as defined in Partial Function Equality. Then $h$ is said to be '''obtained from $f, g_1, g_2, \ldots, g_k$ by substitution'''. Note that $h \left({n_1, n_2, \ldots, n_k}\right)$ is defined only when: * All of $g_1 \left({n_1, n_2, \ldots, n_k}\right), g_2 \left({n_1, n_2, \ldots, n_k}\right), \ldots, g_t \left({n_1, n_2, \ldots, n_k}\right)$ are defined * $f \left({g_1 \left({n_1, n_2, \ldots, n_k}\right), g_2 \left({n_1, n_2, \ldots, n_k}\right), \ldots, g_t \left({n_1, n_2, \ldots, n_k}\right)}\right)$ is defined.  902scgeuilpski6x8f3xrrjypbjez3o
Definition:Legendre's Constant
'''Legendre's constant''' (or the '''Legendre constant''') is a mathematical constant conjectured by {{AuthorRef|Adrien-Marie Legendre}} to specify the prime-counting function $\map \pi n$. Legendre conjectured in $1796$ that $\map \pi n$ satisfies: :$\displaystyle \lim_{n \mathop \to \infty} \map \pi n - \frac n {\map \ln n} = B$ where $B$ is '''Legendre's constant'''. If such a number $B$ exists, then this implies the Prime Number Theorem. {{AuthorRef|Adrien-Marie Legendre|Legendre}}'s guess for $B$ was about $1 \cdotp 08366$. {{OEIS|A228211}} Later, {{AuthorRef|Carl Friedrich Gauss}} looked at this problem and thought that $B$ might actually be lower. In $1896$, {{AuthorRef|Jacques Salomon Hadamard}} and {{AuthorRef|Charles de la Vallée Poussin}} independently proved the Prime Number Theorem and showed that $B$ is in fact equal to $1$. {{AuthorRef|Adrien-Marie Legendre|Legendre}}'s first guess of $1 \cdotp 08366 \ldots$ is still (incorrectly) referred to as '''Legendre's constant''', even though its "correct" value is in fact exactly $1$. Hence it is only now of historical importance. {{NamedforDef|Adrien-Marie Legendre|cat = Legendre}}
Definition:Concatenation of URM Programs
Let $P$ and $Q$ be URM programs. The '''concatenation of $P$ and $Q$''' is denoted $P * Q$. It consists of program $P$ followed immediately by program $Q$. If $P$ and $Q$ compute particular functions, then it is usually necessary to make modifications to either $P$ or $Q$, or both, to make sure that the concatenated program continues to compute a function. The simplest concatenation is illustrated in Composition of One-Variable URM Computable Functions.
Definition:Primitive Recursion
=== Primitive Recursion on Several Variables === {{:Definition:Primitive Recursion/Several Variables}} === Primitive Recursion on One Variable === {{:Definition:Primitive Recursion/One Variable}} === Primitive Recursion on Partial Functions === {{:Definition:Primitive Recursion/Partial Function}}
Definition:Basic Primitive Recursive Function
The '''basic primitive recursive functions''' are: === Zero Function === {{:Definition:Basic Primitive Recursive Function/Zero Function}} === Successor Function === {{:Definition:Basic Primitive Recursive Function/Successor Function}} === Projection Function === {{:Definition:Basic Primitive Recursive Function/Projection Function}} === Identity Function === {{:Definition:Basic Primitive Recursive Function/Identity Function}}
Definition:Primitive Recursive
=== Function === {{:Definition:Primitive Recursive/Function}} === Set === {{:Definition:Primitive Recursive/Set}} === Relation === {{:Definition:Primitive Recursive/Relation}}
Definition:Logarithmic Integral
