id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-2100 | Lagrange's Four Square Theorem | Every positive integer can be expressed as a sum of four squares. | $1$ can trivially be expressed as a sum of four squares:
:$1 = 1^2 + 0^2 + 0^2 + 0^2$
From Product of Sums of Four Squares it is sufficient to show that each prime can be expressed as a sum of four squares.
The prime number $2$ certainly can: $2 = 1^2 + 1^2 + 0^2 + 0^2$.
It remains to consider the odd primes.
=== Exist... | Every [[Definition:Positive Integer|positive integer]] can be expressed as a [[Definition:Sum (Addition)|sum]] of four [[Definition:Square Number|squares]]. | $1$ can trivially be expressed as a [[Definition:Sum (Addition)|sum]] of four [[Definition:Square Number|squares]]:
:$1 = 1^2 + 0^2 + 0^2 + 0^2$
From [[Product of Sums of Four Squares]] it is sufficient to show that each [[Definition:Prime Number|prime]] can be expressed as a [[Definition:Sum (Addition)|sum]] of four... | Lagrange's Four Square Theorem/Proof 1 | https://proofwiki.org/wiki/Lagrange's_Four_Square_Theorem | https://proofwiki.org/wiki/Lagrange's_Four_Square_Theorem/Proof_1 | [
"Lagrange's Four Square Theorem",
"Sums of Squares"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Sum",
"Definition:Square Number"
] | [
"Definition:Addition/Sum",
"Definition:Square Number",
"Product of Sums of Four Squares",
"Definition:Prime Number",
"Definition:Addition/Sum",
"Definition:Square Number",
"Definition:Prime Number",
"Definition:Odd Prime",
"Definition:Odd Prime",
"Definition:Addition/Sum",
"Definition:Square Num... |
proofwiki-2101 | Lagrange's Four Square Theorem | Every positive integer can be expressed as a sum of four squares. | === Proof for Odd Primes ===
Suppose $p$ is an odd prime.
Define:
:$S := \set {\alpha^2 \pmod p: \alpha \in \hointr 0 {\dfrac p 2} \cap \Z}$
Define:
:$S' := \set {-1 - \beta^2 \pmod p: \beta \in \hointr 0 {\dfrac p 2} \cap \Z}$
Suppose for $\alpha, \alpha' \in S$:
:$\alpha^2 \equiv \alpha'^2 \pmod p$
Obviously:
:... | Every [[Definition:Positive Integer|positive integer]] can be expressed as a [[Definition:Sum (Addition)|sum]] of four [[Definition:Square Number|squares]]. | === Proof for Odd Primes ===
Suppose $p$ is an [[Definition:Odd Prime|odd prime]].
Define:
:$S := \set {\alpha^2 \pmod p: \alpha \in \hointr 0 {\dfrac p 2} \cap \Z}$
Define:
:$S' := \set {-1 - \beta^2 \pmod p: \beta \in \hointr 0 {\dfrac p 2} \cap \Z}$
Suppose for $\alpha, \alpha' \in S$:
:$\alpha^2 \equiv ... | Lagrange's Four Square Theorem/Proof 2 | https://proofwiki.org/wiki/Lagrange's_Four_Square_Theorem | https://proofwiki.org/wiki/Lagrange's_Four_Square_Theorem/Proof_2 | [
"Lagrange's Four Square Theorem",
"Sums of Squares"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Sum",
"Definition:Square Number"
] | [
"Definition:Odd Prime",
"Dirichlet's Box Principle/Corollary",
"Two-Step Subgroup Test",
"Definition:Cartesian Product/Coordinate",
"Definition:Linearly Independent/Set",
"Definition:Basis of Vector Space",
"Definition:Point Lattice",
"Definition:Euclidean Metric",
"Definition:Euclidean Metric",
"... |
proofwiki-2102 | Necessary Condition for Existence of BIBD | Let there exist a BIBD with parameters $v, b, r, k, \lambda$.
Then the following are true:
:$(1): \quad b k = r v$
:$(2): \quad \lambda \paren {v - 1} = r \paren {k - 1}$
:$(3): \quad b \dbinom k 2 = \lambda \dbinom v 2$
:$(4): \quad k < v$
:$(5): \quad r > \lambda$
All of $v, b, r, k, \lambda$ are integers.
Some sourc... | :$(1)$: We have by definition of balanced incomplete block design that:
:each treatment is in exactly $r$ blocks
:each block is of size $k$.
We have that $b k$ is the number of blocks times the size of each block.
We also have that $r v$ is the number of treatments times the number of blocks each treatment is in.
The t... | Let there exist a [[Definition:Balanced Incomplete Block Design|BIBD]] with parameters $v, b, r, k, \lambda$.
Then the following are true:
:$(1): \quad b k = r v$
:$(2): \quad \lambda \paren {v - 1} = r \paren {k - 1}$
:$(3): \quad b \dbinom k 2 = \lambda \dbinom v 2$
:$(4): \quad k < v$
:$(5): \quad r > \lambda$... | :$(1)$: We have by definition of [[Definition:Balanced Incomplete Block Design|balanced incomplete block design]] that:
:each [[Definition:Treatment|treatment]] is in exactly $r$ [[Definition:Block (Block Design)|blocks]]
:each [[Definition:Block (Block Design)|block]] is of [[Definition:Size of Block|size]] $k$.
We h... | Necessary Condition for Existence of BIBD | https://proofwiki.org/wiki/Necessary_Condition_for_Existence_of_BIBD | https://proofwiki.org/wiki/Necessary_Condition_for_Existence_of_BIBD | [
"Balanced Incomplete Block Designs"
] | [
"Definition:Balanced Incomplete Block Design",
"Definition:Integer"
] | [
"Definition:Balanced Incomplete Block Design",
"Definition:Treatment",
"Definition:Randomized Block",
"Definition:Randomized Block",
"Definition:Size of Block",
"Definition:Randomized Block",
"Definition:Size of Block",
"Definition:Randomized Block",
"Definition:Treatment",
"Definition:Randomized ... |
proofwiki-2103 | Mapping from Singleton is Injection | Let $f: S \to T$ be a mapping.
Let $S$ be a singleton.
Then $f$ is an injection. | Let $S = \set s$.
For $f$ to be an injection, all we need to do is show:
:$\forall x_1, x_2 \in S: \map f {x_1} = \map f {x_2} \implies x_1 = x_2$
But as $S$ is a singleton, it follows that:
:$x_1 = x_2 = s$
Hence the result.
{{qed}}
Category:Injections
Category:Singletons
8uicojvbyk3yq3v7gffgk7xd1qriew8 | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $S$ be a [[Definition:Singleton|singleton]].
Then $f$ is an [[Definition:Injection|injection]]. | Let $S = \set s$.
For $f$ to be an [[Definition:Injection|injection]], all we need to do is show:
:$\forall x_1, x_2 \in S: \map f {x_1} = \map f {x_2} \implies x_1 = x_2$
But as $S$ is a [[Definition:Singleton|singleton]], it follows that:
:$x_1 = x_2 = s$
Hence the result.
{{qed}}
[[Category:Injections]]
[[Catego... | Mapping from Singleton is Injection | https://proofwiki.org/wiki/Mapping_from_Singleton_is_Injection | https://proofwiki.org/wiki/Mapping_from_Singleton_is_Injection | [
"Injections",
"Singletons"
] | [
"Definition:Mapping",
"Definition:Singleton",
"Definition:Injection"
] | [
"Definition:Injection",
"Definition:Singleton",
"Category:Injections",
"Category:Singletons"
] |
proofwiki-2104 | Mapping to Singleton is Surjection | Let $S$ be a non-empty set.
Let $f: S \to T$ be a mapping.
Let $T$ be a singleton.
Then $f$ is a surjection. | Let $T = \set t$.
For $f$ to be a surjection, all we need to do is show:
:$\forall y \in T: \exists x \in S: \map f x = y$.
As $S \ne \O$, $\exists s \in S$.
As $f: S \to T$ is a mapping, it follows that $\map f s \in T$.
So as $\map f s \in T$ it follows that $t = \map f s$.
As $T = \set t$, it follows that $\forall y... | Let $S$ be a non-[[Definition:Empty Set|empty]] [[Definition:Set|set]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $T$ be a [[Definition:Singleton|singleton]].
Then $f$ is a [[Definition:Surjection|surjection]]. | Let $T = \set t$.
For $f$ to be a [[Definition:Surjection|surjection]], all we need to do is show:
:$\forall y \in T: \exists x \in S: \map f x = y$.
As $S \ne \O$, $\exists s \in S$.
As $f: S \to T$ is a [[Definition:Mapping|mapping]], it follows that $\map f s \in T$.
So as $\map f s \in T$ it follows that $t = ... | Mapping to Singleton is Surjection | https://proofwiki.org/wiki/Mapping_to_Singleton_is_Surjection | https://proofwiki.org/wiki/Mapping_to_Singleton_is_Surjection | [
"Surjections",
"Singletons"
] | [
"Definition:Empty Set",
"Definition:Set",
"Definition:Mapping",
"Definition:Singleton",
"Definition:Surjection"
] | [
"Definition:Surjection",
"Definition:Mapping",
"Category:Surjections",
"Category:Singletons"
] |
proofwiki-2105 | Generating Function for Constant Sequence | Let $\sequence {a_n}$ be the sequence defined as:
:$\forall n \in \N: a_n = r$
for some $r \in \R$.
Then the generating function for $\sequence {a_n}$ is given as:
:$\map G z = \dfrac r {1 - z}$ for $\size z < 1$ | {{begin-eqn}}
{{eqn | l = \map G z
| r = \sum_{n \mathop = 0}^\infty r z^n
| c = {{Defof|Generating Function}}
}}
{{eqn | r = r \sum_{n \mathop = 0}^\infty z^n
| c =
}}
{{eqn | r = \frac r {1 - z}
| c = Sum of Infinite Geometric Sequence
}}
{{end-eqn}}
for $\size z < 1$.
{{qed}} | Let $\sequence {a_n}$ be the [[Definition:Sequence|sequence]] defined as:
:$\forall n \in \N: a_n = r$
for some $r \in \R$.
Then the [[Definition:Generating Function|generating function]] for $\sequence {a_n}$ is given as:
:$\map G z = \dfrac r {1 - z}$ for $\size z < 1$ | {{begin-eqn}}
{{eqn | l = \map G z
| r = \sum_{n \mathop = 0}^\infty r z^n
| c = {{Defof|Generating Function}}
}}
{{eqn | r = r \sum_{n \mathop = 0}^\infty z^n
| c =
}}
{{eqn | r = \frac r {1 - z}
| c = [[Sum of Infinite Geometric Sequence]]
}}
{{end-eqn}}
for $\size z < 1$.
{{qed}} | Generating Function for Constant Sequence | https://proofwiki.org/wiki/Generating_Function_for_Constant_Sequence | https://proofwiki.org/wiki/Generating_Function_for_Constant_Sequence | [
"Generating Function for Constant Sequence",
"Examples of Generating Functions"
] | [
"Definition:Sequence",
"Definition:Generating Function"
] | [
"Sum of Infinite Geometric Sequence"
] |
proofwiki-2106 | Area of Circle | The area $A$ of a circle is given by:
:$A = \pi r^2$
where $r$ is the radius of the circle. | Let the circle of radius $r$ be divided into many sectors:
:400px
If they are made small enough, they can be approximated to triangles whose heights are all $r$.
Let the bases of these triangles be denoted:
:$b_1, b_2, b_3, \ldots$
From Area of Triangle in Terms of Side and Altitude, their areas are:
:$\dfrac {r b_1} 2... | The [[Definition:Area|area]] $A$ of a [[Definition:Circle|circle]] is given by:
:$A = \pi r^2$
where $r$ is the [[Definition:Radius of Circle|radius]] of the circle. | Let the [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $r$ be divided into many [[Definition:Sector of Circle|sectors]]:
:[[File:Area-of-Circle-Kepler's-Proof.png|400px]]
If they are made small enough, they can be approximated to [[Definition:Triangle (Geometry)|triangles]] whose [[Definition:... | Area of Circle/Kepler's Proof | https://proofwiki.org/wiki/Area_of_Circle | https://proofwiki.org/wiki/Area_of_Circle/Kepler's_Proof | [
"Area of Circle",
"Circles",
"Area Formulas"
] | [
"Definition:Area",
"Definition:Circle",
"Definition:Circle/Radius"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Sector of Circle",
"File:Area-of-Circle-Kepler's-Proof.png",
"Definition:Triangle (Geometry)",
"Definition:Triangle (Geometry)/Height",
"Definition:Triangle (Geometry)/Base",
"Definition:Triangle (Geometry)",
"Area of Triangle in Terms of ... |
proofwiki-2107 | Area of Circle | The area $A$ of a circle is given by:
:$A = \pi r^2$
where $r$ is the radius of the circle. | From Equation of Circle:
:$x^2 + y^2 = r^2$
Thus $y = \pm \sqrt {r^2 - x^2}$.
It follows that from the geometric interpretation of the definite integral:
{{begin-eqn}}
{{eqn | l = A
| r = \int_{-r}^r \paren {\sqrt {r^2 - x^2} - \paren {-\sqrt {r^2 - x^2} } } \rd x
}}
{{eqn | r = \int_{-r}^r 2 \sqrt {r^2 - x^2} \r... | The [[Definition:Area|area]] $A$ of a [[Definition:Circle|circle]] is given by:
:$A = \pi r^2$
where $r$ is the [[Definition:Radius of Circle|radius]] of the circle. | From [[Equation of Circle]]:
:$x^2 + y^2 = r^2$
Thus $y = \pm \sqrt {r^2 - x^2}$.
It follows that from the [[Definition:Geometric Interpretation of Definite Integral|geometric interpretation of the definite integral]]:
{{begin-eqn}}
{{eqn | l = A
| r = \int_{-r}^r \paren {\sqrt {r^2 - x^2} - \paren {-\sqrt {r... | Area of Circle/Proof 1 | https://proofwiki.org/wiki/Area_of_Circle | https://proofwiki.org/wiki/Area_of_Circle/Proof_1 | [
"Area of Circle",
"Circles",
"Area Formulas"
] | [
"Definition:Area",
"Definition:Circle",
"Definition:Circle/Radius"
] | [
"Equation of Circle",
"Definition:Darboux Integral/Geometric Interpretation",
"Integration by Substitution",
"Sum of Squares of Sine and Cosine",
"Integral of Constant/Definite",
"Primitive of Cosine Function"
] |
proofwiki-2108 | Area of Circle | The area $A$ of a circle is given by:
:$A = \pi r^2$
where $r$ is the radius of the circle. | Proof by shell integration:
The circle can be divided into a set of infinitesimally thin rings, each of which has area $2 \pi t \rd t$, since the ring has length $2 \pi t$ and thickness $\rd t$.
{{Handwaving|The fact that the above is a valid approximation needs to be established.}}
{{begin-eqn}}
{{eqn | l = A
| ... | The [[Definition:Area|area]] $A$ of a [[Definition:Circle|circle]] is given by:
:$A = \pi r^2$
where $r$ is the [[Definition:Radius of Circle|radius]] of the circle. | Proof by [[Definition:Shell Integration|shell integration]]:
The circle can be divided into a set of infinitesimally thin rings, each of which has area $2 \pi t \rd t$, since [[Perimeter of Circle|the ring has length $2 \pi t$]] and thickness $\rd t$.
{{Handwaving|The fact that the above is a valid approximation need... | Area of Circle/Proof 2 | https://proofwiki.org/wiki/Area_of_Circle | https://proofwiki.org/wiki/Area_of_Circle/Proof_2 | [
"Area of Circle",
"Circles",
"Area Formulas"
] | [
"Definition:Area",
"Definition:Circle",
"Definition:Circle/Radius"
] | [
"Definition:Shell Integration",
"Perimeter of Circle",
"Perimeter of Circle"
] |
proofwiki-2109 | Area of Circle | The area $A$ of a circle is given by:
:$A = \pi r^2$
where $r$ is the radius of the circle. | :400px
Construct a circle with radius $r$ and circumference $c$, whose area is denoted by $C$.
Construct a triangle with height $r$ and base $c$, whose area is denoted by $T$.
=== Lemma $1$ ===
{{:Area of Circle/Proof 3/Lemma 1}}{{qed|lemma}}
=== Lemma $2$ ===
{{:Area of Circle/Proof 3/Lemma 2}}{{qed|lemma}}
=== Lemma ... | The [[Definition:Area|area]] $A$ of a [[Definition:Circle|circle]] is given by:
:$A = \pi r^2$
where $r$ is the [[Definition:Radius of Circle|radius]] of the circle. | :[[File:Area-of-Circle-Proof-3.png|400px]]
Construct a [[Definition:Circle|circle]] with [[Definition:Radius of Circle|radius]] $r$ and [[Definition:Circumference of Circle|circumference]] $c$, whose [[Definition:Area|area]] is denoted by $C$.
Construct a [[Definition:Triangle (Geometry)|triangle]] with [[Definition:... | Area of Circle/Proof 3 | https://proofwiki.org/wiki/Area_of_Circle | https://proofwiki.org/wiki/Area_of_Circle/Proof_3 | [
"Area of Circle",
"Circles",
"Area Formulas"
] | [
"Definition:Area",
"Definition:Circle",
"Definition:Circle/Radius"
] | [
"File:Area-of-Circle-Proof-3.png",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Circumference",
"Definition:Area",
"Definition:Triangle (Geometry)",
"Definition:Altitude of Triangle",
"Definition:Triangle (Geometry)/Base",
"Definition:Area",
"Area of Circle/Proof 3/Lemma 1",... |
proofwiki-2110 | Area of Circle | The area $A$ of a circle is given by:
:$A = \pi r^2$
where $r$ is the radius of the circle. | Expressing the area in polar coordinates:
{{begin-eqn}}
{{eqn | l = \iint \rd A
| r = \int_0^r \int_0^{2 \pi} t \rd t \rd \theta
| c =
}}
{{eqn | r = \intlimits {\int_0^r t \theta} 0 {2 \pi} \rd t
| c =
}}
{{eqn | r = \int_0^r 2 \pi t \rd t
| c =
}}
{{eqn | r = 2 \pi \paren {\intlimits {\frac... | The [[Definition:Area|area]] $A$ of a [[Definition:Circle|circle]] is given by:
:$A = \pi r^2$
where $r$ is the [[Definition:Radius of Circle|radius]] of the circle. | Expressing the [[Definition:Area|area]] in [[Definition:Polar Coordinates|polar coordinates]]:
{{begin-eqn}}
{{eqn | l = \iint \rd A
| r = \int_0^r \int_0^{2 \pi} t \rd t \rd \theta
| c =
}}
{{eqn | r = \intlimits {\int_0^r t \theta} 0 {2 \pi} \rd t
| c =
}}
{{eqn | r = \int_0^r 2 \pi t \rd t
... | Area of Circle/Proof 4 | https://proofwiki.org/wiki/Area_of_Circle | https://proofwiki.org/wiki/Area_of_Circle/Proof_4 | [
"Area of Circle",
"Circles",
"Area Formulas"
] | [
"Definition:Area",
"Definition:Circle",
"Definition:Circle/Radius"
] | [
"Definition:Area",
"Definition:Polar Coordinates"
] |
proofwiki-2111 | Area of Circle | The area $A$ of a circle is given by:
:$A = \pi r^2$
where $r$ is the radius of the circle. | From Equation of Circle:
:$x^2 + y^2 = r^2$
Let $A$ be the area of the circle whose equation is given by $x^2 + y^2 = r^2$.
We have that:
:$y = \pm \sqrt {r^2 - x^2}$
For the upper half of the circle:
:$y = +\sqrt {r^2 - x^2}$
Thus for the right hand half of the upper half of the circle:
{{begin-eqn}}
{{eqn | l = \frac... | The [[Definition:Area|area]] $A$ of a [[Definition:Circle|circle]] is given by:
:$A = \pi r^2$
where $r$ is the [[Definition:Radius of Circle|radius]] of the circle. | From [[Equation of Circle]]:
:$x^2 + y^2 = r^2$
Let $A$ be the area of the [[Definition:Circle|circle]] whose equation is given by $x^2 + y^2 = r^2$.
We have that:
:$y = \pm \sqrt {r^2 - x^2}$
For the upper half of the [[Definition:Circle|circle]]:
:$y = +\sqrt {r^2 - x^2}$
Thus for the right hand half of the uppe... | Area of Circle/Proof 6 | https://proofwiki.org/wiki/Area_of_Circle | https://proofwiki.org/wiki/Area_of_Circle/Proof_6 | [
"Area of Circle",
"Circles",
"Area Formulas"
] | [
"Definition:Area",
"Definition:Circle",
"Definition:Circle/Radius"
] | [
"Equation of Circle",
"Definition:Circle",
"Definition:Circle",
"Definition:Circle",
"Definite Integral from 0 to a of Root of a Squared minus x Squared"
] |
proofwiki-2112 | Area of Circle | The area $A$ of a circle is given by:
:$A = \pi r^2$
where $r$ is the radius of the circle. | By the method of exhaustion:
:800px
== Construction ==
{{tidy}}
{{MissingLinks}}
For step $1$ of the construction, construct a circle $C$ and a diameter of $C$, thereby dividing the circumference of $C$ into $2$ arcs.