The '''logarithmic integral''' is defined as: :$\displaystyle \map {\operatorname {li} } x = \PV_0^x \frac {\d t} {\map \ln t}$ where: :$\ln$ denotes the natural logarithm function :$\operatorname {PV}$ denotes the Cauchy principal value of the proceeding integral. {{mistake|"Proceeding" is wrong here. Suspected malapropism. Recommended a rewrite to make it more accurate.}} That is, as $\dfrac 1 {\ln t}$ has discontinuities at $t = 0$ and $t = 1$: :$\map {\operatorname {li} } x = \begin{cases}\displaystyle \lim_{\varepsilon \to 0^+} \paren {\int_{\varepsilon}^x \frac {\rd t} {\ln t} } & 0 < x < 1 \\ \displaystyle \lim_{\varepsilon \mathop \to 0^+} \paren {\int_\varepsilon^{1 - \varepsilon} \frac {\rd t} {\ln t} + \int_{1 + \varepsilon}^x \frac {\rd t} {\ln t} } & x > 1\end{cases}$ === Eulerian Logarithmic Interval === {{:Definition:Logarithmic Integral/Eulerian}}
Definition:Big-Omega
'''Big-Omega notation''' is a type of order notation for typically comparing 'run-times' or growth rates between two growth functions. Let $f, g$ be two functions. {{explain|Domain and range of $f$ and $g$ to be specified: reals?}} Then: : $f \left({n}\right) \in \Omega \left({g \left({n}\right)}\right)$ iff: : $\exists c > 0, k \ge 0: \forall n > k: f \left({n}\right) \ge c g \left({n}\right)$ This is read as: :'''$f \left({n}\right)$ is big omega of $g \left({n}\right)$'''. {{refactor|Put the below into separate pages, either as a 2nd definition or just an equivalence proof}} Another method of determining the condition is the following limit: :$\displaystyle \lim_{n \to \infty} {\frac{f \left({n}\right)} {g \left({n}\right)}} = c > 0$ where $0 < c \le \infty$. If such a $c$ does exist, then: : $f \left({n}\right) \in \Omega \left({g \left({n}\right)}\right)$ {{refactor|The below into its own page}} To say that $f \left({n}\right) \in \Omega \left({g \left({n}\right)}\right)$ is equivalent to: : $g \left({n}\right) \in \mathcal O \left({f \left({n}\right)}\right)$ where $\mathcal O$ is the big-O notation.
Definition:Big-Theta
'''Big-Theta notation''' is a type of order notation for typically comparing 'run-times' or growth rates between two growth functions. Big-Theta is a stronger statement than big-O and big-omega. Suppose $f: \N \to \R, g: \N \to \R$ are two functions. Then: :$f \left({n}\right) \in \Theta \left({g \left({n}\right)}\right)$  iff: :$(f \left({n}\right) \in O \left({g \left({n}\right)}\right) \land (f \left({n}\right) \in \Omega \left({g \left({n}\right)}\right)$ where $O \left({g \left({n}\right)}\right)$ is big-O and $\Omega \left({g \left({n}\right)}\right)$ is Big-Omega. This is read as: : $f \left({n}\right)$ is '''big-theta''' of $g \left({n}\right)$. Another method of determining the condition is the following limit: :$\displaystyle \lim_{n \to \infty} \frac{f \left({n}\right)}{g \left({n}\right)} = c$, where $0 < c < \infty$ If such a $c$ does exist, then $f \left({n}\right) \in \Theta (g \left({n}\right))$.
Definition:Signum Function
Let $X \subseteq \R$ be a subset of the real numbers. The '''signum function''' $\sgn: X \to \set {-1, 0, 1}$ is defined as: :$\forall x \in X: \map \sgn x := \sqbrk {x > 0} - \sqbrk {x < 0}$ where $\sqbrk {x > 0}$ etc. denotes Iverson's convention. That is: :$\forall x \in X: \map \sgn x := \begin{cases} -1 & : x < 0 \\ 0 & : x = 0 \\ 1 & : x > 0 \end{cases}$
Definition:Hall Subgroup
Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is a Hall subgroup of $G$  the index and order of $H$ in $G$ are coprime: :$\index G H \perp \order H$
Definition:Hall Divisor
Let $n \in \Z$ be an integer. A '''Hall divisor''' of $n$ is a divisor $d$ of $n$ such that $n$ and $\dfrac n d$ are coprime. {{NamedforDef|Philip Hall|cat=Hall}}
Definition:Fermat Number
A '''Fermat number''' is a natural number of the form $2^{\paren {2^n} } + 1$, where $n = 0, 1, 2, \ldots$. The number $2^{\paren {2^n} } + 1$ is, in this context, often denoted $F_n$.
Definition:Bounded Minimization
=== Function === Let $f: \N^{k+1} \to \N$ be a function. Let $\left({n_1, n_2, \ldots, n_k}\right) \in \N^k$ and let $z \in \N$ be fixed. Then the '''bounded minimization operation on $f$''' is written as: :$\mu y \le z \left({f \left({n_1, n_2, \ldots, n_k, y}\right) = 0}\right)$ and is specified as follows: :$\mu y \le z \left({f \left({n_1, n_2, \ldots, n_k, y}\right) = 0}\right) = \begin{cases} \text{the smallest } y \in \N \text{ such that } f \left({n_1, n_2, \ldots, n_k, y}\right) = 0 & : \exists y \in \N: y \le z \\ z + 1 & : \text{otherwise} \end{cases}$ === Relation === Let $\mathcal R \left({n_1, n_2, \ldots, n_k, y}\right) $ be a $k+1$-ary relation on $\N^{k+1}$. Let $\left({n_1, n_2, \ldots, n_k}\right) \in \N^k$ and let $z \in \N$ be fixed. Then the '''bounded minimization operation on $\mathcal R$''' is written as: :$\mu y \le z \mathcal R\left({n_1, n_2, \ldots, n_k, y}\right)$ and is specified as follows: :$\mu y \le z \mathcal R\left({n_1, n_2, \ldots, n_k, y}\right) = \begin{cases} \text{the smallest } y \in \N \text{ for which } \mathcal R \left({n_1, n_2, \ldots, n_k, y}\right) \text{ holds} & : \exists y \in \N: y \le z \\ z + 1 & : \text{otherwise} \end{cases}$ We can consider the definition for a function to be a special case of this. === The no-solution case === The choice of $z + 1$ for the value when there is no solution $y$ less than or equal to $z$ is arbitrary, but convenient. It ensures a well-defined solution for every $z$.  rc6242xnjliugkuy1a959k2vrh36zdn
Definition:Sequence Coding
Let $\left \langle {a_1, a_2, \ldots, a_k}\right \rangle$ be a finite sequence in $\N^*$ (that is, $\forall i \in \left\{{1, 2, \ldots, k}\right\}: a_i > 0$). Let $p_i$ be the $i$th prime number, so that: * $p_1 = 2$ * $p_2 = 3$ * $p_3 = 5$ * $p_4 = 7$ etc. Let $n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$ where $p_i$ is the $i$th prime number. Then $n \in \N$ '''codes the sequence $\left \langle {a_1, a_2, \ldots, a_k}\right \rangle$''', or $n$ is the '''code number for the sequence $\left \langle {a_1, a_2, \ldots, a_k}\right \rangle$'''. The set of all code numbers of finite sequences in $\N$ is denoted $\operatorname{Seq}$. Note that $n \in \operatorname{Seq}$  $n$ is divisible by all the primes $p_1, p_2, \ldots, p_k$ , where $p_k$ is the ''largest'' prime dividing $n$.
Definition:Length of Integer
Let $n \in \Z$ such that $n \ge 2$. The '''length of $n$''' is defined as the number of distinct prime factors of $n$ and is denoted $\operatorname{len} \left({n}\right)$.