Let $A$ be the area of $C$ and $r$ be the radius of $C$.
For $n > 1$, step $n$ of the construction con... | The [[Definition:Area|area]] $A$ of a [[Definition:Circle|circle]] is given by:
:$A = \pi r^2$
where $r$ is the [[Definition:Radius of Circle|radius]] of the circle. | By the [[Definition:Method of Exhaustion|method of exhaustion]]:
:[[File:AreaOfCircleMethodOfExhaustion.png|800px]]
== Construction ==
{{tidy}}
{{MissingLinks}}
For step $1$ of the construction, construct a circle $C$ and a diameter of $C$, thereby dividing the circumference of $C$ into $2$ [[Definition:Arc of Circ... | Area of Circle/Proof 7 | https://proofwiki.org/wiki/Area_of_Circle | https://proofwiki.org/wiki/Area_of_Circle/Proof_7 | [
"Area of Circle",
"Circles",
"Area Formulas"
] | [
"Definition:Area",
"Definition:Circle",
"Definition:Circle/Radius"
] | [
"Definition:Method of Exhaustion",
"File:AreaOfCircleMethodOfExhaustion.png",
"Definition:Circle/Arc",
"Definition:Circle/Arc",
"Definition:Circle/Arc",
"Definition:Circle/Arc",
"File:Inscription.png",
"Definition:Circle/Arc",
"Squeeze Theorem",
"Combination Theorem for Sequences",
"Area of Isos... |
proofwiki-2113 | Distance Formula | The distance $d$ between two points $A = \tuple {x_1, y_1}$ and $B = \tuple {x_2, y_2}$ on a Cartesian plane is:
:$d = \sqrt {\paren {x_1 - x_2}^2 + \paren {y_1 - y_2}^2}$ | The distance in the horizontal direction between $A$ and $B$ is given by $\size {x_1 - x_2}$.
The distance in the vertical direction between $A$ and $B$ is given by $\size {y_1 - y_2}$.
By definition, the angle between a horizontal and a vertical line is a right angle.
So when we place a point $C = \tuple {x_1, y_2}$, ... | The [[Definition:Distance between Points|distance]] $d$ between two [[Definition:Point|points]] $A = \tuple {x_1, y_1}$ and $B = \tuple {x_2, y_2}$ on a [[Definition:Cartesian Plane|Cartesian plane]] is:
:$d = \sqrt {\paren {x_1 - x_2}^2 + \paren {y_1 - y_2}^2}$ | The [[Definition:Distance between Points|distance]] in the [[Definition:Horizontal|horizontal]] direction between $A$ and $B$ is given by $\size {x_1 - x_2}$.
The [[Definition:Distance between Points|distance]] in the [[Definition:Vertical|vertical]] direction between $A$ and $B$ is given by $\size {y_1 - y_2}$.
By ... | Distance Formula | https://proofwiki.org/wiki/Distance_Formula | https://proofwiki.org/wiki/Distance_Formula | [
"Euclidean Geometry",
"Analytic Geometry",
"Distance Formula"
] | [
"Definition:Distance between Points",
"Definition:Point",
"Definition:Cartesian Plane"
] | [
"Definition:Distance between Points",
"Definition:Horizontal",
"Definition:Distance between Points",
"Definition:Vertical",
"Definition:Angle",
"Definition:Horizontal Line",
"Definition:Vertical Line",
"Definition:Right Angle",
"Definition:Point",
"Definition:Triangle (Geometry)/Right-Angled",
"... |
proofwiki-2114 | Null URM Program Computes Identity Function | The null URM program computes the '''identity function''' $I_\N: \N \to \N$, defined as:
:$\forall n \in \N: \map {I_\N} n = n$ | The null URM program by definition has no instructions.
Therefore, the contents of $R_1$ remain unchanged when "running" it.
{{qed}}
Category:Null URM Program
Category:URM Programs
Category:Identity Mappings
65n8m35btsd14erhzyw0kkxv192buj6 | The [[Definition:Null URM Program|null URM program]] computes the '''[[Definition:Identity Mapping|identity function]]''' $I_\N: \N \to \N$, defined as:
:$\forall n \in \N: \map {I_\N} n = n$ | The [[Definition:Null URM Program|null URM program]] by definition has no instructions.
Therefore, the contents of $R_1$ remain unchanged when "running" it.
{{qed}}
[[Category:Null URM Program]]
[[Category:URM Programs]]
[[Category:Identity Mappings]]
65n8m35btsd14erhzyw0kkxv192buj6 | Null URM Program Computes Identity Function | https://proofwiki.org/wiki/Null_URM_Program_Computes_Identity_Function | https://proofwiki.org/wiki/Null_URM_Program_Computes_Identity_Function | [
"Null URM Program",
"URM Programs",
"Identity Mappings"
] | [
"Definition:Unlimited Register Machine/Null Program",
"Definition:Identity Mapping"
] | [
"Definition:Unlimited Register Machine/Null Program",
"Category:Null URM Program",
"Category:URM Programs",
"Category:Identity Mappings"
] |
proofwiki-2115 | Composition of One-Variable URM Computable Functions | Let $f: \N \to \N$ and $g: \N \to \N$ be URM computable functions of one variable.
Let $f \circ g$ be the composition of $f$ and $g$.
Then $f \circ g: \N \to \N$ is a URM computable function. | Let $f: \N \to \N$ and $g: \N \to \N$ be URM computable functions of one variable.
Let $P$ be a URM program which computes $f$.
Let $Q$ be a URM program which computes $g$.
Let $s = \map \lambda Q$ be the number of basic instructions in $Q$.
Let $u = \map \rho Q$ be the number of registers used by $Q$.
In order to comp... | Let $f: \N \to \N$ and $g: \N \to \N$ be [[Definition:URM Computability|URM computable functions]] of one variable.
Let $f \circ g$ be the [[Definition:Composite Function|composition]] of $f$ and $g$.
Then $f \circ g: \N \to \N$ is a [[Definition:URM Computability|URM computable function]]. | Let $f: \N \to \N$ and $g: \N \to \N$ be [[Definition:URM Computability|URM computable functions]] of one variable.
Let $P$ be a [[Definition:URM Program|URM program]] which computes $f$.
Let $Q$ be a [[Definition:URM Program|URM program]] which computes $g$.
Let $s = \map \lambda Q$ be the [[Definition:Unlimited R... | Composition of One-Variable URM Computable Functions | https://proofwiki.org/wiki/Composition_of_One-Variable_URM_Computable_Functions | https://proofwiki.org/wiki/Composition_of_One-Variable_URM_Computable_Functions | [
"URM Programs"
] | [
"Definition:URM Computability",
"Definition:Composition of Mappings",
"Definition:URM Computability"
] | [
"Definition:URM Computability",
"Definition:Unlimited Register Machine/Program",
"Definition:Unlimited Register Machine/Program",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine",
"Definit... |
proofwiki-2116 | Concatenation of URM Programs is Associative | Let $P, Q, R$ be one-variable URM programs.
Then the concatenated URM programs $P * \paren {Q * R}$ and $\paren {P * Q} * R$ are the same. | For ease of analysis, let us assume that:
:Each of $P$ and $Q$ have already had the appropriate Clear Registers Program $\map Z {2, \map \rho P}$ and $\map Z {2, \map \rho Q}$ appended to them
:Each of $P$ and $Q$ have already had the appropriate amendments made to their exit jumps so as to lead to the first line of th... | Let $P, Q, R$ be one-variable [[Definition:URM Program|URM programs]].
Then the [[Composition of One-Variable URM Computable Functions|concatenated URM programs]] $P * \paren {Q * R}$ and $\paren {P * Q} * R$ are the same. | For ease of analysis, let us assume that:
:Each of $P$ and $Q$ have already had the appropriate [[Clear Registers Program]] $\map Z {2, \map \rho P}$ and $\map Z {2, \map \rho Q}$ appended to them
:Each of $P$ and $Q$ have already had the appropriate amendments made to their [[Definition:Unlimited Register Machine/Prog... | Concatenation of URM Programs is Associative | https://proofwiki.org/wiki/Concatenation_of_URM_Programs_is_Associative | https://proofwiki.org/wiki/Concatenation_of_URM_Programs_is_Associative | [
"URM Programs"
] | [
"Definition:Unlimited Register Machine/Program",
"Composition of One-Variable URM Computable Functions"
] | [
"Clear Registers Program",
"Definition:Unlimited Register Machine/Program/Termination",
"Clear Registers Program",
"Definition:Unlimited Register Machine",
"Composition of One-Variable URM Computable Functions",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine",
"Categor... |
proofwiki-2117 | Primitive of Cosine Function | :$\ds \int \cos x \rd x = \sin x + C$ | From Derivative of Sine Function:
:$\dfrac \d {\d x} \sin x = \cos x$
The result follows from the definition of primitive.
{{Qed}} | :$\ds \int \cos x \rd x = \sin x + C$ | From [[Derivative of Sine Function]]:
:$\dfrac \d {\d x} \sin x = \cos x$
The result follows from the definition of [[Definition:Primitive (Calculus)|primitive]].
{{Qed}} | Primitive of Cosine Function/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Cosine_Function | https://proofwiki.org/wiki/Primitive_of_Cosine_Function/Proof_1 | [
"Primitive of Cosine Function",
"Primitives of Trigonometric Functions",
"Primitives involving Cosine Function",
"Cosine Function"
] | [] | [
"Derivative of Sine Function",
"Definition:Primitive (Calculus)"
] |
proofwiki-2118 | Primitive of Cosine Function | :$\ds \int \cos x \rd x = \sin x + C$ | {{begin-eqn}}
{{eqn | l = \int \cos x \rd x
| r = \frac 1 2 \int \paren {e^{i x} + e^{-i x} } \rd x
| c = Euler's Cosine Identity
}}
{{eqn | r = \frac 1 {2 i} \paren {e^{i x} - e^{-i x} } + C
| c = Primitive of Exponential of a x
}}
{{eqn | r = \sin x + C
| c = Euler's Sine Identity
}}
{{end-eqn... | :$\ds \int \cos x \rd x = \sin x + C$ | {{begin-eqn}}
{{eqn | l = \int \cos x \rd x
| r = \frac 1 2 \int \paren {e^{i x} + e^{-i x} } \rd x
| c = [[Euler's Cosine Identity]]
}}
{{eqn | r = \frac 1 {2 i} \paren {e^{i x} - e^{-i x} } + C
| c = [[Primitive of Exponential of a x]]
}}
{{eqn | r = \sin x + C
| c = [[Euler's Sine Identity]]
... | Primitive of Cosine Function/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Cosine_Function | https://proofwiki.org/wiki/Primitive_of_Cosine_Function/Proof_2 | [
"Primitive of Cosine Function",
"Primitives of Trigonometric Functions",
"Primitives involving Cosine Function",
"Cosine Function"
] | [] | [
"Euler's Cosine Identity",
"Primitive of Exponential of a x",
"Euler's Sine Identity"
] |
proofwiki-2119 | Primitive of Sine Function | :$\ds \int \sin x \rd x = -\cos x + C$ | From Derivative of Cosine Function:
:$\map {\dfrac \d {\d x} } {-\cos x} = \sin x$
The result follows from the definition of primitive.
{{qed}} | :$\ds \int \sin x \rd x = -\cos x + C$ | From [[Derivative of Cosine Function]]:
:$\map {\dfrac \d {\d x} } {-\cos x} = \sin x$
The result follows from the definition of [[Definition:Primitive (Calculus)|primitive]].
{{qed}} | Primitive of Sine Function/Proof | https://proofwiki.org/wiki/Primitive_of_Sine_Function | https://proofwiki.org/wiki/Primitive_of_Sine_Function/Proof | [
"Primitive of Sine Function",
"Primitives of Trigonometric Functions",
"Primitives involving Sine Function",
"Sine Function"
] | [] | [
"Derivative of Cosine Function",
"Definition:Primitive (Calculus)"
] |
proofwiki-2120 | Function Obtained by Substitution from URM Computable Functions | Let the functions $f: \N^t \to \N, g_1: \N^k \to \N, g_2: \N^k \to \N, \ldots, g_t: \N^k \to \N$ all be URM computable functions.
Let $h: \N^k \to \N$ be defined from $f, g_1, g_2, \ldots, g_t$ by substitution.
Then $h$ is also URM computable. | From the definition:
:$\map h {n_1, n_2, \ldots, n_k} = \map f {\map {g_1} {n_1, n_2, \ldots, n_k}, \map {g_2} {n_1, n_2, \ldots, n_k}, \ldots, \map {g_t} {n_1, n_2, \ldots, n_k} }$
Let $P, Q_1, Q_2, \ldots, Q_t$ be normalized URM programs which compute $f, g_1, g_2, \ldots, g_t$ respectively.
Let $\map u = \max \set {... | Let the [[Definition:Function|functions]] $f: \N^t \to \N, g_1: \N^k \to \N, g_2: \N^k \to \N, \ldots, g_t: \N^k \to \N$ all be [[Definition:URM Computability|URM computable functions]].
Let $h: \N^k \to \N$ be defined from $f, g_1, g_2, \ldots, g_t$ by [[Definition:Substitution (Mathematical Logic)|substitution]].
... | From the [[Definition:Substitution (Mathematical Logic)|definition]]:
:$\map h {n_1, n_2, \ldots, n_k} = \map f {\map {g_1} {n_1, n_2, \ldots, n_k}, \map {g_2} {n_1, n_2, \ldots, n_k}, \ldots, \map {g_t} {n_1, n_2, \ldots, n_k} }$
Let $P, Q_1, Q_2, \ldots, Q_t$ be [[Normalized URM Program|normalized URM programs]] wh... | Function Obtained by Substitution from URM Computable Functions | https://proofwiki.org/wiki/Function_Obtained_by_Substitution_from_URM_Computable_Functions | https://proofwiki.org/wiki/Function_Obtained_by_Substitution_from_URM_Computable_Functions | [
"URM Programs"
] | [
"Definition:Function",
"Definition:URM Computability",
"Definition:Substitution (Mathematical Logic)",
"Definition:URM Computability"
] | [
"Definition:Substitution (Mathematical Logic)",
"Normalized URM Program",
"Definition:URM Computability",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register M... |
proofwiki-2121 | Normalized URM Program | Let $P$ be a URM program.
Let $l = \map \lambda P$ be the number of basic instructions in $P$.
Let $u = \map \rho P$ be the number of registers used by $P$.
Then $P$ can be modified as follows:
:Every <tt>Jump</tt> of the form $\map J {m, n, q}$ where $q > l$ may be replaced by $\map J {m, n, l + 1}$
:If $u > 0$, a Cle... | Each <tt>Jump</tt> of the form $\map J {m, n, q}$ where $q > l$ leads to a line which does not contain an instruction.
The line $\map J {m, n, l + 1}$ likewise contains no instructions, by definition.
Therefore, when jumping to $\map J {m, n, l + 1}$ the program behaves in exactly the same way: that is, it stops when t... | Let $P$ be a [[Definition:URM Program|URM program]].
Let $l = \map \lambda P$ be the [[Definition:Unlimited Register Machine#Length of Program|number of basic instructions]] in $P$.
Let $u = \map \rho P$ be the [[Definition:Unlimited Register Machine#Number of Registers Used|number of registers used]] by $P$.
Then ... | Each <tt>Jump</tt> of the form $\map J {m, n, q}$ where $q > l$ leads to a line which does not contain an instruction.
The line $\map J {m, n, l + 1}$ likewise contains no instructions, by definition.
Therefore, when jumping to $\map J {m, n, l + 1}$ the program behaves in exactly the same way: that is, it stops when... | Normalized URM Program | https://proofwiki.org/wiki/Normalized_URM_Program | https://proofwiki.org/wiki/Normalized_URM_Program | [
"URM Programs"
] | [
"Definition:Unlimited Register Machine/Program",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine",
"Clear Registers Program",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine",
"Definition:Concatenation of URM Programs",
"Definition:Unlimi... | [
"Definition:Unlimited Register Machine/Program",
"Definition:Unlimited Register Machine/Program/Termination",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine/Program",
"Definition:Unlimited Register Machine/Program/Termination",
"Definition:Unlimited Register Machine/Progra... |
proofwiki-2122 | Block Copy Program | Let $k, m, n \in \N$ be natural numbers such that:
* $k \ge 1$;
* $\size {m - n} \ge k$.
The URM program defined as:
{|
|-
! align="right" | Line !!
! align="left" | Command
|-
| align="right" | $1$ ||
| align="left" | $\map C {m, n}$
|-
| align="right" | $2$ ||
| align="left" | $\map C {m + 1, n + 1}$
|-
| align="righ... | Immediate.
{{Qed}}
Category:URM Programs
6xd5p0svdom0u5gch47nnjtclz687el | Let $k, m, n \in \N$ be [[Definition:Natural Numbers|natural numbers]] such that:
* $k \ge 1$;
* $\size {m - n} \ge k$.
The [[Definition:URM Program|URM program]] defined as:
{|
|-
! align="right" | Line !!
! align="left" | Command
|-
| align="right" | $1$ ||
| align="left" | $\map C {m, n}$
|-
| align="right" | $2$ ... | Immediate.
{{Qed}}
[[Category:URM Programs]]
6xd5p0svdom0u5gch47nnjtclz687el | Block Copy Program | https://proofwiki.org/wiki/Block_Copy_Program | https://proofwiki.org/wiki/Block_Copy_Program | [
"URM Programs"
] | [
"Definition:Natural Numbers",
"Definition:Unlimited Register Machine/Program",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine"
] | [
"Category:URM Programs"
] |
proofwiki-2123 | Function Obtained by Primitive Recursion from URM Computable Functions | Let the functions $f: \N^k \to \N$ and $g: \N^{k + 2} \to \N$ be URM computable functions.
Let $h: \N^{k + 1} \to \N$ be obtained from $f$ and $g$ by primitive recursion.
Then $h$ is also URM computable. | From the definition:
:$\forall n \in \N: \map h {n_1, n_2, \ldots, n_k, n} = \begin {cases}
\map f {n_1, n_2, \ldots, n_k} & : n = 0 \\
\map g {n_1, n_2, \ldots, n_k, n - 1, \map h {n_1, n_2, \ldots, n_k, n - 1} } & : n > 0
\end {cases}$
Let $P$ and $Q$ be normalized URM programs which compute $f$ and $g$ respectively... | Let the [[Definition:Function|functions]] $f: \N^k \to \N$ and $g: \N^{k + 2} \to \N$ be [[Definition:URM Computability|URM computable functions]].
Let $h: \N^{k + 1} \to \N$ be obtained from $f$ and $g$ by [[Definition:Primitive Recursion|primitive recursion]].
Then $h$ is also [[Definition:URM Computability|URM co... | From the [[Definition:Primitive Recursion|definition]]:
:$\forall n \in \N: \map h {n_1, n_2, \ldots, n_k, n} = \begin {cases}
\map f {n_1, n_2, \ldots, n_k} & : n = 0 \\
\map g {n_1, n_2, \ldots, n_k, n - 1, \map h {n_1, n_2, \ldots, n_k, n - 1} } & : n > 0
\end {cases}$
Let $P$ and $Q$ be [[Normalized URM Program|... | Function Obtained by Primitive Recursion from URM Computable Functions | https://proofwiki.org/wiki/Function_Obtained_by_Primitive_Recursion_from_URM_Computable_Functions | https://proofwiki.org/wiki/Function_Obtained_by_Primitive_Recursion_from_URM_Computable_Functions | [
"URM Programs"
] | [
"Definition:Function",
"Definition:URM Computability",
"Definition:Primitive Recursion",
"Definition:URM Computability"
] | [
"Definition:Primitive Recursion",
"Normalized URM Program",
"Definition:URM Computability",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine",
"Definition:Algorithm",
"Definition:Unlimited Register Machine/Program",
"Definition:... |
proofwiki-2124 | Primitive Recursive Function is URM Computable | Every primitive recursive function is URM computable. | This follows immediately from:
* The fact that the basic primitive recursive functions are URM computable;
* Functions obtained by substitution from URM computable functions are URM computable;
* Functions obtained by primitive recursion from URM computable functions are URM computable;
* The definition of primitive re... | Every [[Definition:Primitive Recursive Function|primitive recursive function]] is [[Definition:URM Computability|URM computable]]. | This follows immediately from:
* The fact that the [[Single Instruction URM Programs#Basic Primitive Recursive Functions |basic primitive recursive functions are URM computable]];
* [[Function Obtained by Substitution from URM Computable Functions|Functions obtained by substitution from URM computable functions are URM... | Primitive Recursive Function is URM Computable | https://proofwiki.org/wiki/Primitive_Recursive_Function_is_URM_Computable | https://proofwiki.org/wiki/Primitive_Recursive_Function_is_URM_Computable | [
"Primitive Recursive Functions"
] | [
"Definition:Primitive Recursive/Function",
"Definition:URM Computability"
] | [
"Single Instruction URM Programs",
"Function Obtained by Substitution from URM Computable Functions",
"Function Obtained by Primitive Recursion from URM Computable Functions",
"Definition:Primitive Recursive/Function"
] |
proofwiki-2125 | Constant Function is Primitive Recursive | The constant function $f_c: \N \to \N$, defined as:
:$\map {f_c} n = c$ where $c \in \N$
is primitive recursive. | The proof proceeds by the Principle of Mathematical Induction. | The [[Definition:Constant Mapping|constant function]] $f_c: \N \to \N$, defined as:
:$\map {f_c} n = c$ where $c \in \N$
is [[Definition:Primitive Recursive Function|primitive recursive]]. | The proof proceeds by the [[Principle of Mathematical Induction]]. | Constant Function is Primitive Recursive | https://proofwiki.org/wiki/Constant_Function_is_Primitive_Recursive | https://proofwiki.org/wiki/Constant_Function_is_Primitive_Recursive | [
"Primitive Recursive Functions",
"Constant Mappings"
] | [
"Definition:Constant Mapping",
"Definition:Primitive Recursive/Function"
] | [
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-2126 | Heron's Formula | Let $\triangle ABC$ be a triangle with sides $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.
Then the area $\AA$ of $\triangle ABC$ is given by:
:$\AA = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$. | Construct the altitude from $A$.
Let the length of the altitude be $h$ and the foot of the altitude be $D$.
Let the distance from $D$ to $B$ be $z$.
:300px
From Pythagoras's Theorem:
:$\paren 1: \quad h^2 + \paren {a - z}^2 = b^2$
and:
:$\paren 2: \quad h^2 + z^2 = c^2$
By subtracting $\paren 1$ from $\paren 2$:
:$2 a ... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] with [[Definition:Side of Polygon|sides]] $a$, $b$ and $c$ [[Definition:Opposite (in Triangle)|opposite]] [[Definition:Vertex of Polygon|vertices]] $A$, $B$ and $C$ respectively.
Then the [[Definition:Area|area]] $\AA$ of $\triangle ABC$ is given by... | Construct the [[Definition:Altitude of Triangle|altitude]] from $A$.
Let the [[Definition:Length (Linear Measure)|length]] of the [[Definition:Altitude of Triangle|altitude]] be $h$ and the [[Definition:Foot of Altitude|foot]] of the [[Definition:Altitude of Triangle|altitude]] be $D$.
Let the [[Definition:Distance b... | Heron's Formula/Proof 1 | https://proofwiki.org/wiki/Heron's_Formula | https://proofwiki.org/wiki/Heron's_Formula/Proof_1 | [
"Heron's Formula",
"Areas of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Polygon/Vertex",
"Definition:Area",
"Definition:Semiperimeter"
] | [
"Definition:Altitude of Triangle",
"Definition:Linear Measure/Length",
"Definition:Altitude of Triangle",
"Definition:Altitude of Triangle/Foot",
"Definition:Altitude of Triangle",
"Definition:Distance between Points",
"File:Heron1.png",
"Pythagoras's Theorem",
"Area of Triangle in Terms of Side and... |
proofwiki-2127 | Heron's Formula | Let $\triangle ABC$ be a triangle with sides $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.
Then the area $\AA$ of $\triangle ABC$ is given by:
:$\AA = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$. | A triangle can be considered as a cyclic quadrilateral one of whose sides has degenerated to zero.
From Brahmagupta's Formula, the area of a cyclic quadrilateral is given by:
:$\sqrt {\paren {s - a} \paren {s - b} \paren {s - c} \paren {s - d}}$
where $s$ is the semiperimeter:
:$s = \dfrac {a + b + c + d} 2$
The result... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] with [[Definition:Side of Polygon|sides]] $a$, $b$ and $c$ [[Definition:Opposite (in Triangle)|opposite]] [[Definition:Vertex of Polygon|vertices]] $A$, $B$ and $C$ respectively.
Then the [[Definition:Area|area]] $\AA$ of $\triangle ABC$ is given by... | A [[Definition:Triangle (Geometry)|triangle]] can be considered as a [[Definition:Cyclic Quadrilateral|cyclic quadrilateral]] one of whose sides has degenerated to zero.
From [[Brahmagupta's Formula]], the [[Definition:Area|area]] of a [[Definition:Cyclic Quadrilateral|cyclic quadrilateral]] is given by:
:$\sqrt {\par... | Heron's Formula/Proof 2 | https://proofwiki.org/wiki/Heron's_Formula | https://proofwiki.org/wiki/Heron's_Formula/Proof_2 | [
"Heron's Formula",
"Areas of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Polygon/Vertex",
"Definition:Area",
"Definition:Semiperimeter"
] | [
"Definition:Triangle (Geometry)",
"Definition:Cyclic Quadrilateral",
"Brahmagupta's Formula",
"Definition:Area",
"Definition:Cyclic Quadrilateral",
"Definition:Semiperimeter"
] |
proofwiki-2128 | Heron's Formula | Let $\triangle ABC$ be a triangle with sides $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.
Then the area $\AA$ of $\triangle ABC$ is given by:
:$\AA = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$. | :700px
Let $\AA$ be the area of $\triangle ABC$.
Construct the incircle of $\triangle ABC$.
Let the incenter of $\triangle ABC$ be $M$.
Let the inradius of $\triangle ABC$ be $r$.
$\triangle ABC$ is made up of three triangles: $\triangle AMB$, $\triangle BMC$ and $\triangle CMA$.
From Area of Triangle in Terms of Side ... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] with [[Definition:Side of Polygon|sides]] $a$, $b$ and $c$ [[Definition:Opposite (in Triangle)|opposite]] [[Definition:Vertex of Polygon|vertices]] $A$, $B$ and $C$ respectively.
Then the [[Definition:Area|area]] $\AA$ of $\triangle ABC$ is given by... | :[[File:Heron3.png|700px]]
Let $\AA$ be the [[Definition:Area|area]] of $\triangle ABC$.
Construct the [[Definition:Incircle of Triangle|incircle]] of $\triangle ABC$.
Let the [[Definition:Incenter of Triangle|incenter]] of $\triangle ABC$ be $M$.
Let the [[Definition:Inradius of Triangle|inradius]] of $\triangle A... | Heron's Formula/Proof 3 | https://proofwiki.org/wiki/Heron's_Formula | https://proofwiki.org/wiki/Heron's_Formula/Proof_3 | [
"Heron's Formula",
"Areas of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Polygon/Vertex",
"Definition:Area",
"Definition:Semiperimeter"
] | [
"File:Heron3.png",
"Definition:Area",
"Definition:Incircle of Triangle",
"Definition:Incircle of Triangle/Incenter",
"Definition:Incircle of Triangle/Inradius",
"Definition:Triangle (Geometry)",
"Area of Triangle in Terms of Side and Altitude",
"Definition:Area",
"Definition:Semiperimeter",
"Defin... |
proofwiki-2129 | Heron's Formula | Let $\triangle ABC$ be a triangle with sides $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.
Then the area $\AA$ of $\triangle ABC$ is given by:
:$\AA = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$. | {{begin-eqn}}
{{eqn | l = \AA
| r = \dfrac {a b \sin C} 2
| c = Area of Triangle in Terms of Two Sides and Angle
}}
{{eqn | r = \dfrac {a b} 2 \cdot 2 \sin \dfrac C 2 \cos \dfrac C 2
| c = Double Angle Formula for Sine
}}
{{eqn | r = a b \sqrt {\dfrac {\paren {s - a} \paren {s - b} } {a b} } \sqrt {\d... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] with [[Definition:Side of Polygon|sides]] $a$, $b$ and $c$ [[Definition:Opposite (in Triangle)|opposite]] [[Definition:Vertex of Polygon|vertices]] $A$, $B$ and $C$ respectively.
Then the [[Definition:Area|area]] $\AA$ of $\triangle ABC$ is given by... | {{begin-eqn}}
{{eqn | l = \AA
| r = \dfrac {a b \sin C} 2
| c = [[Area of Triangle in Terms of Two Sides and Angle]]
}}
{{eqn | r = \dfrac {a b} 2 \cdot 2 \sin \dfrac C 2 \cos \dfrac C 2
| c = [[Double Angle Formula for Sine]]
}}
{{eqn | r = a b \sqrt {\dfrac {\paren {s - a} \paren {s - b} } {a b} } \... | Heron's Formula/Proof 4 | https://proofwiki.org/wiki/Heron's_Formula | https://proofwiki.org/wiki/Heron's_Formula/Proof_4 | [
"Heron's Formula",
"Areas of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Polygon/Vertex",
"Definition:Area",
"Definition:Semiperimeter"
] | [
"Area of Triangle in Terms of Two Sides and Angle",
"Double Angle Formulas/Sine",
"Sine of Half Angle in Triangle",
"Cosine of Half Angle in Triangle"
] |
proofwiki-2130 | Addition is Primitive Recursive | The function $\Add: \N^2 \to \N$, defined as:
:$\map \Add {n, m} = n + m$
is primitive recursive. | We observe that:
:$\map \Add {n, 0} = n + 0 = n$
and that
:$\map \Add {n, m + 1} = n + \paren {m + 1} = \paren {n + m} + 1 = \map \Succ {\map \Add {n, m} }$
where $\Succ$ is the successor function, which is a basic primitive recursive function.
We are to show that $\Add$ is defined by primitive recursion.
So we need to... | The [[Definition:Function|function]] $\Add: \N^2 \to \N$, defined as:
:$\map \Add {n, m} = n + m$
is [[Definition:Primitive Recursive Function|primitive recursive]]. | We observe that:
:$\map \Add {n, 0} = n + 0 = n$
and that
:$\map \Add {n, m + 1} = n + \paren {m + 1} = \paren {n + m} + 1 = \map \Succ {\map \Add {n, m} }$
where $\Succ$ is the [[Definition:Successor Function|successor function]], which is a [[Definition:Basic Primitive Recursive Function|basic primitive recursive fun... | Addition is Primitive Recursive | https://proofwiki.org/wiki/Addition_is_Primitive_Recursive | https://proofwiki.org/wiki/Addition_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Function",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Basic Primitive Recursive Function/Successor Function",
"Definition:Basic Primitive Recursive Function",
"Definition:Primitive Recursion",
"Definition:Primitive Recursive/Function",
"Definition:Basic Primitive Recursive Function",
"Definition:Ordered Tuple",
"Definition:Primitive Recursive/F... |
proofwiki-2131 | Exact Form of Prime-Counting Function | Let:
:$\ds \map \Pi x = \map \Li x - \sum_\rho \map \Li {x^\rho} - \map \ln 2 + \int_x^\infty \frac {\d t} {t \paren {t^2 - 1} \map \ln t}$
where:
:$\map \Li x$ is the offset logarithmic integral
:the sum $\ds \sum_\rho$ is taken over all $0 < \rho \in \R$ such that the zeta function $\map \zeta {\alpha + i \rho} = 0... | {{proof wanted}}
Category:Number Theory
9yaebhxvtysdkgbw2vyfjvgca7a6f15 | Let:
:$\ds \map \Pi x = \map \Li x - \sum_\rho \map \Li {x^\rho} - \map \ln 2 + \int_x^\infty \frac {\d t} {t \paren {t^2 - 1} \map \ln t}$
where:
:$\map \Li x$ is the [[Definition:Offset Logarithmic Integral|offset logarithmic integral]]
:the [[Definition:Summation|sum]] $\ds \sum_\rho$ is taken over all $0 < \rho \... | {{proof wanted}}
[[Category:Number Theory]]
9yaebhxvtysdkgbw2vyfjvgca7a6f15 | Exact Form of Prime-Counting Function | https://proofwiki.org/wiki/Exact_Form_of_Prime-Counting_Function | https://proofwiki.org/wiki/Exact_Form_of_Prime-Counting_Function | [
"Number Theory"
] | [
"Definition:Logarithmic Integral/Eulerian",
"Definition:Summation",
"Definition:Riemann Zeta Function",
"Definition:Prime-Counting Function",
"Definition:Möbius Function"
] | [
"Category:Number Theory"
] |
proofwiki-2132 | Multiplication is Primitive Recursive | The function $\operatorname{mult}: \N^2 \to \N$, defined as:
:$\map \Mult {n, m} = n \times m$
is primitive recursive. | We observe that:
:$\map \Mult {n, 0} = n \times 0 = 0$
and that
:$\map \Mult {n, m + 1} = n \times \paren {m + 1} = \paren {n \times m} + n = \map \Add {\map \Mult {n, m}, n}$.
We are to show that $\Mult$ is obtained by primitive recursion from known primitive recursive functions.
First we note that:
:$\map \Mult {n, 0... | The [[Definition:Function|function]] $\operatorname{mult}: \N^2 \to \N$, defined as:
:$\map \Mult {n, m} = n \times m$
is [[Definition:Primitive Recursive Function|primitive recursive]]. | We observe that:
:$\map \Mult {n, 0} = n \times 0 = 0$
and that
:$\map \Mult {n, m + 1} = n \times \paren {m + 1} = \paren {n \times m} + n = \map \Add {\map \Mult {n, m}, n}$.
We are to show that $\Mult$ is obtained by [[Definition:Primitive Recursion|primitive recursion]] from known [[Definition:Primitive Recursive... | Multiplication is Primitive Recursive | https://proofwiki.org/wiki/Multiplication_is_Primitive_Recursive | https://proofwiki.org/wiki/Multiplication_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Function",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Primitive Recursion",
"Definition:Primitive Recursive/Function",
"Definition:Basic Primitive Recursive Function/Zero Function",
"Definition:Basic Primitive Recursive Function",
"Definition:Primitive Recursive/Function",
"Definition:Basic Primitive Recursive Function/Projection Function",
"De... |
proofwiki-2133 | Linear Function is Primitive Recursive | The function $f: \N \to \N$, defined as:
:$\map f n = a n + b$
where $a$ and $b$ are constants, is primitive recursive. | We have that:
{{begin-eqn}}
{{eqn | l = a n + b
| r = \map \Add {\map \Mult {a, n}, b}
| c =
}}
{{eqn | r = \map \Add {\map \Mult {a, n}, \map {f_b} n}
| c =
}}
{{eqn | r = \map \Add {\map \Mult {\map {f_a} n, \map {\pr_1^1} n}, \map {f_b} n}
| c =
}}
{{end-eqn}}
where:
* $\Mult$ is primitive... | The [[Definition:Function|function]] $f: \N \to \N$, defined as:
:$\map f n = a n + b$
where $a$ and $b$ are [[Definition:Constant|constants]], is [[Definition:Primitive Recursive Function|primitive recursive]]. | We have that:
{{begin-eqn}}
{{eqn | l = a n + b
| r = \map \Add {\map \Mult {a, n}, b}
| c =
}}
{{eqn | r = \map \Add {\map \Mult {a, n}, \map {f_b} n}
| c =
}}
{{eqn | r = \map \Add {\map \Mult {\map {f_a} n, \map {\pr_1^1} n}, \map {f_b} n}
| c =
}}
{{end-eqn}}
where:
* [[Multiplication is... | Linear Function is Primitive Recursive | https://proofwiki.org/wiki/Linear_Function_is_Primitive_Recursive | https://proofwiki.org/wiki/Linear_Function_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Function",
"Definition:Constant",
"Definition:Primitive Recursive/Function"
] | [
"Multiplication is Primitive Recursive",
"Addition is Primitive Recursive",
"Constant Function is Primitive Recursive",
"Definition:Basic Primitive Recursive Function/Projection Function",
"Definition:Basic Primitive Recursive Function",
"Definition:Basic Primitive Recursive Function/Projection Function",... |
proofwiki-2134 | Substitution of Constant yields Primitive Recursive Function | Let $f: \N^{k + 1} \to \N$ be a primitive recursive function.
Then $g: \N^k \to \N$ given by:
:$\map g {n_1, n_2, \ldots, n_k} = \map f {n_1, n_2, \ldots, n_{i - 1}, a, n_i \ldots, n_k}$
is primitive recursive. | Let $n = \tuple {n_1, n_2, \ldots, n_{i - 1}, n_i \ldots, n_k}$.
We see that:
:$\map g {n_1, n_2, \ldots, n_k} = \map f {\map {\pr^k_1} n, \map {\pr^k_2} n, \ldots, \map {\pr^k_{i - 1} } n, \map {f_a} n, \map {\pr^k_i} n, \ldots, \map {\pr^k_k} n}$
We have that:
* $\pr^k_j$ is a basic primitive recursive function for a... | Let $f: \N^{k + 1} \to \N$ be a [[Definition:Primitive Recursive Function|primitive recursive function]].
Then $g: \N^k \to \N$ given by:
:$\map g {n_1, n_2, \ldots, n_k} = \map f {n_1, n_2, \ldots, n_{i - 1}, a, n_i \ldots, n_k}$
is [[Definition:Primitive Recursive Function|primitive recursive]]. | Let $n = \tuple {n_1, n_2, \ldots, n_{i - 1}, n_i \ldots, n_k}$.
We see that:
:$\map g {n_1, n_2, \ldots, n_k} = \map f {\map {\pr^k_1} n, \map {\pr^k_2} n, \ldots, \map {\pr^k_{i - 1} } n, \map {f_a} n, \map {\pr^k_i} n, \ldots, \map {\pr^k_k} n}$
We have that:
* $\pr^k_j$ is a [[Definition:Basic Primitive Recursive... | Substitution of Constant yields Primitive Recursive Function | https://proofwiki.org/wiki/Substitution_of_Constant_yields_Primitive_Recursive_Function | https://proofwiki.org/wiki/Substitution_of_Constant_yields_Primitive_Recursive_Function | [
"Primitive Recursive Functions"
] | [
"Definition:Primitive Recursive/Function",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Basic Primitive Recursive Function/Projection Function",
"Constant Function is Primitive Recursive",
"Definition:Substitution (Mathematical Logic)",
"Definition:Primitive Recursive/Function",
"Definition:Primitive Recursive/Function",
"Category:Primitive Recursive Functions"
] |
proofwiki-2135 | Exponentiation is Primitive Recursive | The function $\exp: \N^2 \to \N$, defined as:
:$\map \exp {n, m} = n^m$
is primitive recursive. | We observe that:
:$\map \exp {n, 0} = n^0 = 1$
and that:
:$\map \exp {n, m + 1} = n^\paren {m + 1} = \paren {n^m} \times n = \map {\mathrm {mult} } {\map \exp {n, m}, n}$.
Thus $\exp$ is defined by primitive recursion from the primitive recursive function $\mathrm {mult}$.
Hence the result.
{{qed}}
Category:Primitive R... | The [[Definition:Function|function]] $\exp: \N^2 \to \N$, defined as:
:$\map \exp {n, m} = n^m$
is [[Definition:Primitive Recursive Function|primitive recursive]]. | We observe that:
:$\map \exp {n, 0} = n^0 = 1$
and that:
:$\map \exp {n, m + 1} = n^\paren {m + 1} = \paren {n^m} \times n = \map {\mathrm {mult} } {\map \exp {n, m}, n}$.
Thus $\exp$ is defined by [[Definition:Primitive Recursion|primitive recursion]] from the [[Multiplication is Primitive Recursive|primitive recurs... | Exponentiation is Primitive Recursive | https://proofwiki.org/wiki/Exponentiation_is_Primitive_Recursive | https://proofwiki.org/wiki/Exponentiation_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Function",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Primitive Recursion",
"Multiplication is Primitive Recursive",
"Category:Primitive Recursive Functions"
] |
proofwiki-2136 | Bézout's Identity | Let $a, b \in \Z$ such that $a$ and $b$ are not both zero.
Let $\gcd \set {a, b}$ be the greatest common divisor of $a$ and $b$.
Then:
:$\exists x, y \in \Z: a x + b y = \gcd \set {a, b}$
That is, $\gcd \set {a, b}$ is an integer combination (or linear combination) of $a$ and $b$.
Furthermore, $\gcd \set {a, b}$ is the... | Work the Euclidean Division Algorithm backwards.