Definition:Prime Enumeration Function
Let the function $p: \N \to \N$ be defined as: {| | align="left" | $p(0) = 1$, |  |- | align="left" | $p(n) =$ the $n^\text{th}$ prime number, | align="left" | if $n > 0$ |} Thus for example: * $p \left({0}\right) = 1$ * $p \left({1}\right) = 2$ * $p \left({2}\right) = 3$ * $p \left({3}\right) = 5 \ldots$ This function is called the '''prime enumeration function'''. Note, of course, that although $p \left({0}\right) = 1$, there is no suggestion of treating $1$ as prime (it definitely isn't).  5znwmmg9j7fhvw5u2zzwk6wi00uvbsc
Definition:Prime Exponent Function
Let $n \in \N$ be a natural number. Let the prime decomposition of $n$ be given as: :$\displaystyle n = \prod_{j \mathop = 1}^k \left({p \left({j}\right)}\right)^{a_j}$ where $p \left({j}\right)$ is the prime enumeration function. Then the exponent $a_j$ of $p \left({j}\right)$ in $n$ is denoted $\left({n}\right)_j$. If $p \left({j}\right)$ does not divide $n$, then $\left({n}\right)_j = 0$. We also define: : $\forall n \in \N: \left({n}\right)_0 = 0$ : $\forall j \in \N: \left({0}\right)_j = 0$ : $\forall j \in \N: \left({1}\right)_j = 0$
Definition:Partial Function
Let $S \subset \N^k$. Let $f: S \to \N$ be a function. Suppose that $\forall x \in \N^k \setminus S$, $f$ is undefined at $x$. Then $f$ is known as a '''partial function from $\N^k$ to $\N$'''. Thus we can specify a function that has values for some, but not all, elements of $\N$. It can be seen that the definition of a '''partial function''' as given here is compatible with that of a partial mapping.
Definition:Fibonacci Prime
A '''Fibonacci prime''' is a Fibonacci number which happens to be prime.
Definition:Minimization
=== Function === {{:Definition:Minimization/Function}} === Partial Function === {{:Definition:Minimization/Partial Function}} === Relation === {{:Definition:Minimization/Relation}}
Definition:Total Function
Let $f: \N^k \to \N$ be a partial function such that $f$ is defined on $S \subseteq \N^k$. Now let $S = \N^k$. That is, the domain of $f$ is then the ''whole'' of $\N^k$. Then $f$ is a '''total function'''.  9cdhchwm3nvl55y2i8wd6ja4wi7ryyk
Definition:Partial Function Equality
Let $g: \N^k \to \N$ and $h: \N^k \to \N$ be partial functions. We write: :$\map g {n_1, n_2, \ldots, n_k} \approx \map h {n_1, n_2, \ldots, n_k}$  either: :both $\map g {n_1, n_2, \ldots, n_k}$ and $\map h {n_1, n_2, \ldots, n_k}$ are defined and equal or: :neither $\map g {n_1, n_2, \ldots, n_k}$ nor $\map h {n_1, n_2, \ldots, n_k}$ are defined. That is,  $\map g {n_1, n_2, \ldots, n_k} = \map h {n_1, n_2, \ldots, n_k}$ wherever either are defined. Thus, '''$g$ is equal to $h$''', and we can write $g = h$, : :$\forall x \in \N^k: \map g x \approx \map h x$
Definition:Degree (Vertex)
Let $G = \struct {V, E}$ be an undirected graph. Let $v \in V$ be a vertex of $G$. The '''degree of $v$ in $G$''' is the number of edges to which it is incident. It is denoted $\map {\deg_G} v$, or just $\map \deg v$ if it is clear from the context which graph is being referred to. That is: :$\map {\deg_G} v = \card {\set {u \in V : \set {u, v} \in E} }$
Definition:Recursive
=== Function === {{:Definition:Recursive/Function}} === Set === {{:Definition:Recursive/Set}} === Relation === {{:Definition:Recursive/Relation}}
Definition:Total Recursive Function
In general, a recursive function is a partial function. A '''total recursive function''' is a recursive function which is total.  d62xoedqu8ncmk22ggtbm02u9t3xw08
Definition:Trace Table
Let $P$ be a URM program. The '''trace table''' of $P$ consists of: * The stage of computation; * The number of the instruction of $P$ that is about to be performed; * A list of the contents of all the registers used by $P$ at this point. Thus the trace table is a list of the states of the URM program at each stage.