{{qed}} | Let $a, b \in \Z$ such that $a$ and $b$ are not both [[Definition:Zero (Number)|zero]].
Let $\gcd \set {a, b}$ be the [[Definition:Greatest Common Divisor of Integers|greatest common divisor]] of $a$ and $b$.
Then:
:$\exists x, y \in \Z: a x + b y = \gcd \set {a, b}$
That is, $\gcd \set {a, b}$ is an [[Definition:... | Work the [[Euclidean Algorithm|Euclidean Division Algorithm]] backwards.
{{qed}} | Bézout's Identity/Proof 1 | https://proofwiki.org/wiki/Bézout's_Identity | https://proofwiki.org/wiki/Bézout's_Identity/Proof_1 | [
"Bézout's Identity",
"Greatest Common Divisor",
"Number Theory"
] | [
"Definition:Zero (Number)",
"Definition:Greatest Common Divisor/Integers",
"Definition:Integer Combination",
"Definition:Linear Combination",
"Definition:Smallest Element",
"Definition:Positive/Integer",
"Definition:Integer Combination"
] | [
"Euclidean Algorithm"
] |
proofwiki-2137 | Bézout's Identity | Let $a, b \in \Z$ such that $a$ and $b$ are not both zero.
Let $\gcd \set {a, b}$ be the greatest common divisor of $a$ and $b$.
Then:
:$\exists x, y \in \Z: a x + b y = \gcd \set {a, b}$
That is, $\gcd \set {a, b}$ is an integer combination (or linear combination) of $a$ and $b$.
Furthermore, $\gcd \set {a, b}$ is the... | Let $a, b \in \Z$ such that $a$ and $b$ are not both zero.
Let $S$ be the set of all positive integer combinations of $a$ and $b$:
:$S = \set {x \in \Z, x > 0: x = m a + n b: m, n \in \Z}$
First we establish that $S \ne \O$.
We have:
{{begin-eqn}}
{{eqn | l = a > 0
| o = \implies
| r = \size a = 1 \times a ... | Let $a, b \in \Z$ such that $a$ and $b$ are not both [[Definition:Zero (Number)|zero]].
Let $\gcd \set {a, b}$ be the [[Definition:Greatest Common Divisor of Integers|greatest common divisor]] of $a$ and $b$.
Then:
:$\exists x, y \in \Z: a x + b y = \gcd \set {a, b}$
That is, $\gcd \set {a, b}$ is an [[Definition:... | Let $a, b \in \Z$ such that $a$ and $b$ are not both [[Definition:Zero (Number)|zero]].
Let $S$ be the [[Definition:Set|set]] of all [[Definition:Positive Integer|positive]] [[Definition:Integer Combination|integer combinations]] of $a$ and $b$:
:$S = \set {x \in \Z, x > 0: x = m a + n b: m, n \in \Z}$
First we est... | Bézout's Identity/Proof 2 | https://proofwiki.org/wiki/Bézout's_Identity | https://proofwiki.org/wiki/Bézout's_Identity/Proof_2 | [
"Bézout's Identity",
"Greatest Common Divisor",
"Number Theory"
] | [
"Definition:Zero (Number)",
"Definition:Greatest Common Divisor/Integers",
"Definition:Integer Combination",
"Definition:Linear Combination",
"Definition:Smallest Element",
"Definition:Positive/Integer",
"Definition:Integer Combination"
] | [
"Definition:Zero (Number)",
"Definition:Set",
"Definition:Positive/Integer",
"Definition:Integer Combination",
"Definition:Positive/Integer",
"Definition:Bounded Below Set",
"Set of Integers Bounded Below by Integer has Smallest Element",
"Division Theorem",
"Common Divisor Divides Integer Combinati... |
proofwiki-2138 | Bézout's Identity | Let $a, b \in \Z$ such that $a$ and $b$ are not both zero.
Let $\gcd \set {a, b}$ be the greatest common divisor of $a$ and $b$.
Then:
:$\exists x, y \in \Z: a x + b y = \gcd \set {a, b}$
That is, $\gcd \set {a, b}$ is an integer combination (or linear combination) of $a$ and $b$.
Furthermore, $\gcd \set {a, b}$ is the... | Consider the Euclidean algorithm in action:
{{begin-eqn}}
{{eqn | l = a
| r = q_1 b + r_1
| c =
}}
{{eqn | l = b
| r = q_2 r_1 + r_2
| c =
}}
{{eqn | l = r_1
| r = q_3 r_2 + r_3
| c =
}}
{{eqn | l = \cdots
| o =
| c =
}}
{{eqn | l = r_{n - 2}
| r = q_n r_{n - 1... | Let $a, b \in \Z$ such that $a$ and $b$ are not both [[Definition:Zero (Number)|zero]].
Let $\gcd \set {a, b}$ be the [[Definition:Greatest Common Divisor of Integers|greatest common divisor]] of $a$ and $b$.
Then:
:$\exists x, y \in \Z: a x + b y = \gcd \set {a, b}$
That is, $\gcd \set {a, b}$ is an [[Definition:... | Consider the [[Euclidean Algorithm/Demonstration|Euclidean algorithm]] in action:
{{begin-eqn}}
{{eqn | l = a
| r = q_1 b + r_1
| c =
}}
{{eqn | l = b
| r = q_2 r_1 + r_2
| c =
}}
{{eqn | l = r_1
| r = q_3 r_2 + r_3
| c =
}}
{{eqn | l = \cdots
| o =
| c =
}}
{{eqn |... | Bézout's Identity/Proof 3 | https://proofwiki.org/wiki/Bézout's_Identity | https://proofwiki.org/wiki/Bézout's_Identity/Proof_3 | [
"Bézout's Identity",
"Greatest Common Divisor",
"Number Theory"
] | [
"Definition:Zero (Number)",
"Definition:Greatest Common Divisor/Integers",
"Definition:Integer Combination",
"Definition:Linear Combination",
"Definition:Smallest Element",
"Definition:Positive/Integer",
"Definition:Integer Combination"
] | [
"Euclidean Algorithm/Demonstration",
"Principle of Mathematical Induction",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Integer",
"Bézout's Identity/Proof 3",
"Principle of Mathematical Induction"
] |
proofwiki-2139 | Bézout's Identity | Let $a, b \in \Z$ such that $a$ and $b$ are not both zero.
Let $\gcd \set {a, b}$ be the greatest common divisor of $a$ and $b$.
Then:
:$\exists x, y \in \Z: a x + b y = \gcd \set {a, b}$
That is, $\gcd \set {a, b}$ is an integer combination (or linear combination) of $a$ and $b$.
Furthermore, $\gcd \set {a, b}$ is the... | Let $a, b \in \Z$ such that $a$ and $b$ are not both zero.
Let $J$ be the set of all integer combinations of $a$ and $b$:
:$J = \set {x: x = m a + n b: m, n \in \Z}$
First we show that $J$ is an ideal of $\Z$
Let $\alpha = m_1 a + n_1 b$ and $\beta = m_2 a + n_2 b$, and let $c \in \Z$
Then $\alpha,\beta \in J$ and :
{{... | Let $a, b \in \Z$ such that $a$ and $b$ are not both [[Definition:Zero (Number)|zero]].
Let $\gcd \set {a, b}$ be the [[Definition:Greatest Common Divisor of Integers|greatest common divisor]] of $a$ and $b$.
Then:
:$\exists x, y \in \Z: a x + b y = \gcd \set {a, b}$
That is, $\gcd \set {a, b}$ is an [[Definition:... | Let $a, b \in \Z$ such that $a$ and $b$ are not both [[Definition:Zero (Number)|zero]].
Let $J$ be the [[Definition:Set|set]] of all [[Definition:Integer Combination|integer combinations]] of $a$ and $b$:
:$J = \set {x: x = m a + n b: m, n \in \Z}$
First we show that $J$ is an ideal of $\Z$
Let $\alpha = m_1 a + n... | Bézout's Identity/Proof 4 | https://proofwiki.org/wiki/Bézout's_Identity | https://proofwiki.org/wiki/Bézout's_Identity/Proof_4 | [
"Bézout's Identity",
"Greatest Common Divisor",
"Number Theory"
] | [
"Definition:Zero (Number)",
"Definition:Greatest Common Divisor/Integers",
"Definition:Integer Combination",
"Definition:Linear Combination",
"Definition:Smallest Element",
"Definition:Positive/Integer",
"Definition:Integer Combination"
] | [
"Definition:Zero (Number)",
"Definition:Set",
"Definition:Integer Combination",
"Definition:Integral Ideal",
"Definition:Zero (Number)"
] |
proofwiki-2140 | Bézout's Identity | Let $a, b \in \Z$ such that $a$ and $b$ are not both zero.
Let $\gcd \set {a, b}$ be the greatest common divisor of $a$ and $b$.
Then:
:$\exists x, y \in \Z: a x + b y = \gcd \set {a, b}$
That is, $\gcd \set {a, b}$ is an integer combination (or linear combination) of $a$ and $b$.
Furthermore, $\gcd \set {a, b}$ is the... | Let $\gcd \set {a, b} = d$.
Let $\dfrac a d = p$ and $\dfrac b d = q$.
From Integers Divided by GCD are Coprime:
:$\gcd \left\{{p, q}\right\} = 1$
From Integer Combination of Coprime Integers:
:$\exists x, y \in \Z: p x + q y = 1$
The result follows by multiplying both sides by $d$.
{{qed}} | Let $a, b \in \Z$ such that $a$ and $b$ are not both [[Definition:Zero (Number)|zero]].
Let $\gcd \set {a, b}$ be the [[Definition:Greatest Common Divisor of Integers|greatest common divisor]] of $a$ and $b$.
Then:
:$\exists x, y \in \Z: a x + b y = \gcd \set {a, b}$
That is, $\gcd \set {a, b}$ is an [[Definition:... | Let $\gcd \set {a, b} = d$.
Let $\dfrac a d = p$ and $\dfrac b d = q$.
From [[Integers Divided by GCD are Coprime]]:
:$\gcd \left\{{p, q}\right\} = 1$
From [[Integer Combination of Coprime Integers]]:
:$\exists x, y \in \Z: p x + q y = 1$
The result follows by multiplying both sides by $d$.
{{qed}} | Bézout's Identity/Proof 5 | https://proofwiki.org/wiki/Bézout's_Identity | https://proofwiki.org/wiki/Bézout's_Identity/Proof_5 | [
"Bézout's Identity",
"Greatest Common Divisor",
"Number Theory"
] | [
"Definition:Zero (Number)",
"Definition:Greatest Common Divisor/Integers",
"Definition:Integer Combination",
"Definition:Linear Combination",
"Definition:Smallest Element",
"Definition:Positive/Integer",
"Definition:Integer Combination"
] | [
"Integers Divided by GCD are Coprime",
"Integer Combination of Coprime Integers"
] |
proofwiki-2141 | Bézout's Identity | Let $a, b \in \Z$ such that $a$ and $b$ are not both zero.
Let $\gcd \set {a, b}$ be the greatest common divisor of $a$ and $b$.
Then:
:$\exists x, y \in \Z: a x + b y = \gcd \set {a, b}$
That is, $\gcd \set {a, b}$ is an integer combination (or linear combination) of $a$ and $b$.
Furthermore, $\gcd \set {a, b}$ is the... | We have that Integers are Euclidean Domain, where the Euclidean valuation $\nu$ is defined as:
:$\map \nu x = \size x$
The result follows from Bézout's Identity on Euclidean Domain.
{{qed}} | Let $a, b \in \Z$ such that $a$ and $b$ are not both [[Definition:Zero (Number)|zero]].
Let $\gcd \set {a, b}$ be the [[Definition:Greatest Common Divisor of Integers|greatest common divisor]] of $a$ and $b$.
Then:
:$\exists x, y \in \Z: a x + b y = \gcd \set {a, b}$
That is, $\gcd \set {a, b}$ is an [[Definition:... | We have that [[Integers are Euclidean Domain]], where the [[Definition:Euclidean Valuation|Euclidean valuation]] $\nu$ is defined as:
:$\map \nu x = \size x$
The result follows from [[Bézout's Identity on Euclidean Domain]].
{{qed}} | Bézout's Identity/Proof 6 | https://proofwiki.org/wiki/Bézout's_Identity | https://proofwiki.org/wiki/Bézout's_Identity/Proof_6 | [
"Bézout's Identity",
"Greatest Common Divisor",
"Number Theory"
] | [
"Definition:Zero (Number)",
"Definition:Greatest Common Divisor/Integers",
"Definition:Integer Combination",
"Definition:Linear Combination",
"Definition:Smallest Element",
"Definition:Positive/Integer",
"Definition:Integer Combination"
] | [
"Integers are Euclidean Domain",
"Definition:Euclidean Domain/Valuation",
"Bézout's Identity/Euclidean Domain"
] |
proofwiki-2142 | Predecessor Function is Primitive Recursive | The '''predecessor function''' $\operatorname{pred}: \N \to \N$ defined as:
:$\map {\operatorname{pred} } n = \begin{cases}
0 & : n = 0 \\
n-1 & : n > 0
\end{cases}$
is primitive recursive. | We can use Primitive Recursion on One Variable to find $g: \N^2 \to \N$ and $h: \N \to \N$ such that:
:$\map h n = \begin{cases}
\map {\operatorname{zero} } n & : n = 0 \\
\map g {n - 1, \map h {n - 1} } & : n > 0
\end{cases}
$
By setting:
:$\map g {n, m} = \map {\pr^2_1} {n, m}$
we see that setting $h = \operatorname... | The '''predecessor function''' $\operatorname{pred}: \N \to \N$ defined as:
:$\map {\operatorname{pred} } n = \begin{cases}
0 & : n = 0 \\
n-1 & : n > 0
\end{cases}$
is [[Definition:Primitive Recursive Function|primitive recursive]]. | We can use [[Definition:Primitive Recursion/One Variable|Primitive Recursion on One Variable]] to find $g: \N^2 \to \N$ and $h: \N \to \N$ such that:
:$\map h n = \begin{cases}
\map {\operatorname{zero} } n & : n = 0 \\
\map g {n - 1, \map h {n - 1} } & : n > 0
\end{cases}
$
By setting:
:$\map g {n, m} = \map {\pr^2... | Predecessor Function is Primitive Recursive | https://proofwiki.org/wiki/Predecessor_Function_is_Primitive_Recursive | https://proofwiki.org/wiki/Predecessor_Function_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Primitive Recursive/Function"
] | [
"Definition:Primitive Recursion/One Variable",
"Definition:Basic Primitive Recursive Function",
"Category:Primitive Recursive Functions"
] |
proofwiki-2143 | Cut-Off Subtraction is Primitive Recursive | The partial subtraction function, defined as:
:<nowiki>$\forall \tuple {n, m} \in \N^2: n \mathop {\dot -} m = \begin{cases}
0 & : n < m \\
n - m & : n \ge m
\end{cases}$</nowiki>
is primitive recursive. | We see that:
:<nowiki>$n \mathop {\dot -} \paren {m + 1} = \begin{cases}
0 & : n \mathop {\dot -} m = 0 \\
\paren {n \mathop {\dot -} m} - 1 & : n \mathop {\dot -} m > 0
\end{cases}$</nowiki>
Hence we can define partial subtraction as:
:<nowiki>$n \mathop {\dot -} m = \begin{cases}
n & : m = 0 \\
\operatorname{pred} \p... | The [[Definition:Partial Subtraction|partial subtraction]] function, defined as:
:<nowiki>$\forall \tuple {n, m} \in \N^2: n \mathop {\dot -} m = \begin{cases}
0 & : n < m \\
n - m & : n \ge m
\end{cases}$</nowiki>
is [[Definition:Primitive Recursive Function|primitive recursive]]. | We see that:
:<nowiki>$n \mathop {\dot -} \paren {m + 1} = \begin{cases}
0 & : n \mathop {\dot -} m = 0 \\
\paren {n \mathop {\dot -} m} - 1 & : n \mathop {\dot -} m > 0
\end{cases}$</nowiki>
Hence we can define [[Definition:Partial Subtraction|partial subtraction]] as:
:<nowiki>$n \mathop {\dot -} m = \begin{cases}
n... | Cut-Off Subtraction is Primitive Recursive | https://proofwiki.org/wiki/Cut-Off_Subtraction_is_Primitive_Recursive | https://proofwiki.org/wiki/Cut-Off_Subtraction_is_Primitive_Recursive | [
"Partial Subtraction",
"Primitive Recursive Functions"
] | [
"Definition:Partial Subtraction",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Partial Subtraction",
"Definition:Primitive Recursion",
"Predecessor Function is Primitive Recursive"
] |
proofwiki-2144 | Maximum Function is Primitive Recursive | The maximum function $\max: \N^2 \to \N$, defined as:
:<nowiki>$\map \max {n, m} = \begin{cases}
m: & n \le m \\
n: & m \le n
\end{cases}$</nowiki>
is primitive recursive. | We see that:
:$\map \max {n, m} = \paren {n \ \dot - \ m} + m$
where $\dot -$ denotes the partial subtraction because:
:$(1):\quad n > m \implies \paren {n \ \dot - \ m} + m = n - m + m = n$
:$(2):\quad n < m \implies \paren {n \ \dot - \ m} + m = 0 + m = m$
:$(3):\quad n = m \implies \paren {n \ \dot - \ m} + m = 0 + ... | The [[Definition:Max Operation|maximum function]] $\max: \N^2 \to \N$, defined as:
:<nowiki>$\map \max {n, m} = \begin{cases}
m: & n \le m \\
n: & m \le n
\end{cases}$</nowiki>
is [[Definition:Primitive Recursive Function|primitive recursive]]. | We see that:
:$\map \max {n, m} = \paren {n \ \dot - \ m} + m$
where $\dot -$ denotes the [[Definition:Partial Subtraction|partial subtraction]] because:
:$(1):\quad n > m \implies \paren {n \ \dot - \ m} + m = n - m + m = n$
:$(2):\quad n < m \implies \paren {n \ \dot - \ m} + m = 0 + m = m$
:$(3):\quad n = m \impli... | Maximum Function is Primitive Recursive | https://proofwiki.org/wiki/Maximum_Function_is_Primitive_Recursive | https://proofwiki.org/wiki/Maximum_Function_is_Primitive_Recursive | [
"Max Operation",
"Primitive Recursive Functions"
] | [
"Definition:Max Operation",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Partial Subtraction",
"Definition:Substitution (Mathematical Logic)",
"Cut-Off Subtraction is Primitive Recursive",
"Category:Max Operation",
"Category:Primitive Recursive Functions"
] |
proofwiki-2145 | Sum of Maximum and Minimum | For all numbers $a, b$ where $a, b$ in $\N, \Z, \Q$ or $\R$:
:$a + b = \max \set {a, b} + \min \set {a, b}$ | From the definitions of max and min:
:<nowiki>$\max \set {a, b} = \begin{cases}
b: & a \le b \\
a: & b \le a
\end{cases}$</nowiki>
and
:<nowiki>$\min \set {a, b} = \begin{cases}
a: & a \le b \\
b: & b \le a
\end{cases}$</nowiki>
Let $a < b$.
Then:
:$\max \set {a, b} + \min \set {a, b} = b + a$
Let $a > b$.
Then:
:$\max... | For all [[Definition:Number|numbers]] $a, b$ where $a, b$ in $\N, \Z, \Q$ or $\R$:
:$a + b = \max \set {a, b} + \min \set {a, b}$ | From the definitions of [[Definition:Max Operation|max]] and [[Definition:Min Operation|min]]:
:<nowiki>$\max \set {a, b} = \begin{cases}
b: & a \le b \\
a: & b \le a
\end{cases}$</nowiki>
and
:<nowiki>$\min \set {a, b} = \begin{cases}
a: & a \le b \\
b: & b \le a
\end{cases}$</nowiki>
Let $a < b$.
Then:
:$\max \set... | Sum of Maximum and Minimum | https://proofwiki.org/wiki/Sum_of_Maximum_and_Minimum | https://proofwiki.org/wiki/Sum_of_Maximum_and_Minimum | [
"Algebra"
] | [
"Definition:Number"
] | [
"Definition:Max Operation",
"Definition:Min Operation",
"Definition:Complex Number",
"Category:Algebra"
] |
proofwiki-2146 | Minimum Function is Primitive Recursive | The minimum function $\min: \N^2 \to \N$, defined as:
:$\map \min {n, m} = \begin{cases}
n: & n \le m \\
m: & m \le n
\end{cases}$
is primitive recursive. | From Sum Less Maximum is Minimum we have that:
:$\map \min {n, m} = n + m - \map \max {n, m}$.
As $n + m \ge \map \max {n, m}$, we have that:
:$\map \min {n, m} = n + m \ \dot - \ \map \max {n, m}$
Hence we see that $\min$ is obtained by substitution from:
* the primitive recursive function $n \ \dot - \ m$
* the prim... | The [[Definition:Min Operation|minimum function]] $\min: \N^2 \to \N$, defined as:
:$\map \min {n, m} = \begin{cases}
n: & n \le m \\
m: & m \le n
\end{cases}$
is [[Definition:Primitive Recursive Function|primitive recursive]]. | From [[Sum Less Maximum is Minimum]] we have that:
:$\map \min {n, m} = n + m - \map \max {n, m}$.
As $n + m \ge \map \max {n, m}$, we have that:
:$\map \min {n, m} = n + m \ \dot - \ \map \max {n, m}$
Hence we see that $\min$ is obtained by [[Definition:Substitution (Mathematical Logic)|substitution]] from:
* the ... | Minimum Function is Primitive Recursive | https://proofwiki.org/wiki/Minimum_Function_is_Primitive_Recursive | https://proofwiki.org/wiki/Minimum_Function_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Min Operation",
"Definition:Primitive Recursive/Function"
] | [
"Sum Less Maximum is Minimum",
"Definition:Substitution (Mathematical Logic)",
"Cut-Off Subtraction is Primitive Recursive",
"Maximum Function is Primitive Recursive",
"Category:Primitive Recursive Functions"
] |
proofwiki-2147 | Absolute Difference Function is Primitive Recursive | The '''absolute difference''' function $\operatorname {adf}: \N^2 \to \N$, defined as:
:$\map {\operatorname {adf} } {n, m} = \size {n - m}$
where $\size a$ is defined as the absolute value of $a$, is primitive recursive. | We note that:
:$\size {n - m} = \paren {n \mathop {\dot -} m} + \paren {m \mathop {\dot -} n} = \map {\operatorname {add} } {\paren {n \mathop {\dot -} m}, \paren {m \mathop {\dot -} n} }$
Next we note that:
:$m \mathop {\dot -} n = \map {\pr^2_2} {n, m} \mathop {\dot -} \map {\pr^2_1} {n, m}$
where $\pr^2_k$ is the pr... | The '''[[Definition:Absolute Difference|absolute difference]]''' function $\operatorname {adf}: \N^2 \to \N$, defined as:
:$\map {\operatorname {adf} } {n, m} = \size {n - m}$
where $\size a$ is defined as the [[Definition:Absolute Value|absolute value]] of $a$, is [[Definition:Primitive Recursive Function|primitive re... | We note that:
:$\size {n - m} = \paren {n \mathop {\dot -} m} + \paren {m \mathop {\dot -} n} = \map {\operatorname {add} } {\paren {n \mathop {\dot -} m}, \paren {m \mathop {\dot -} n} }$
Next we note that:
:$m \mathop {\dot -} n = \map {\pr^2_2} {n, m} \mathop {\dot -} \map {\pr^2_1} {n, m}$
where $\pr^2_k$ is the [... | Absolute Difference Function is Primitive Recursive | https://proofwiki.org/wiki/Absolute_Difference_Function_is_Primitive_Recursive | https://proofwiki.org/wiki/Absolute_Difference_Function_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Absolute Difference",
"Definition:Absolute Value",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Basic Primitive Recursive Function/Projection Function",
"Definition:Substitution (Mathematical Logic)",
"Cut-Off Subtraction is Primitive Recursive",
"Addition is Primitive Recursive",
"Definition:Basic Primitive Recursive Function/Projection Function",
"Category:Primitive Recursive Functions... |
proofwiki-2148 | Primitive Recursive Set is URM Computable | Every primitive recursive set is URM computable. | This follows immediately from:
* a set is primitive recursive if its characteristic function is primitive recursive
* the fact that every Primitive Recursive Function is URM Computable.
{{qed}} | Every [[Definition:Primitive Recursive Set|primitive recursive set]] is [[Definition:URM Computability#Set|URM computable]]. | This follows immediately from:
* a [[Definition:Set|set]] is [[Definition:Primitive Recursive Set|primitive recursive]] if its [[Definition:Characteristic Function of Set|characteristic function]] is [[Definition:Primitive Recursive Function|primitive recursive]]
* the fact that every [[Primitive Recursive Function is ... | Primitive Recursive Set is URM Computable | https://proofwiki.org/wiki/Primitive_Recursive_Set_is_URM_Computable | https://proofwiki.org/wiki/Primitive_Recursive_Set_is_URM_Computable | [
"Primitive Recursive Functions"
] | [
"Definition:Primitive Recursive/Set",
"Definition:URM Computability"
] | [
"Definition:Set",
"Definition:Primitive Recursive/Set",
"Definition:Characteristic Function (Set Theory)/Set",
"Definition:Primitive Recursive/Function",
"Primitive Recursive Function is URM Computable"
] |
proofwiki-2149 | Set Containing Only Zero is Primitive Recursive | The subset $\left\{{0}\right\} \subset \N$ is primitive recursive. | We note that:
:$1 \mathop {\dot -} n = \begin{cases}
1 & : n = 0 \\
0 & : n > 0
\end{cases}$
and so the characteristic function $\chi_{\left\{{0}\right\}}$ is given by $\chi_{\left\{{0}\right\}} \left({n}\right) = 1 \mathop {\dot -} n$.
So $\chi_{\left\{{0}\right\}}$ is obtained by substitution from the primitive recur... | The [[Definition:Subset|subset]] $\left\{{0}\right\} \subset \N$ is [[Definition:Primitive Recursive Set|primitive recursive]]. | We note that:
:$1 \mathop {\dot -} n = \begin{cases}
1 & : n = 0 \\
0 & : n > 0
\end{cases}$
and so the [[Definition:Characteristic Function of Set|characteristic function]] $\chi_{\left\{{0}\right\}}$ is given by $\chi_{\left\{{0}\right\}} \left({n}\right) = 1 \mathop {\dot -} n$.
So $\chi_{\left\{{0}\right\}}$ is ob... | Set Containing Only Zero is Primitive Recursive | https://proofwiki.org/wiki/Set_Containing_Only_Zero_is_Primitive_Recursive | https://proofwiki.org/wiki/Set_Containing_Only_Zero_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Subset",
"Definition:Primitive Recursive/Set"
] | [
"Definition:Characteristic Function (Set Theory)/Set",
"Definition:Substitution (Mathematical Logic)",
"Cut-Off Subtraction is Primitive Recursive",
"Constant Function is Primitive Recursive",
"Category:Primitive Recursive Functions"
] |
proofwiki-2150 | Set of Non-Zero Natural Numbers is Primitive Recursive | Let $\N^*$ be defined as $\N^* = \N \setminus \set 0$.
The subset $\N^* \subset \N$ is primitive recursive. | We have that the characteristic function $\chi_{\set 0}$ of $\set 0$ is primitive recursive.
We note that:
:If $n = 0$ then $\map {\chi_{\set 0} } n = 1$ therefore $\map {\chi_{\set 0} } {\map {\chi_{\set 0} } n} = 0$.
:If $n > 0$ then $\map {\chi_{\set 0} } n = 0$ therefore $\map {\chi_{\set 0} } {\map {\chi_{\set 0} ... | Let $\N^*$ be defined as $\N^* = \N \setminus \set 0$.
The [[Definition:Subset|subset]] $\N^* \subset \N$ is [[Definition:Primitive Recursive Set|primitive recursive]]. | We have that the [[Definition:Characteristic Function of Set|characteristic function]] $\chi_{\set 0}$ of $\set 0$ is [[Set Containing Only Zero is Primitive Recursive|primitive recursive]].
We note that:
:If $n = 0$ then $\map {\chi_{\set 0} } n = 1$ therefore $\map {\chi_{\set 0} } {\map {\chi_{\set 0} } n} = 0$.
... | Set of Non-Zero Natural Numbers is Primitive Recursive | https://proofwiki.org/wiki/Set_of_Non-Zero_Natural_Numbers_is_Primitive_Recursive | https://proofwiki.org/wiki/Set_of_Non-Zero_Natural_Numbers_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Subset",
"Definition:Primitive Recursive/Set"
] | [
"Definition:Characteristic Function (Set Theory)/Set",
"Set Containing Only Zero is Primitive Recursive",
"Definition:Substitution (Mathematical Logic)",
"Set Containing Only Zero is Primitive Recursive",
"Category:Primitive Recursive Functions"
] |
proofwiki-2151 | Signum Function is Primitive Recursive | Let $\sgn: \N \to \N$ be defined as the signum function.
Then $\sgn$ is primitive recursive. | By Signum Function on Natural Numbers as Characteristic Function, $\map \sgn n = \chi_{\N^*}$, where $\N^* = \N \setminus \set 0$.
By Set of Non-Zero Natural Numbers is Primitive Recursive, $\N^*$ is primitive recursive.
Thus $\sgn$ is primitive recursive by definition of Primitive Recursive Set.
{{qed}}
Category:Signu... | Let $\sgn: \N \to \N$ be defined as the [[Definition:Signum Function/Natural Numbers|signum function]].
Then $\sgn$ is [[Definition:Primitive Recursive Set|primitive recursive]]. | By [[Signum Function on Natural Numbers as Characteristic Function]], $\map \sgn n = \chi_{\N^*}$, where $\N^* = \N \setminus \set 0$.
By [[Set of Non-Zero Natural Numbers is Primitive Recursive]], $\N^*$ is [[Definition:Primitive Recursive Set|primitive recursive]].
Thus $\sgn$ is [[Definition:Primitive Recursive Se... | Signum Function is Primitive Recursive | https://proofwiki.org/wiki/Signum_Function_is_Primitive_Recursive | https://proofwiki.org/wiki/Signum_Function_is_Primitive_Recursive | [
"Signum Function",
"Primitive Recursive Functions"
] | [
"Definition:Signum Function/Natural Numbers",
"Definition:Primitive Recursive/Set"
] | [
"Signum Function on Natural Numbers as Characteristic Function",
"Set of Non-Zero Natural Numbers is Primitive Recursive",
"Definition:Primitive Recursive/Set",
"Definition:Primitive Recursive/Set",
"Definition:Primitive Recursive/Set",
"Category:Signum Function",
"Category:Primitive Recursive Functions... |
proofwiki-2152 | Set of Natural Numbers is Primitive Recursive | The set of natural numbers $\N$ is primitive recursive. | The characteristic function $\chi_\N: \N \to \N$ is defined as:
:$\forall n \in \N: \chi_\N \left({n}\right) = 1$.
So:
: $\chi_\N \left({n}\right) = f^1_1 \left({n}\right)$
The constant function $f^1_1$ is primitive recursive.
Hence the result.
{{qed}}
Category:Primitive Recursive Functions
Category:Natural Numbers
10v... | The [[Definition:Natural Numbers|set of natural numbers]] $\N$ is [[Definition:Primitive Recursive Set|primitive recursive]]. | The [[Definition:Characteristic Function of Set|characteristic function]] $\chi_\N: \N \to \N$ is defined as:
:$\forall n \in \N: \chi_\N \left({n}\right) = 1$.
So:
: $\chi_\N \left({n}\right) = f^1_1 \left({n}\right)$
The [[Constant Function is Primitive Recursive|constant function $f^1_1$ is primitive recursive]].
... | Set of Natural Numbers is Primitive Recursive | https://proofwiki.org/wiki/Set_of_Natural_Numbers_is_Primitive_Recursive | https://proofwiki.org/wiki/Set_of_Natural_Numbers_is_Primitive_Recursive | [
"Primitive Recursive Functions",
"Natural Numbers"
] | [
"Definition:Natural Numbers",
"Definition:Primitive Recursive/Set"
] | [
"Definition:Characteristic Function (Set Theory)/Set",
"Constant Function is Primitive Recursive",
"Category:Primitive Recursive Functions",
"Category:Natural Numbers"
] |
proofwiki-2153 | Complement of Primitive Recursive Set | Let $S \subseteq \N$ be primitive recursive.
Then its relative complement $\N \setminus S$ of $S$ in $\N$ is primitive recursive. | By definition, we have that the characteristic function $\map {\chi_{\N \mathop \setminus S} } n = 1$ {{iff}} $\map {\chi_S} n = 0$.
So:
:$\map {\chi_{\N \mathop \setminus S} } n = \map {\chi_{\set 0} } {\map {\chi_S} n}$
Thus $\chi_{\N \mathop \setminus S}$ is obtained by substitution from $\chi_{\set 0}$ and $\chi_S$... | Let $S \subseteq \N$ be [[Definition:Primitive Recursive Set|primitive recursive]].
Then its [[Definition:Relative Complement|relative complement]] $\N \setminus S$ of $S$ in $\N$ is [[Definition:Primitive Recursive Set|primitive recursive]]. | By definition, we have that the [[Definition:Characteristic Function of Set|characteristic function]] $\map {\chi_{\N \mathop \setminus S} } n = 1$ {{iff}} $\map {\chi_S} n = 0$.
So:
:$\map {\chi_{\N \mathop \setminus S} } n = \map {\chi_{\set 0} } {\map {\chi_S} n}$
Thus $\chi_{\N \mathop \setminus S}$ is obtained b... | Complement of Primitive Recursive Set | https://proofwiki.org/wiki/Complement_of_Primitive_Recursive_Set | https://proofwiki.org/wiki/Complement_of_Primitive_Recursive_Set | [
"Primitive Recursive Functions",
"Relative Complement"
] | [
"Definition:Primitive Recursive/Set",
"Definition:Relative Complement",
"Definition:Primitive Recursive/Set"
] | [
"Definition:Characteristic Function (Set Theory)/Set",
"Definition:Substitution (Mathematical Logic)",
"Set Containing Only Zero is Primitive Recursive",
"Category:Primitive Recursive Functions",
"Category:Relative Complement"
] |
proofwiki-2154 | Intersection of Primitive Recursive Sets | Let $A, B \subseteq \N$ be subsets of the set of natural numbers $\N$.
Let $A$ and $B$ both be primitive recursive.
Then $A \cap B$, the intersection of $A$ and $B$, is primitive recursive. | $A$ and $B$ are primitive recursive, therefore so are their [Definition:Characteristic Function of Set|characteristic functions]] $\chi_A$ and $\chi_B$.
Let $n \in \N$ be a natural number.
From Characteristic Function of Intersection: Variant 1:
:$\chi_{A \cap B} \left({n}\right) = \chi_A \left({n}\right) \times \chi_B... | Let $A, B \subseteq \N$ be [[Definition:Subset|subsets]] of the [[Definition:Natural Numbers|set of natural numbers]] $\N$.
Let $A$ and $B$ both be [[Definition:Primitive Recursive Set|primitive recursive]].
Then $A \cap B$, the [[Definition:Set Intersection|intersection]] of $A$ and $B$, is [[Definition:Primitive R... | $A$ and $B$ are [[Definition:Primitive Recursive Set|primitive recursive]], therefore so are their [Definition:Characteristic Function of Set|characteristic functions]] $\chi_A$ and $\chi_B$.
Let $n \in \N$ be a [[Definition:Natural Numbers|natural number]].
From [[Characteristic Function of Intersection/Variant 1|Ch... | Intersection of Primitive Recursive Sets | https://proofwiki.org/wiki/Intersection_of_Primitive_Recursive_Sets | https://proofwiki.org/wiki/Intersection_of_Primitive_Recursive_Sets | [
"Intersection",
"Set Intersection",
"Primitive Recursive Functions",
"Set Intersection"
] | [
"Definition:Subset",
"Definition:Natural Numbers",
"Definition:Primitive Recursive/Set",
"Definition:Set Intersection",
"Definition:Primitive Recursive/Set"
] | [
"Definition:Primitive Recursive/Set",
"Definition:Natural Numbers",
"Characteristic Function of Intersection/Variant 1",
"Definition:Substitution (Mathematical Logic)",
"Definition:Primitive Recursive/Function",
"Category:Primitive Recursive Functions",
"Category:Set Intersection"
] |
proofwiki-2155 | Union of Primitive Recursive Sets | Let $A, B \subseteq \N$ be subsets of the set of natural numbers $\N$.
Let $A$ and $B$ both be primitive recursive.
Then $A \cup B$, the union of $A$ and $B$, is primitive recursive. | $A$ and $B$ are primitive recursive, therefore so are their characteristic functions $\chi_A$ and $\chi_B$.
Let $n \in \N$ be a natural number.
Then $n \in A \cup B \iff \map {\chi_A} n + \map {\chi_B} n > 0$.
So:
{{begin-eqn}}
{{eqn | l = \map {\chi_{A \cup B} } n
| r = \map \sgn {\map {\chi_A} n + \map {\chi_B}... | Let $A, B \subseteq \N$ be [[Definition:Subset|subsets]] of the [[Definition:Natural Numbers|set of natural numbers]] $\N$.
Let $A$ and $B$ both be [[Definition:Primitive Recursive Set|primitive recursive]].
Then $A \cup B$, the [[Definition:Set Union|union]] of $A$ and $B$, is [[Definition:Primitive Recursive Set|pr... | $A$ and $B$ are [[Definition:Primitive Recursive Set|primitive recursive]], therefore so are their [[Definition:Characteristic Function of Set|characteristic functions]] $\chi_A$ and $\chi_B$.
Let $n \in \N$ be a [[Definition:Natural Numbers|natural number]].
Then $n \in A \cup B \iff \map {\chi_A} n + \map {\chi_B} ... | Union of Primitive Recursive Sets | https://proofwiki.org/wiki/Union_of_Primitive_Recursive_Sets | https://proofwiki.org/wiki/Union_of_Primitive_Recursive_Sets | [
"Primitive Recursive Functions",
"Set Union"
] | [
"Definition:Subset",
"Definition:Natural Numbers",
"Definition:Primitive Recursive/Set",
"Definition:Set Union",
"Definition:Primitive Recursive/Set"
] | [
"Definition:Primitive Recursive/Set",
"Definition:Characteristic Function (Set Theory)/Set",
"Definition:Natural Numbers",
"Signum Function is Primitive Recursive",
"Addition is Primitive Recursive",
"Definition:Substitution (Mathematical Logic)",
"Definition:Primitive Recursive/Function",
"Category:P... |
proofwiki-2156 | Set of Even Numbers is Primitive Recursive | Let $E \subseteq \N$ be the set of all even natural numbers.
Then $E$ is primitive recursive. | If $n \in E$ then $n$ is of the form $n = 2 k$ where $k \in \N$.
We have that:
* if the characteristic function $\map {\chi_E} n = 1$ then $\map {\chi_E} {n + 1} = 0$.
* if the characteristic function $\map {\chi_E} n = 0$ then $\map {\chi_E} {n + 1} = 1$.
So $\chi_E$ can be defined by:
:<nowiki>$\map {\chi_E} n = \beg... | Let $E \subseteq \N$ be the [[Definition:Set|set]] of all [[Definition:Even Integer|even]] [[Definition:Natural Numbers|natural numbers]].
Then $E$ is [[Definition:Primitive Recursive Set|primitive recursive]]. | If $n \in E$ then $n$ is of the form $n = 2 k$ where $k \in \N$.
We have that:
* if the [[Definition:Characteristic Function of Set|characteristic function]] $\map {\chi_E} n = 1$ then $\map {\chi_E} {n + 1} = 0$.
* if the [[Definition:Characteristic Function of Set|characteristic function]] $\map {\chi_E} n = 0$ then... | Set of Even Numbers is Primitive Recursive | https://proofwiki.org/wiki/Set_of_Even_Numbers_is_Primitive_Recursive | https://proofwiki.org/wiki/Set_of_Even_Numbers_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Set",
"Definition:Even Integer",
"Definition:Natural Numbers",
"Definition:Primitive Recursive/Set"
] | [
"Definition:Characteristic Function (Set Theory)/Set",
"Definition:Characteristic Function (Set Theory)/Set",
"Definition:Primitive Recursion",
"Definition:Constant Mapping",
"Definition:Signum Function",
"Constant Function is Primitive Recursive",
"Signum Function is Primitive Recursive",
"Category:P... |
proofwiki-2157 | Primitive Recursive Relation is URM Computable | Every primitive recursive relation is URM computable. | This follows immediately from:
* a relation is primitive recursive if its characteristic function is a primitive recursive
* the fact that every Primitive Recursive Function is URM Computable.