Definition:Algorithmic Computability
The concept of '''algorithmic computability''' is an intuitive one. An '''algorithmically computable function''' is a function which can be carried by means of an algorithm, theoretically by a person using pencil and paper. The concept arose in the decades before the invention of digital computers. Much of the theoretical groundwork was done in the 1930s by such as {{AuthorRef|Alan Mathison Turing}}, {{AuthorRef|Alonzo Church}} and {{AuthorRef|Stephen Cole Kleene}}. The term used by {{AuthorRef|Alonzo Church|Church}} when he discussed the issue in his how famous Church's Thesis was '''effectively calculable'''.
Definition:Mistake
A '''mistake''' (as opposed to an error) is an inaccuracy or (deduced) falsehood caused by (ultimately) human fault. Synonyms include such terms as '''boob''', '''bloomer''', '''blooper''', '''blunder''', '''cock-up''', '''bug''' (computer science), '''mess-up''', '''oversight''', '''howler''' and other less polite words frequently encountered at the backs of schoolrooms worldwide. Please feel free to add your own.
Definition:Error
Let $x$ be an approximation to a (true) value $X$. The '''error''' $\varepsilon$ is a measure of how much difference there is between $x$ and $X$.
Definition:Transversal (Geometry)
A '''transversal''' of two lines lying in the same plane is a line which intersects them in two different points. The transversal is said to '''cut''' the two lines that it crosses. :400px In the above diagram, $EF$ is a '''transversal''' of the lines $AB$ and $CD$.
Definition:Supplementary Angles
:500px Let $\angle ACB$ be a straight angle. Let $\angle BCD + \angle DCA = \angle ACB$. That is, $\angle DCA = \angle ACB - \angle BCD$. Then $\angle DCA$ is the '''supplement''' of $\angle BCD$. Hence, for any angle $\alpha$ (whether less than a straight angle or not), the '''supplement''' of $\alpha$ is $\pi - \alpha$. Measured in degrees, the '''supplement''' of $\alpha$ is $180^\circ - \alpha$. If $\alpha$ is the '''supplement''' of $\beta$, then it follows that $\beta$ is the '''supplement''' of $\alpha$. Hence we can say that $\alpha$ and $\beta$ are '''supplementary'''. It can be seen from this that the '''supplement''' of a reflex angle is negative. Thus, '''supplementary angles''' are two angles whose measures add up to the measure of $2$ right angles. That is, their measurements add up to $180$ degrees or $\pi$ radians. Another (equivalent) definition is to say that two angles are '''supplementary''' which, when set next to each other, form a straight angle.
Definition:Complementary Angles
:300px Let $\angle BAC$ be a right angle. Let $\angle BAD + \angle DAC = \angle BAC$. That is, $\angle DAC = \angle BAC - \angle BAD$. Then $\angle DAC$ is the '''complement''' of $\angle BAD$. Hence, for any angle $\alpha$ (whether less than a right angle or not), the complement of $\alpha$ is $\dfrac \pi 2 - \alpha$. Measured in degrees, the complement of $\alpha$ is $90^\circ - \alpha$. If $\alpha$ is the complement of $\beta$, then it follows that $\beta$ is the complement of $\alpha$. Hence we can say that $\alpha$ and $\beta$ are '''complementary'''. It can be seen from this that the '''complement''' of an angle greater than a right angle is negative. Thus '''complementary angles''' are two angles whose measures add up to the measure of a right angle. That is, their measurements add up to $90$ degrees or $\dfrac \pi 2$ radians.
Definition:Lucas Number
=== Definition 1 === {{:Definition:Lucas Number/Definition 1}} === Definition 2 === {{:Definition:Lucas Number/Definition 2}} === Sequence === {{:Definition:Lucas Number/Sequence}}
Definition:Exclusive Or
'''Exclusive Or''' is a binary connective which can be written symbolically as $p \oplus q$ whose behaviour is as follows: :$p \oplus q$ means: : '''Either $p$ is true ''or'' $q$ is true ''but not both''.''' or symbolically: : $p \oplus q := \paren {p \lor q} \land \neg \paren {p \land q}$ where $\land$ denotes the ''and'' operator and $\lor$ denotes the ''or'' operator.
Definition:Doubleton
A '''doubleton''' is a set that contains exactly two elements.