{{qed}} | Every [[Definition:Primitive Recursive Relation|primitive recursive relation]] is [[Definition:URM Computability#Set|URM computable]]. | This follows immediately from:
* a relation is [[Definition:Primitive Recursive Relation|primitive recursive]] if its [[Definition:Characteristic Function of Relation|characteristic function]] is a [[Definition:Primitive Recursive Function|primitive recursive]]
* the fact that every [[Primitive Recursive Function is UR... | Primitive Recursive Relation is URM Computable | https://proofwiki.org/wiki/Primitive_Recursive_Relation_is_URM_Computable | https://proofwiki.org/wiki/Primitive_Recursive_Relation_is_URM_Computable | [
"Primitive Recursive Functions"
] | [
"Definition:Primitive Recursive/Relation",
"Definition:URM Computability"
] | [
"Definition:Primitive Recursive/Relation",
"Definition:Characteristic Function (Set Theory)/Relation",
"Definition:Primitive Recursive/Function",
"Primitive Recursive Function is URM Computable"
] |
proofwiki-2158 | Equality Relation is Primitive Recursive | The relation $\operatorname{eq} \subseteq \N^2$, defined as:
:$\map {\operatorname {eq} } {n, m} \iff n = m$
is primitive recursive. | We note that:
:$n = m \iff \size {n - m} = 0$
:$n \ne m \iff \size {n - m} > 0$
So it can be seen that the characteristic function of $\operatorname{eq}$ is given by:
:$\map {\chi_{\operatorname {eq} } } {n, m} = \overline {\map \sgn {\map {\operatorname {adf} } {n, m} } }$.
So $\map {\chi_{\operatorname {eq} } } {n, m... | The [[Definition:Relation|relation]] $\operatorname{eq} \subseteq \N^2$, defined as:
:$\map {\operatorname {eq} } {n, m} \iff n = m$
is [[Definition:Primitive Recursive Relation|primitive recursive]]. | We note that:
:$n = m \iff \size {n - m} = 0$
:$n \ne m \iff \size {n - m} > 0$
So it can be seen that the [[Definition:Characteristic Function of Relation|characteristic function]] of $\operatorname{eq}$ is given by:
:$\map {\chi_{\operatorname {eq} } } {n, m} = \overline {\map \sgn {\map {\operatorname {adf} } {n, m... | Equality Relation is Primitive Recursive | https://proofwiki.org/wiki/Equality_Relation_is_Primitive_Recursive | https://proofwiki.org/wiki/Equality_Relation_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Relation",
"Definition:Primitive Recursive/Relation"
] | [
"Definition:Characteristic Function (Set Theory)/Relation",
"Definition:Substitution (Mathematical Logic)",
"Signum Function is Primitive Recursive",
"Absolute Difference Function is Primitive Recursive",
"Definition:Primitive Recursive/Function",
"Category:Primitive Recursive Functions"
] |
proofwiki-2159 | Sylow Subgroup is Hall Subgroup | Let $G$ be a group.
Let $H$ be a Sylow $p$-subgroup of $G$.
Then $H$ is a Hall subgroup of $G$. | Let $p$ be prime.
Let $G$ be a finite group such that $\order G = k p^n$ where $p \nmid k$.
By definition, a Sylow $p$-subgroup $H$ of $G$ is a subgroup of $G$ of order $p^n$.
By Lagrange's Theorem, the index of $H$ in $G$ is given by:
:$\index G H = \dfrac {\order G} {\order H}$
So in this case:
:$\index G H = \dfrac ... | Let $G$ be a [[Definition:Group|group]].
Let $H$ be a [[Definition:Sylow p-Subgroup|Sylow $p$-subgroup]] of $G$.
Then $H$ is a [[Definition:Hall Subgroup|Hall subgroup]] of $G$. | Let $p$ be [[Definition:Prime Number|prime]].
Let $G$ be a [[Definition:Finite Group|finite group]] such that $\order G = k p^n$ where $p \nmid k$.
By definition, a [[Definition:Sylow p-Subgroup|Sylow $p$-subgroup]] $H$ of $G$ is a [[Definition:Subgroup|subgroup]] of $G$ of [[Definition:Order of Group|order]] $p^n$.
... | Sylow Subgroup is Hall Subgroup | https://proofwiki.org/wiki/Sylow_Subgroup_is_Hall_Subgroup | https://proofwiki.org/wiki/Sylow_Subgroup_is_Hall_Subgroup | [
"Sylow p-Subgroups",
"Hall Subgroups"
] | [
"Definition:Group",
"Definition:Sylow p-Subgroup",
"Definition:Hall Subgroup"
] | [
"Definition:Prime Number",
"Definition:Finite Group",
"Definition:Sylow p-Subgroup",
"Definition:Subgroup",
"Definition:Order of Structure",
"Lagrange's Theorem (Group Theory)",
"Definition:Index of Subgroup",
"Prime not Divisor implies Coprime",
"Definition:Hall Subgroup",
"Category:Sylow p-Subgr... |
proofwiki-2160 | Set Operations on Primitive Recursive Relations | Let $\RR_1 \subseteq N^k$ and $\RR_2 \subseteq N^k$ be $k$-ary relations on $N^k$.
Let $\RR_1$ and $\RR_2$ be primitive recursive.
Then the following are all primitive recursive relations:
:$\TT = \neg \RR_1$
:$\UU = \RR_1 \land \RR_2$
:$\VV = \RR_1 \lor \RR_2$ | By hypothesis, the characteristic functions $\chi_{\RR_1}, \chi_{\RR_2}$ of $\RR_1$ and $\RR_2$ are primitive recursive.
Then we have that the characteristic functions of $\TT, \UU, \VV$ are given by:
:$\chi_\TT = \map {\overline \sgn} {\chi_{\RR_1} }$
:$\chi_\UU = \chi_{\RR_1} \times \chi_{\RR_2}$
:$\chi_\VV = \map {\... | Let $\RR_1 \subseteq N^k$ and $\RR_2 \subseteq N^k$ be [[Definition:General Relation|$k$-ary relations]] on $N^k$.
Let $\RR_1$ and $\RR_2$ be [[Definition:Primitive Recursive Relation|primitive recursive]].
Then the following are all [[Definition:Primitive Recursive Relation|primitive recursive relations]]:
:$\TT = \... | [[Definition:By Hypothesis|By hypothesis]], the [[Definition:Characteristic Function of Relation|characteristic functions]] $\chi_{\RR_1}, \chi_{\RR_2}$ of $\RR_1$ and $\RR_2$ are [[Definition:Primitive Recursive Function|primitive recursive]].
Then we have that the [[Definition:Characteristic Function of Relation|cha... | Set Operations on Primitive Recursive Relations | https://proofwiki.org/wiki/Set_Operations_on_Primitive_Recursive_Relations | https://proofwiki.org/wiki/Set_Operations_on_Primitive_Recursive_Relations | [
"Primitive Recursive Functions"
] | [
"Definition:Relation/General Definition",
"Definition:Primitive Recursive/Relation",
"Definition:Primitive Recursive/Relation"
] | [
"Definition:By Hypothesis",
"Definition:Characteristic Function (Set Theory)/Relation",
"Definition:Primitive Recursive/Function",
"Definition:Characteristic Function (Set Theory)/Relation",
"Complement of Primitive Recursive Set",
"Intersection of Primitive Recursive Sets",
"Union of Primitive Recursiv... |
proofwiki-2161 | Ordering Relations are Primitive Recursive | The ordering relations on $\N^2$:
* $n < m$
* $n \le m$
* $n \ge m$
* $n > m$
are all primitive recursive. | We note that:
: $n < m \iff m \mathop{\dot -} n > 0$
: $n \ge m \iff m \mathop{\dot -} n = 0$
So it can be seen that the characteristic function of $<$ is given by:
:$\map {\chi_<} {n, m} = \map \sgn {m \mathop{\dot -} n}$
So $\chi_<$ is defined by substitution from the signum and the partial subtraction.
From Signum F... | The [[Definition:Ordering|ordering relations]] on $\N^2$:
* $n < m$
* $n \le m$
* $n \ge m$
* $n > m$
are all [[Definition:Primitive Recursive Relation|primitive recursive]]. | We note that:
: $n < m \iff m \mathop{\dot -} n > 0$
: $n \ge m \iff m \mathop{\dot -} n = 0$
So it can be seen that the [[Definition:Characteristic Function of Relation|characteristic function]] of $<$ is given by:
:$\map {\chi_<} {n, m} = \map \sgn {m \mathop{\dot -} n}$
So $\chi_<$ is defined by [[Definition:Subst... | Ordering Relations are Primitive Recursive | https://proofwiki.org/wiki/Ordering_Relations_are_Primitive_Recursive | https://proofwiki.org/wiki/Ordering_Relations_are_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Ordering",
"Definition:Primitive Recursive/Relation"
] | [
"Definition:Characteristic Function (Set Theory)/Relation",
"Definition:Substitution (Mathematical Logic)",
"Definition:Signum Function",
"Definition:Partial Subtraction",
"Signum Function is Primitive Recursive",
"Cut-Off Subtraction is Primitive Recursive",
"Definition:Primitive Recursive/Function",
... |
proofwiki-2162 | Permutation of Variables of Primitive Recursive Function | Let $f: \N^k \to \N$ be a primitive recursive function.
Let $\sigma$ be a permutation of $\set {1, 2, \ldots, k}$.
Then the function $h: \N^k \to \N$ defined as:
:$\map h {n_1, n_2, \ldots, n_k} = \map f {n_{\map \sigma 1}, n_{\map \sigma 2}, \ldots, n_{\map \sigma k} }$
is also primitive recursive. | We have that:
:$\forall j \in \set {1, 2, \ldots, k}: n_{\map \sigma j} = \pr^k_{\map \sigma j}$.
Thus $h$ is obtained by substitution from $f$ and the projection functions $\pr^k_{\map \sigma j}$.
The result follows.
{{qed}}
It follows that if a function $h$ can be obtained from known primitive recursive functions by ... | Let $f: \N^k \to \N$ be a [[Definition:Primitive Recursive Function|primitive recursive function]].
Let $\sigma$ be a [[Definition:Permutation|permutation]] of $\set {1, 2, \ldots, k}$.
Then the [[Definition:Function|function]] $h: \N^k \to \N$ defined as:
:$\map h {n_1, n_2, \ldots, n_k} = \map f {n_{\map \sigma 1}... | We have that:
:$\forall j \in \set {1, 2, \ldots, k}: n_{\map \sigma j} = \pr^k_{\map \sigma j}$.
Thus $h$ is obtained by [[Definition:Substitution (Mathematical Logic)|substitution]] from $f$ and the [[Definition:Basic Primitive Recursive Function/Projection Function|projection functions]] $\pr^k_{\map \sigma j}$.
T... | Permutation of Variables of Primitive Recursive Function | https://proofwiki.org/wiki/Permutation_of_Variables_of_Primitive_Recursive_Function | https://proofwiki.org/wiki/Permutation_of_Variables_of_Primitive_Recursive_Function | [
"Primitive Recursive Functions"
] | [
"Definition:Primitive Recursive/Function",
"Definition:Permutation",
"Definition:Function",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Substitution (Mathematical Logic)",
"Definition:Basic Primitive Recursive Function/Projection Function",
"Definition:Function",
"Definition:Primitive Recursive/Function",
"Definition:Primitive Recursion",
"Definition:Primitive Recursive/Function",
"Category:Primitive Recursive Functions"
] |
proofwiki-2163 | Definition by Cases is Primitive Recursive | Let $\RR_1, \RR_2, \ldots, \RR_k$ be primitive recursive relations on $\N^l$ such that:
:$\forall i, j \in \set{1, 2, \ldots, k}: \RR_i \implies \lnot \RR_j$, that is, all relations are mutually exclusive
:$\forall \tuple {n_1, n_2, \ldots, n_l} \in \N^l: \exists i \in \set {1, 2, \ldots, k}: \map {\RR_i} {n_1, n_2, \l... | We have:
{{begin-eqn}}
{{eqn | l = \map f {n_1, n_2, \ldots, n_l}
| r = \map {g_1} {n_1, n_2, \ldots, n_l} \times \map {\chi_{\RR_1} } {n_1, n_2, \ldots, n_l}
| c =
}}
{{eqn | o =
| ro= +
| r = \map {g_2} {n_1, n_2, \ldots, n_l} \times \map {\chi_{\RR_2} } {n_1, n_2, \ldots, n_l}
| ... | Let $\RR_1, \RR_2, \ldots, \RR_k$ be [[Definition:Primitive Recursive Relation|primitive recursive relations]] on $\N^l$ such that:
:$\forall i, j \in \set{1, 2, \ldots, k}: \RR_i \implies \lnot \RR_j$, that is, all relations are mutually exclusive
:$\forall \tuple {n_1, n_2, \ldots, n_l} \in \N^l: \exists i \in \set {... | We have:
{{begin-eqn}}
{{eqn | l = \map f {n_1, n_2, \ldots, n_l}
| r = \map {g_1} {n_1, n_2, \ldots, n_l} \times \map {\chi_{\RR_1} } {n_1, n_2, \ldots, n_l}
| c =
}}
{{eqn | o =
| ro= +
| r = \map {g_2} {n_1, n_2, \ldots, n_l} \times \map {\chi_{\RR_2} } {n_1, n_2, \ldots, n_l}
|... | Definition by Cases is Primitive Recursive | https://proofwiki.org/wiki/Definition_by_Cases_is_Primitive_Recursive | https://proofwiki.org/wiki/Definition_by_Cases_is_Primitive_Recursive | [
"Primitive Recursive Functions",
"Definition by Cases is Primitive Recursive"
] | [
"Definition:Primitive Recursive/Relation",
"Definition:Primitive Recursive/Function",
"Definition:Function",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Primitive Recursive/Relation",
"Definition:Primitive Recursive/Function",
"Definition:Substitution (Mathematical Logic)",
"Addition is Primitive Recursive",
"Definition:Primitive Recursive/Function",
"Definition:Primitive Recursive/Function",
"Category:Primitive Recursive Functions",
"Cate... |
proofwiki-2164 | Factorial is Primitive Recursive | The factorial function $\operatorname{fac}: \N \to \N$ defined as:
:$\map {\operatorname{fac} } n = n!$
is primitive recursive. | From the definition of the factorial, we have that:
:$\map {\operatorname{fac} } n = \begin{cases}
1 & : n = 0 \\
\map {\operatorname{mult} } {n, \map {\operatorname{fac} } {n - 1} } & : n > 0
\end{cases}$
Thus $\operatorname{fac}$ is obtained by primitive recursion from the constant $1$ and the primitive recursive fun... | The [[Definition:Factorial|factorial function]] $\operatorname{fac}: \N \to \N$ defined as:
:$\map {\operatorname{fac} } n = n!$
is [[Definition:Primitive Recursive Function|primitive recursive]]. | From the definition of the [[Definition:Factorial|factorial]], we have that:
:$\map {\operatorname{fac} } n = \begin{cases}
1 & : n = 0 \\
\map {\operatorname{mult} } {n, \map {\operatorname{fac} } {n - 1} } & : n > 0
\end{cases}$
Thus $\operatorname{fac}$ is obtained by [[Definition:Primitive Recursion|primitive rec... | Factorial is Primitive Recursive | https://proofwiki.org/wiki/Factorial_is_Primitive_Recursive | https://proofwiki.org/wiki/Factorial_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Factorial",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Factorial",
"Definition:Primitive Recursion",
"Constant Function is Primitive Recursive",
"Multiplication is Primitive Recursive",
"Definition:Primitive Recursive/Function",
"Category:Primitive Recursive Functions"
] |
proofwiki-2165 | Domain of Injection to Countable Set is Countable | Let $X$ be a set, and let $Y$ be a countable set.
Let $f: X \to Y$ be an injection.
Then $X$ is also countable. | Since $Y$ is countable, there exists an injection $g: Y \to \N$.
From Composite of Injections is Injection, $g \circ f: X \to \N$ is also an injection.
That is, $X$ is countable.
{{qed}}
Category:Injections
Category:Countable Sets
s54rmrwnmbz2900fs883itl41t5hk5a | Let $X$ be a [[Definition:Set|set]], and let $Y$ be a [[Definition:Countable Set|countable set]].
Let $f: X \to Y$ be an [[Definition:Injection|injection]].
Then $X$ is also [[Definition:Countable Set|countable]]. | Since $Y$ is [[Definition:Countable Set|countable]], there exists an [[Definition:Injection|injection]] $g: Y \to \N$.
From [[Composite of Injections is Injection]], $g \circ f: X \to \N$ is also an [[Definition:Injection|injection]].
That is, $X$ is [[Definition:Countable Set|countable]].
{{qed}}
[[Category:Inject... | Domain of Injection to Countable Set is Countable | https://proofwiki.org/wiki/Domain_of_Injection_to_Countable_Set_is_Countable | https://proofwiki.org/wiki/Domain_of_Injection_to_Countable_Set_is_Countable | [
"Injections",
"Countable Sets"
] | [
"Definition:Set",
"Definition:Countable Set",
"Definition:Injection",
"Definition:Countable Set"
] | [
"Definition:Countable Set",
"Definition:Injection",
"Composite of Injections is Injection",
"Definition:Injection",
"Definition:Countable Set",
"Category:Injections",
"Category:Countable Sets"
] |
proofwiki-2166 | Unique Code for URM Instruction | Each basic instruction $I$ in a URM Program can be identified with a unique '''code number''' $\beta \left({I}\right)$.
We also define the following sets:
* $\operatorname{Zinstr}$ is the set of codes of all the <tt>Zero</tt> instructions
* $\operatorname{Sinstr}$ is the set of codes of all the <tt>Successor</tt> instr... | Each basic URM instruction is of one of the following forms:
{| border="1"
|-
| <tt>Zero</tt>
| $Z \left({n}\right)$
|-
| <tt>Successor</tt>
| $S \left({n}\right)$
|-
| <tt>Copy</tt>
| $C \left({m, n}\right)$
|-
| <tt>Jump</tt>
| $J \left({m, n, q}\right)$
|}
Let $\Bbb I$ be the set of all basic URM instructions.
W... | Each [[Definition:Unlimited Register Machine|basic instruction]] $I$ in a [[Definition:URM Program|URM Program]] can be identified with a unique '''code number''' $\beta \left({I}\right)$.
We also define the following sets:
* $\operatorname{Zinstr}$ is the [[Definition:Set|set]] of codes of all the <tt>Zero</tt> inst... | Each [[Definition:Unlimited Register Machine|basic URM instruction]] is of one of the following forms:
{| border="1"
|-
| <tt>Zero</tt>
| $Z \left({n}\right)$
|-
| <tt>Successor</tt>
| $S \left({n}\right)$
|-
| <tt>Copy</tt>
| $C \left({m, n}\right)$
|-
| <tt>Jump</tt>
| $J \left({m, n, q}\right)$
|}
Let $\Bbb I... | Unique Code for URM Instruction | https://proofwiki.org/wiki/Unique_Code_for_URM_Instruction | https://proofwiki.org/wiki/Unique_Code_for_URM_Instruction | [
"URM Programs"
] | [
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine/Program",
"Definition:Set",
"Definition:Set",
"Definition:Set",
"Definition:Set",
"Definition:Set",
"Definition:Unlimited Register Machine"
] | [
"Definition:Unlimited Register Machine",
"Definition:Set",
"Definition:Unlimited Register Machine",
"Fundamental Theorem of Arithmetic",
"Definition:Unlimited Register Machine",
"Definition:Injection"
] |
proofwiki-2167 | URM Instructions are Countably Infinite | The set $\Bbb I$ of all basic URM instructions is countably infinite. | We can immediately see that $\Bbb I$ is infinite as, for example, $\phi: \N \to \Bbb I$ defined as:
:$\phi \left({n}\right) = Z \left({n}\right)$
is definitely injective.
From Unique Code for URM Instruction, we see that $\beta: \Bbb I \to \N$ is also an injection.
The result follows from Domain of Injection to Countab... | The [[Definition:Set|set]] $\Bbb I$ of all [[Definition:Unlimited Register Machine|basic URM instructions]] is [[Definition:Countable|countably infinite]]. | We can immediately see that $\Bbb I$ is [[Definition:Infinite|infinite]] as, for example, $\phi: \N \to \Bbb I$ defined as:
:$\phi \left({n}\right) = Z \left({n}\right)$
is definitely [[Definition:Injection|injective]].
From [[Unique Code for URM Instruction]], we see that $\beta: \Bbb I \to \N$ is also an [[Definiti... | URM Instructions are Countably Infinite | https://proofwiki.org/wiki/URM_Instructions_are_Countably_Infinite | https://proofwiki.org/wiki/URM_Instructions_are_Countably_Infinite | [
"URM Programs",
"Countable Sets"
] | [
"Definition:Set",
"Definition:Unlimited Register Machine",
"Definition:Countable Set"
] | [
"Definition:Infinite",
"Definition:Injection",
"Unique Code for URM Instruction",
"Definition:Injection",
"Domain of Injection to Countable Set is Countable",
"Category:URM Programs",
"Category:Countable Sets"
] |
proofwiki-2168 | URM Programs are Countably Infinite | The set $\mathbf P$ of all URM programs is countably infinite. | We can immediately see that $\mathbf P$ is infinite as the number of URM instructions is infinite.
From Unique Code for URM Program, we see that $\gamma: \mathbf P \to \N$ is also an injection.
The result follows from Domain of Injection to Countable Set is Countable.
{{qed}}
Category:URM Programs
Category:Countable Se... | The [[Definition:Set|set]] $\mathbf P$ of all [[Definition:URM Program|URM programs]] is [[Definition:Countable|countably infinite]]. | We can immediately see that $\mathbf P$ is [[Definition:Infinite|infinite]] as the number of [[URM Instructions are Countably Infinite|URM instructions is infinite]].
From [[Unique Code for URM Program]], we see that $\gamma: \mathbf P \to \N$ is also an [[Definition:Injection|injection]].
The result follows from [[... | URM Programs are Countably Infinite | https://proofwiki.org/wiki/URM_Programs_are_Countably_Infinite | https://proofwiki.org/wiki/URM_Programs_are_Countably_Infinite | [
"URM Programs",
"Countable Sets"
] | [
"Definition:Set",
"Definition:Unlimited Register Machine/Program",
"Definition:Countable Set"
] | [
"Definition:Infinite",
"URM Instructions are Countably Infinite",
"Unique Code for URM Program",
"Definition:Injection",
"Domain of Injection to Countable Set is Countable",
"Category:URM Programs",
"Category:Countable Sets"
] |
proofwiki-2169 | Unique Code for URM Program | Any URM program can be assigned a unique '''code number'''. | Let $\mathbf P$ be the set of all URM programs.
Let $P \in \mathbf P$ be a URM program with $k$ basic instructions:
{|
|-
! align="right" | Line !!
! align="left" | Command !!
|-
| align="right" | $1$ ||
| align="left" | $I_1$ ||
|-
| align="right" | $2$ ||
| align="left" | $I_2$ ||
|-
| align="right" | $\vdots$ ||
| a... | Any [[Definition:URM Program|URM program]] can be assigned a unique '''code number'''. | Let $\mathbf P$ be the [[Definition:Set|set]] of all [[Definition:URM Program|URM programs]].
Let $P \in \mathbf P$ be a [[Definition:URM Program|URM program]] with $k$ [[Definition:Unlimited Register Machine#Basic Instruction|basic instructions]]:
{|
|-
! align="right" | Line !!
! align="left" | Command !!
|-
| alig... | Unique Code for URM Program | https://proofwiki.org/wiki/Unique_Code_for_URM_Program | https://proofwiki.org/wiki/Unique_Code_for_URM_Program | [
"URM Programs"
] | [
"Definition:Unlimited Register Machine/Program"
] | [
"Definition:Set",
"Definition:Unlimited Register Machine/Program",
"Definition:Unlimited Register Machine/Program",
"Definition:Unlimited Register Machine",
"Definition:Prime Number",
"Unique Code for URM Instruction",
"Fundamental Theorem of Arithmetic",
"Definition:Unlimited Register Machine/Program... |
proofwiki-2170 | URM Computable Functions of One Variable is Countably Infinite | The set $\mathbf U$ of all URM computable functions of $1$ variable is countably infinite. | Let $\mathbf U$ be the set of all URM computable functions.
For each $f \in \mathbf U$, let $P_f$ be a URM program which computes $f$.
Such a program is very probably not unique, so in order to be definite about it, we can pick $P_f$ to be the URM program with the smallest code $\gamma \left({P_f}\right)$.
This is poss... | The [[Definition:Set|set]] $\mathbf U$ of all [[Definition:URM Computability|URM computable functions]] of $1$ variable is [[Definition:Countable|countably infinite]]. | Let $\mathbf U$ be the [[Definition:Set|set]] of all [[Definition:URM Computability|URM computable functions]].
For each $f \in \mathbf U$, let $P_f$ be a [[Definition:URM Program|URM program]] which computes $f$.
Such a program is very probably not unique, so in order to be definite about it, we can pick $P_f$ to be... | URM Computable Functions of One Variable is Countably Infinite | https://proofwiki.org/wiki/URM_Computable_Functions_of_One_Variable_is_Countably_Infinite | https://proofwiki.org/wiki/URM_Computable_Functions_of_One_Variable_is_Countably_Infinite | [
"URM Programs",
"Countable Sets"
] | [
"Definition:Set",
"Definition:URM Computability",
"Definition:Countable Set"
] | [
"Definition:Set",
"Definition:URM Computability",
"Definition:Unlimited Register Machine/Program",
"Definition:Unlimited Register Machine/Program",
"Unique Code for URM Program",
"Well-Ordering Principle",
"Definition:Function",
"Definition:Unlimited Register Machine/Program",
"Definition:Injection"... |
proofwiki-2171 | Natural Number Functions are Uncountable | The set of all natural number one-variable functions $\set {f: \N \to \N}$ is uncountably infinite. | Let $\Bbb F$ be the set of all functions from $\N$ to $\N$.
Clearly $\Bbb F$ is infinite because it contains for each $k \in \N$ the constant function $f_k: \N \to \N: \map {f_k} n = k$ and these are all different and (trivially) countably infinite in number.
Let $\Phi: \N \to \Bbb F$ be a function.
For each $n \in \N$... | The [[Definition:Set|set]] of all [[Definition:Natural Numbers|natural number]] one-variable [[Definition:Function|functions]] $\set {f: \N \to \N}$ is [[Definition:Uncountable Set|uncountably infinite]]. | Let $\Bbb F$ be the [[Definition:Set|set]] of all [[Definition:Function|functions]] from $\N$ to $\N$.
Clearly $\Bbb F$ is [[Definition:Infinite Set|infinite]] because it contains for each $k \in \N$ the [[Definition:Constant Mapping|constant function]] $f_k: \N \to \N: \map {f_k} n = k$ and these are all different an... | Natural Number Functions are Uncountable | https://proofwiki.org/wiki/Natural_Number_Functions_are_Uncountable | https://proofwiki.org/wiki/Natural_Number_Functions_are_Uncountable | [
"Mapping Theory",
"Uncountable Sets"
] | [
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Function",
"Definition:Uncountable/Set"
] | [
"Definition:Set",
"Definition:Function",
"Definition:Infinite Set",
"Definition:Constant Mapping",
"Definition:Countably Infinite/Set",
"Definition:Function",
"Definition:Function",
"Definition:Surjection",
"Definition:Bijection",
"Definition:Bijection",
"Definition:Set Equivalence",
"Countabl... |
proofwiki-2172 | Not All Natural Number Functions are Primitive Recursive | Not all functions $f: \N \to \N$ are primitive recursive. | All primitive recursive functions are URM computable.
The set of $\mathbf U$ of URM programs is countably infinite.
The set of $\Bbb F$ of natural number functions is uncountably infinite.
Hence there is no surjection from $\mathbf U \to \Bbb F$.
Hence $\mathbf U \subsetneq \Bbb F$.
Hence $\exists f \in \Bbb F: f \noti... | Not all [[Definition:Function|functions]] $f: \N \to \N$ are [[Definition:Primitive Recursive Function|primitive recursive]]. | All [[Primitive Recursive Function is URM Computable|primitive recursive functions are URM computable]].
The set of $\mathbf U$ of [[URM Programs are Countably Infinite|URM programs is countably infinite]].
The set of $\Bbb F$ of [[Natural Number Functions are Uncountable|natural number functions is uncountably infin... | Not All Natural Number Functions are Primitive Recursive | https://proofwiki.org/wiki/Not_All_Natural_Number_Functions_are_Primitive_Recursive | https://proofwiki.org/wiki/Not_All_Natural_Number_Functions_are_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Function",
"Definition:Primitive Recursive/Function"
] | [
"Primitive Recursive Function is URM Computable",
"URM Programs are Countably Infinite",
"Natural Number Functions are Uncountable",
"Definition:Surjection",
"Category:Primitive Recursive Functions"
] |
proofwiki-2173 | Bounded Summation is Primitive Recursive | Let the function $f: \N^{k + 1} \to \N$ be primitive recursive.
Then so is the function $g: \N^{k + 1} \to \N$ defined as:
:<nowiki>$\ds \map g {n_1, n_2, \ldots, n_k, z} = \begin{cases}
0 & : z = 0 \\
\ds \sum_{y \mathop = 1}^z \map f {n_1, n_2, \ldots, n_k, y} & : z > 0
\end{cases}$</nowiki> | The function $g$ satisfies:
:$\map g {n_1, n_2, \ldots, n_k, 0} = 0$
:$\map g {n_1, n_2, \ldots, n_k, z + 1} = \map g {n_1, n_2, \ldots, n_k, z} + \map f {n_1, n_2, \ldots, n_k, z + 1}$.
Hence $g$ is defined by primitive recursion from:
:the primitive recursive function $\Add$
:$f$, which is primitive recursive
:consta... | Let the [[Definition:Function|function]] $f: \N^{k + 1} \to \N$ be [[Definition:Primitive Recursive Function|primitive recursive]].
Then so is the function $g: \N^{k + 1} \to \N$ defined as:
:<nowiki>$\ds \map g {n_1, n_2, \ldots, n_k, z} = \begin{cases}
0 & : z = 0 \\
\ds \sum_{y \mathop = 1}^z \map f {n_1, n_2, \ldo... | The function $g$ satisfies:
:$\map g {n_1, n_2, \ldots, n_k, 0} = 0$
:$\map g {n_1, n_2, \ldots, n_k, z + 1} = \map g {n_1, n_2, \ldots, n_k, z} + \map f {n_1, n_2, \ldots, n_k, z + 1}$.
Hence $g$ is defined by [[Definition:Primitive Recursion|primitive recursion]] from:
:the [[Addition is Primitive Recursive|primitiv... | Bounded Summation is Primitive Recursive | https://proofwiki.org/wiki/Bounded_Summation_is_Primitive_Recursive | https://proofwiki.org/wiki/Bounded_Summation_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Function",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Primitive Recursion",
"Addition is Primitive Recursive",
"Definition:Primitive Recursive/Function",
"Constant Function is Primitive Recursive"
] |
proofwiki-2174 | Bounded Product is Primitive Recursive | Let the function $f: \N^{k + 1} \to \N$ be primitive recursive.
Then so is the function $g: \N^{k + 1} \to \N$ defined as:
:<nowiki>$\map g {n_1, n_2, \ldots, n_k, z} = \begin{cases}
1 & : z = 0 \\
\ds \prod_{y \mathop = 1}^z \map f {n_1, n_2, \ldots, n_k, y} & : z > 0
\end{cases}$</nowiki> | The function $g$ satisfies:
:$\map g {n_1, n_2, \ldots, n_k, z} = 0$
:$\map g {n_1, n_2, \ldots, n_k, z + 1} = \map g {n_1, n_2, \ldots, n_k, z} \times \map f {n_1, n_2, \ldots, n_k, z + 1}$
Hence $g$ is defined by primitive recursion from:
:the primitive recursive function $\Add$
:$f$, which is primitive recursive
:co... | Let the [[Definition:Function|function]] $f: \N^{k + 1} \to \N$ be [[Definition:Primitive Recursive Function|primitive recursive]].
Then so is the function $g: \N^{k + 1} \to \N$ defined as:
:<nowiki>$\map g {n_1, n_2, \ldots, n_k, z} = \begin{cases}
1 & : z = 0 \\
\ds \prod_{y \mathop = 1}^z \map f {n_1, n_2, \ldots,... | The function $g$ satisfies:
:$\map g {n_1, n_2, \ldots, n_k, z} = 0$
:$\map g {n_1, n_2, \ldots, n_k, z + 1} = \map g {n_1, n_2, \ldots, n_k, z} \times \map f {n_1, n_2, \ldots, n_k, z + 1}$
Hence $g$ is defined by [[Definition:Primitive Recursion|primitive recursion]] from:
:the [[Addition is Primitive Recursive|prim... | Bounded Product is Primitive Recursive | https://proofwiki.org/wiki/Bounded_Product_is_Primitive_Recursive | https://proofwiki.org/wiki/Bounded_Product_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Function",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Primitive Recursion",
"Addition is Primitive Recursive",
"Definition:Primitive Recursive/Function",
"Constant Function is Primitive Recursive"
] |
proofwiki-2175 | Divisor Relation is Primitive Recursive | The divisor relation $m \divides n$ in $\N^2$ is primitive recursive. | We note that $m \divides n \iff n = q m$ where $q \in \Z$.
So we see that $m \divides n \iff \map \rem {n, m} = 0$ (see Remainder is Primitive Recursive).
Thus we define the function $\operatorname{div}: \N^2 \to \N$ as:
:$\map {\operatorname {div} } {n, m} = \map {\chi_{\operatorname {eq} } } {\map \rem {n, m}, 0}$
wh... | The [[Definition:Divisor of Integer|divisor relation]] $m \divides n$ in $\N^2$ is [[Definition:Primitive Recursive Relation|primitive recursive]]. | We note that $m \divides n \iff n = q m$ where $q \in \Z$.
So we see that $m \divides n \iff \map \rem {n, m} = 0$ (see [[Remainder is Primitive Recursive]]).
Thus we define the [[Definition:Function|function]] $\operatorname{div}: \N^2 \to \N$ as:
:$\map {\operatorname {div} } {n, m} = \map {\chi_{\operatorname {eq}... | Divisor Relation is Primitive Recursive | https://proofwiki.org/wiki/Divisor_Relation_is_Primitive_Recursive | https://proofwiki.org/wiki/Divisor_Relation_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Primitive Recursive/Relation"
] | [
"Remainder is Primitive Recursive",
"Definition:Function",
"Definition:Characteristic Function (Set Theory)/Relation",
"Definition:Substitution (Mathematical Logic)",
"Remainder is Primitive Recursive",
"Equality Relation is Primitive Recursive",
"Constant Function is Primitive Recursive",
"Definition... |
proofwiki-2176 | Divisor Count Function is Primitive Recursive | The divisor count function is primitive recursive. | The divisor count function $\sigma_0: \N \to \N$ is defined as:
:$\ds \map {\sigma_0} n = \sum_{d \mathop \divides n} 1$
where $\ds \sum_{d \mathop \divides n}$ is the sum over all divisors of $n$.
Thus we can define $\map {\sigma_0} n$ as:
:$\ds \map {\sigma_0} n = \sum_{y \mathop = 1}^n \map {\operatorname {div} } {n... | The [[Definition:Divisor Count Function|divisor count function]] is [[Definition:Primitive Recursive Function|primitive recursive]]. | The [[Definition:Divisor Count Function|divisor count function]] $\sigma_0: \N \to \N$ is defined as:
:$\ds \map {\sigma_0} n = \sum_{d \mathop \divides n} 1$
where $\ds \sum_{d \mathop \divides n}$ is the [[Definition:Sum Over Divisors|sum over all divisors of $n$]].
Thus we can define $\map {\sigma_0} n$ as:
:$\ds \... | Divisor Count Function is Primitive Recursive | https://proofwiki.org/wiki/Divisor_Count_Function_is_Primitive_Recursive | https://proofwiki.org/wiki/Divisor_Count_Function_is_Primitive_Recursive | [
"Primitive Recursive Functions",
"Divisor Count Function"
] | [
"Definition:Divisor Count Function",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Divisor Count Function",
"Definition:Sum Over Divisors",
"Definition:Substitution (Mathematical Logic)",
"Divisor Relation is Primitive Recursive",
"Bounded Summation is Primitive Recursive",
"Category:Primitive Recursive Functions",
"Category:Divisor Count Function"
] |
proofwiki-2177 | Set of Prime Numbers is Primitive Recursive | The set $\Bbb P$ of prime numbers is primitive recursive. | A prime number is defined as an element of $\N$ with '''exactly two''' positive divisors.
So, we have that $n > 0$ is prime {{iff}} $\map \tau n = 2$, where $\tau: \N \to \N$ is the divisor count function.
Thus we can define the characteristic function of the set of prime numbers $\Bbb P$ as:
:$\forall n > 0: \map {\ch... | The set $\Bbb P$ of [[Definition:Prime Number|prime numbers]] is [[Definition:Primitive Recursive Set|primitive recursive]]. | A [[Definition:Prime Number|prime number]] is defined as an element of $\N$ with '''exactly two''' positive [[Definition:Divisor of Integer|divisors]].
So, we have that $n > 0$ is [[Definition:Prime Number|prime]] {{iff}} $\map \tau n = 2$, where $\tau: \N \to \N$ is the [[Definition:Divisor Count Function|divisor co... | Set of Prime Numbers is Primitive Recursive | https://proofwiki.org/wiki/Set_of_Prime_Numbers_is_Primitive_Recursive | https://proofwiki.org/wiki/Set_of_Prime_Numbers_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Prime Number",
"Definition:Primitive Recursive/Set"
] | [
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Prime Number",
"Definition:Divisor Count Function",
"Definition:Characteristic Function (Set Theory)/Set",
"Definition:Prime Number",
"Definition:Function",
"Divisor Relation is Primitive Recursive",
"Bounded Summation is... |
proofwiki-2178 | Prime Enumeration Function is Primitive Recursive | Let the function $p: \N \to \N$ be the prime enumeration function, defined as:
:$\map p n = \begin{cases}
1 & : n = 0 \\
\mbox{the } n \mbox{th prime number} & : n > 0
\end{cases}$
Then $p$ is primitive recursive. | We can define $p$ recursively by:
:$\map p {n + 1} = \text{the smallest } y \in \N \text { such that } y \text { is prime and } \map p n < y$
Hence we can express it as:
:$\map p {n + 1} = \map {\mu y} {\map {\chi_\Bbb P} y = 1 \land \map p n < y}$
where:
* $\map {\chi_\Bbb P} y$ is the characteristic function of the s... | Let the [[Definition:Function|function]] $p: \N \to \N$ be the [[Definition:Prime Enumeration Function|prime enumeration function]], defined as:
:$\map p n = \begin{cases}
1 & : n = 0 \\
\mbox{the } n \mbox{th prime number} & : n > 0
\end{cases}$
Then $p$ is [[Definition:Primitive Recursive Function|primitive recursi... | We can define $p$ recursively by:
:$\map p {n + 1} = \text{the smallest } y \in \N \text { such that } y \text { is prime and } \map p n < y$
Hence we can express it as:
:$\map p {n + 1} = \map {\mu y} {\map {\chi_\Bbb P} y = 1 \land \map p n < y}$
where:
* $\map {\chi_\Bbb P} y$ is the [[Definition:Characteristic Fun... | Prime Enumeration Function is Primitive Recursive | https://proofwiki.org/wiki/Prime_Enumeration_Function_is_Primitive_Recursive | https://proofwiki.org/wiki/Prime_Enumeration_Function_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Function",
"Definition:Prime Enumeration Function",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Characteristic Function (Set Theory)/Set",
"Definition:Set",
"Definition:Prime Number",
"Definition:Relation",
"Definition:Relation",
"Definition:Primitive Recursive/Function",
"Definition:Substitution (Mathematical Logic)",
"Equality Relation is Primitive Recursive",
"Set of Prime Numbe... |
proofwiki-2179 | Length Function is Primitive Recursive | Let $n \in \N$.
Let $\map \len n$ denote the length of $n$.
Then the function $\len: \N \to \N$ is primitive recursive. | Clearly $\map \len 0 = 0$.
For $n > 0$, we have:
:$\ds \map \len n = \sum_{y \mathop = 1}^n \map {\operatorname {div} } {n, \map p y}$
where:
:$\map {\operatorname {div} } {n, m}$ is defined as:
::$\map {\operatorname {div} } {n, y} = \begin{cases}
1 & : y \divides n \\
0 & : y \nmid n
\end{cases}$
:$\map p y$ is the $... | Let $n \in \N$.
Let $\map \len n$ denote the [[Definition:Length of an Integer|length]] of $n$.
Then the [[Definition:Function|function]] $\len: \N \to \N$ is [[Definition:Primitive Recursive Function|primitive recursive]]. | Clearly $\map \len 0 = 0$.
For $n > 0$, we have:
:$\ds \map \len n = \sum_{y \mathop = 1}^n \map {\operatorname {div} } {n, \map p y}$
where:
:$\map {\operatorname {div} } {n, m}$ is defined as:
::$\map {\operatorname {div} } {n, y} = \begin{cases}
1 & : y \divides n \\
0 & : y \nmid n
\end{cases}$
:$\map p y$ is the ... | Length Function is Primitive Recursive | https://proofwiki.org/wiki/Length_Function_is_Primitive_Recursive | https://proofwiki.org/wiki/Length_Function_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Length of Integer",
"Definition:Function",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Prime Number",
"Definition:Function",
"Divisor Relation is Primitive Recursive",
"Prime Enumeration Function is Primitive Recursive",
"Bounded Summation is Primitive Recursive",
"Definition:Primitive Recursive/Function",
"Definition:Primitive Recursive/Function",
"Category:Primitive Recurs... |
proofwiki-2180 | Prime Exponent Function is Primitive Recursive | Let $n \in \N$ be a natural number.
Let $\tuple {n, j}: \N^2 \to \N$ be defined as:
:$\tuple {n, j} = \paren n_j$
where $\paren n_j$ is the prime exponent function.
Then $\tuple {n, j}$ is primitive recursive. | Let $\map p j$ be the prime enumeration function.
For $n \ne 0$ and $j \ne 0$, we see that $\paren n_j$ is the largest value of $k$ for which $\map p j^k$ is a divisor of $n$.
Thus $\paren n_j$ is the ''smallest'' value of $k$ for which $\map p j^{k + 1}$ is ''not'' a divisor of $n$.
We note that if $r \ge n$ and $j \n... | Let $n \in \N$ be a [[Definition:Natural Numbers|natural number]].
Let $\tuple {n, j}: \N^2 \to \N$ be defined as:
:$\tuple {n, j} = \paren n_j$
where $\paren n_j$ is the [[Definition:Prime Exponent Function|prime exponent function]].
Then $\tuple {n, j}$ is [[Definition:Primitive Recursive Function|primitive recursi... | Let $\map p j$ be the [[Definition:Prime Enumeration Function|prime enumeration function]].
For $n \ne 0$ and $j \ne 0$, we see that $\paren n_j$ is the largest value of $k$ for which $\map p j^k$ is a [[Definition:Divisor of Integer|divisor]] of $n$.
Thus $\paren n_j$ is the ''smallest'' value of $k$ for which $\map... | Prime Exponent Function is Primitive Recursive | https://proofwiki.org/wiki/Prime_Exponent_Function_is_Primitive_Recursive | https://proofwiki.org/wiki/Prime_Exponent_Function_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Natural Numbers",
"Definition:Prime Exponent Function",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Prime Enumeration Function",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Upper Bound of Mapping",
"Definition:Divisor (Algebra)/Integer",
"Divisor Relation is Primitive Recursive",
"Equality Relation is Primitive Recursive",
"Prime Enumeratio... |
proofwiki-2181 | Set of Codes for URM Instructions is Primitive Recursive | The set $\operatorname{Instr}$ of codes of all basic URM instructions is primitive recursive. | Since the Union of Primitive Recursive Sets is itself primitive recursive, all we need to do is show that each of $\operatorname{Zinstr}$, $\operatorname{Sinstr}$, $\operatorname{Cinstr}$ and $\operatorname{Jinstr}$ are primitive recursive.
First we consider $\operatorname{Zinstr}$.
:$\operatorname{Zinstr} = \left\{{\b... | The set $\operatorname{Instr}$ of [[Unique Code for URM Instruction|codes of all basic URM instructions]] is [[Definition:Primitive Recursive Set|primitive recursive]]. | Since the [[Union of Primitive Recursive Sets]] is itself [[Definition:Primitive Recursive Set|primitive recursive]], all we need to do is show that each of $\operatorname{Zinstr}$, $\operatorname{Sinstr}$, $\operatorname{Cinstr}$ and $\operatorname{Jinstr}$ are [[Definition:Primitive Recursive Set|primitive recursive]... | Set of Codes for URM Instructions is Primitive Recursive | https://proofwiki.org/wiki/Set_of_Codes_for_URM_Instructions_is_Primitive_Recursive | https://proofwiki.org/wiki/Set_of_Codes_for_URM_Instructions_is_Primitive_Recursive | [
"URM Programs",
"Primitive Recursive Functions"
] | [
"Unique Code for URM Instruction",
"Definition:Primitive Recursive/Set"
] | [
"Union of Primitive Recursive Sets",
"Definition:Primitive Recursive/Set",
"Definition:Primitive Recursive/Set",
"Definition:Natural Numbers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Characteristic Function (Set Theory)/Set",
"Divisor Relation is Primitive Recursive",
"Signum Function is P... |
proofwiki-2182 | Set of Sequence Codes is Primitive Recursive | Let $\operatorname{Seq}$ be the set of all code numbers of finite sequences in $\N$.
Then $\operatorname{Seq}$ is primitive recursive. | By the definition of a primitive recursive set, it is sufficient to show that the characteristic function $\chi_{\operatorname{Seq}}$ of $\operatorname{Seq}$ is primitive recursive.
Let $p: \N \to \N$ be the prime enumeration function.
Let $\map {\operatorname{len} } n$ be the length of $n$.
We note that $\map {\chi_{\... | Let $\operatorname{Seq}$ be the [[Definition:Set|set]] of all [[Definition:Sequence Coding|code numbers]] of [[Definition:Finite Sequence|finite sequences]] in $\N$.
Then $\operatorname{Seq}$ is [[Definition:Primitive Recursive Set|primitive recursive]]. | By the definition of a [[Definition:Primitive Recursive Set|primitive recursive set]], it is sufficient to show that the [[Definition:Characteristic Function of Set|characteristic function]] $\chi_{\operatorname{Seq}}$ of $\operatorname{Seq}$ is [[Definition:Primitive Recursive Function|primitive recursive]].
Let $p: ... | Set of Sequence Codes is Primitive Recursive | https://proofwiki.org/wiki/Set_of_Sequence_Codes_is_Primitive_Recursive | https://proofwiki.org/wiki/Set_of_Sequence_Codes_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Set",
"Definition:Sequence Coding",
"Definition:Finite Sequence",
"Definition:Primitive Recursive/Set"
] | [
"Definition:Primitive Recursive/Set",
"Definition:Characteristic Function (Set Theory)/Set",
"Definition:Primitive Recursive/Function",
"Definition:Prime Enumeration Function",
"Definition:Length of Integer",
"Definition:Divisor (Algebra)/Integer",
"Divisor Relation is Primitive Recursive",
"Bounded P... |
proofwiki-2183 | Set of Codes for URM Programs is Primitive Recursive | Let $\operatorname{Prog}$ be the set of all code numbers of URM programs.
Then $\operatorname{Prog}$ is a primitive recursive set. | A natural number $n$ codes a URM program {{iff}} it codes a sequence of positive integers which are the code numbers of URM instructions.
Suppose $n$ codes such a sequence.
Then $\map \len n$ is the number of terms in this sequence, where $\map \len n$ is the length of $n$.
Also, for $1 \le j \le \map \len n$, $\paren ... | Let $\operatorname{Prog}$ be the [[Definition:Set|set]] of all [[Unique Code for URM Program|code numbers of URM programs]].
Then $\operatorname{Prog}$ is a [[Definition:Primitive Recursive Set|primitive recursive set]]. | A [[Definition:Natural Numbers|natural number]] $n$ codes a [[Definition:URM Program|URM program]] {{iff}} it [[Definition:Sequence Coding|codes a sequence]] of [[Definition:Positive Integer|positive integers]] which are the [[Unique Code for URM Instruction|code numbers of URM instructions]].
Suppose $n$ codes such a... | Set of Codes for URM Programs is Primitive Recursive | https://proofwiki.org/wiki/Set_of_Codes_for_URM_Programs_is_Primitive_Recursive | https://proofwiki.org/wiki/Set_of_Codes_for_URM_Programs_is_Primitive_Recursive | [
"URM Programs",
"Primitive Recursive Functions"
] | [
"Definition:Set",
"Unique Code for URM Program",
"Definition:Primitive Recursive/Set"
] | [
"Definition:Natural Numbers",
"Definition:Unlimited Register Machine/Program",
"Definition:Sequence Coding",
"Definition:Positive/Integer",
"Unique Code for URM Instruction",
"Definition:Length of Integer",
"Definition:Prime Exponent Function",
"Definition:Prime Decomposition",
"Definition:Sequence ... |
proofwiki-2184 | Minimization on Relation Equivalent to Minimization on Function | Let $\RR$ be a $k + 1$-ary relation on $\N^{k + 1}$.
Then the function $g: \N^{k + 1} \to \N$ defined as:
:$\map g {n_1, n_2, \ldots, n_k, z} = \mu y \ \map \RR {n_1, n_2, \ldots, n_k, y}$
where $\mu y \ \map \RR {n_1, n_2, \ldots, n_k, y}$ is the minimization operation on $\RR$
is equivalent to minimization on a total... | We have that $\map \RR {n_1, n_2, \ldots, n_k, y}$ holds {{iff}} $\map {\chi_\RR} {n_1, n_2, \ldots, n_k, y} = 1$, from the definition of the characteristic function of a relation.
This in turn holds {{iff}} $\map {\overline \sgn} {\map {\chi_\RR} {n_1, n_2, \ldots, n_k, y} } = 0$, where $\overline \sgn$ is the signum ... | Let $\RR$ be a [[Definition:Relation|$k + 1$-ary relation]] on $\N^{k + 1}$.
Then the [[Definition:Function|function]] $g: \N^{k + 1} \to \N$ defined as:
:$\map g {n_1, n_2, \ldots, n_k, z} = \mu y \ \map \RR {n_1, n_2, \ldots, n_k, y}$
where $\mu y \ \map \RR {n_1, n_2, \ldots, n_k, y}$ is the [[Definition:Minimizati... | We have that $\map \RR {n_1, n_2, \ldots, n_k, y}$ holds {{iff}} $\map {\chi_\RR} {n_1, n_2, \ldots, n_k, y} = 1$, from the definition of the [[Definition:Characteristic Function of Relation|characteristic function of a relation]].
This in turn holds {{iff}} $\map {\overline \sgn} {\map {\chi_\RR} {n_1, n_2, \ldots, n... | Minimization on Relation Equivalent to Minimization on Function | https://proofwiki.org/wiki/Minimization_on_Relation_Equivalent_to_Minimization_on_Function | https://proofwiki.org/wiki/Minimization_on_Relation_Equivalent_to_Minimization_on_Function | [
"Primitive Recursive Functions",
"Recursive Functions"
] | [
"Definition:Relation",
"Definition:Function",
"Definition:Minimization/Relation",
"Definition:Minimization/Relation",
"Definition:Function"
] | [
"Definition:Characteristic Function (Set Theory)/Relation",
"Definition:Signum Function/Signum Complement",
"Definition:Total Function",
"Category:Primitive Recursive Functions",
"Category:Recursive Functions"
] |
proofwiki-2185 | Primitive Recursive Function is Total Recursive Function | Every primitive recursive function is a total recursive function. | A primitive recursive function is a total function, which is apparent from its method of definition.
As the processes for generate a primitive recursive function are a subset of those to generate a recursive function, it follows that a primitive recursive function is also a recursive function.
The result follows from t... | Every [[Definition:Primitive Recursive Function|primitive recursive function]] is a [[Definition:Total Recursive Function|total recursive function]]. | A [[Definition:Primitive Recursive Function|primitive recursive function]] is a [[Definition:Total Function|total function]], which is apparent from its method of definition.
As the processes for generate a [[Definition:Primitive Recursive Function|primitive recursive function]] are a subset of those to generate a [[D... | Primitive Recursive Function is Total Recursive Function | https://proofwiki.org/wiki/Primitive_Recursive_Function_is_Total_Recursive_Function | https://proofwiki.org/wiki/Primitive_Recursive_Function_is_Total_Recursive_Function | [
"Primitive Recursive Functions",
"Recursive Functions"
] | [
"Definition:Primitive Recursive/Function",
"Definition:Total Recursive Function"
] | [
"Definition:Primitive Recursive/Function",
"Definition:Total Function",
"Definition:Primitive Recursive/Function",
"Definition:Recursive/Function",
"Definition:Primitive Recursive/Function",
"Definition:Recursive/Function",
"Definition:Total Recursive Function",
"Category:Primitive Recursive Functions... |
proofwiki-2186 | Function Obtained by Minimization from URM Computable Functions | Let the function $f: \N^{k+1} \to \N$ be a URM computable function.
Let $g: \N^k \to \N$ be the function obtained by minimization from $f$ thus:
:$\map g {n_1, n_2, \ldots, n_k} \approx \map {\mu y} {\map f {n_1, n_2, \ldots, n_k, y} = 0}$
Then $g$ is also URM computable. | Let $f: \N^{k+1} \to \N$ be a URM computable function.
Let $P$ be a URM program which computes $f$.
Let $u = \map \rho P$ be the number of registers used by $P$.
Let $s = \map \lambda P$ be the length of $P$.
We can use:
* the registers $R_{u+1}, R_{u+2}, \ldots, R_{u+k}$ to store the input $\tuple {n_1, n_2, \ldots, n... | Let the [[Definition:Function|function]] $f: \N^{k+1} \to \N$ be a [[Definition:URM Computability|URM computable function]].
Let $g: \N^k \to \N$ be the [[Definition:Function|function]] obtained by [[Definition:Minimization|minimization]] from $f$ thus:
:$\map g {n_1, n_2, \ldots, n_k} \approx \map {\mu y} {\map f {n_... | Let $f: \N^{k+1} \to \N$ be a [[Definition:URM Computability|URM computable function]].
Let $P$ be a [[Definition:URM Program|URM program]] which computes $f$.
Let $u = \map \rho P$ be the [[Definition:Unlimited Register Machine#Number of Registers Used|number of registers used]] by $P$.
Let $s = \map \lambda P$ be ... | Function Obtained by Minimization from URM Computable Functions | https://proofwiki.org/wiki/Function_Obtained_by_Minimization_from_URM_Computable_Functions | https://proofwiki.org/wiki/Function_Obtained_by_Minimization_from_URM_Computable_Functions | [
"URM Programs"
] | [
"Definition:Function",
"Definition:URM Computability",
"Definition:Function",
"Definition:Minimization",
"Definition:URM Computability"
] | [
"Definition:URM Computability",
"Definition:Unlimited Register Machine/Program",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine",
"Definition:Unli... |
proofwiki-2187 | Function Obtained by Minimization from URM Computable Relations | Let $\RR$ be a URM computable $k+1$-ary relation on $\N^{k + 1}$.
Let the function $f: \N^{k + 1} \to \N$ be a URM computable function.
Let $g: \N^k \to \N$ be the function obtained by minimization from $f$ thus:
:$\map g {n_1, n_2, \ldots, n_k} \approx \mu y \map \RR {n_1, n_2, \ldots, n_k, y}$
Then $g$ is also URM co... | From Minimization on Relation Equivalent to Minimization on Function, minimization on $\RR$ is equivalent to minimization on $\overline \sgn \circ \chi_\RR$.
We have that a Primitive Recursive Function is URM Computable.
By definition, if $\RR$ is URM computable then so is its characteristic function $\chi_\RR$.
We hav... | Let $\RR$ be a [[Definition:URM Computability|URM computable]] [[Definition:Relation|$k+1$-ary relation]] on $\N^{k + 1}$.
Let the [[Definition:Function|function]] $f: \N^{k + 1} \to \N$ be a [[Definition:URM Computability|URM computable function]].
Let $g: \N^k \to \N$ be the [[Definition:Function|function]] obtaine... | From [[Minimization on Relation Equivalent to Minimization on Function]], minimization on $\RR$ is equivalent to minimization on $\overline \sgn \circ \chi_\RR$.
We have that a [[Primitive Recursive Function is URM Computable]].
By definition, if $\RR$ is [[Definition:URM Computability|URM computable]] then so is its... | Function Obtained by Minimization from URM Computable Relations | https://proofwiki.org/wiki/Function_Obtained_by_Minimization_from_URM_Computable_Relations | https://proofwiki.org/wiki/Function_Obtained_by_Minimization_from_URM_Computable_Relations | [
"URM Programs"
] | [
"Definition:URM Computability",
"Definition:Relation",
"Definition:Function",
"Definition:URM Computability",
"Definition:Function",
"Definition:Minimization",
"Definition:URM Computability"
] | [
"Minimization on Relation Equivalent to Minimization on Function",
"Primitive Recursive Function is URM Computable",
"Definition:URM Computability",
"Definition:Characteristic Function (Set Theory)/Relation",
"Signum Function is Primitive Recursive",
"Definition:URM Computability",
"Function Obtained by... |
proofwiki-2188 | Recursive Function is URM Computable | Every recursive function is URM computable. | From:
* Functions obtained by minimization from URM computable functions are URM computable
* Functions obtained by minimization from URM computable relations are URM computable
* Functions obtained by primitive recursion from URM computable functions are URM computable
* Functions obtained by substitution from URM com... | Every [[Definition:Recursive Function|recursive function]] is [[Definition:URM Computability|URM computable]]. | From:
* [[Function Obtained by Minimization from URM Computable Functions|Functions obtained by minimization from URM computable functions are URM computable]]
* [[Function Obtained by Minimization from URM Computable Relations|Functions obtained by minimization from URM computable relations are URM computable]]
* [[Fu... | Recursive Function is URM Computable | https://proofwiki.org/wiki/Recursive_Function_is_URM_Computable | https://proofwiki.org/wiki/Recursive_Function_is_URM_Computable | [
"Recursive Functions",
"URM Programs"
] | [
"Definition:Recursive/Function",
"Definition:URM Computability"
] | [
"Function Obtained by Minimization from URM Computable Functions",
"Function Obtained by Minimization from URM Computable Relations",
"Function Obtained by Primitive Recursion from URM Computable Functions",
"Function Obtained by Substitution from URM Computable Functions",
"Primitive Recursive Function is ... |
proofwiki-2189 | Unique Code for State of URM Program | Every state of a URM program can be assigned a unique '''code number'''.
This code number is called the '''state code''' (or '''situation code'''). | The state of a URM program at a particular point in time is defined as:
:the value of the instruction pointer
:the value, at that point, of each of the registers that are used by the program.
Let $P$ be a URM program.
Suppose that, at a given stage of computation:
:the value of the instruction pointer is $a$;
:the valu... | Every [[Definition:Unlimited Register Machine#State|state]] of a [[Definition:URM Program|URM program]] can be assigned a unique '''code number'''.
This code number is called the '''state code''' (or '''situation code'''). | The [[Definition:Unlimited Register Machine#State|state]] of a [[Definition:URM Program|URM program]] at a particular point in time is defined as:
:the value of the [[Definition:Unlimited Register Machine#Instruction Pointer|instruction pointer]]
:the value, at that point, of each of the [[Definition:Unlimited Register... | Unique Code for State of URM Program | https://proofwiki.org/wiki/Unique_Code_for_State_of_URM_Program | https://proofwiki.org/wiki/Unique_Code_for_State_of_URM_Program | [
"URM Programs"
] | [
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine/Program"
] | [
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine/Program",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine/Program",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine",
... |
proofwiki-2190 | State Code Function is Primitive Recursive | Let $k \in \N^*$.
Let $e = \map \gamma P$ be the code number of a URM program $P$.
Let $\tuple {n_1, n_2, \ldots, n_k}$ be the input of $P$.
Let $S_k: \N^{k + 2} \to \N$ be the function defined as:
:$\map {S_k} {e, n_1, n_2, \ldots, n_k, t}$ is the state code for $P$ at stage $t$ of computation of $P$.
If $e$ does not ... | It can easily be seen that $S_k$ is a total function.
Suppose $e = \map \gamma P$ for some URM program $P$.
At stage $0$, we are about to carry out instruction $1$ with the input $\tuple {n_1, n_2, \ldots, n_k}$.
So we have:
:$\map {S_k} {e, n_1, n_2, \ldots, n_k, 0} = \begin{cases}
2^1 3^{n_1} 5^{n_2} \cdots p_{k + 1}... | Let $k \in \N^*$.
Let $e = \map \gamma P$ be the [[Unique Code for URM Program|code number]] of a [[Definition:URM Program|URM program]] $P$.
Let $\tuple {n_1, n_2, \ldots, n_k}$ be the [[Definition:Unlimited Register Machine#Input|input]] of $P$.
Let $S_k: \N^{k + 2} \to \N$ be the [[Definition:Function|function]]... | It can easily be seen that $S_k$ is a [[Definition:Total Function|total function]].
Suppose $e = \map \gamma P$ for some [[Definition:URM Program|URM program]] $P$.
At stage $0$, we are about to carry out [[Definition:Unlimited Register Machine#Basic Instructions|instruction]] $1$ with the [[Definition:Unlimited Regi... | State Code Function is Primitive Recursive | https://proofwiki.org/wiki/State_Code_Function_is_Primitive_Recursive | https://proofwiki.org/wiki/State_Code_Function_is_Primitive_Recursive | [
"Primitive Recursive Functions",
"URM Programs"
] | [
"Unique Code for URM Program",
"Definition:Unlimited Register Machine/Program",
"Definition:Unlimited Register Machine",
"Definition:Function",
"Unique Code for State of URM Program",
"Definition:Unlimited Register Machine",
"Unique Code for URM Program",
"Definition:Unlimited Register Machine/Program... | [
"Definition:Total Function",
"Definition:Unlimited Register Machine/Program",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine/Program",
"Set of Codes for URM Programs is Primitive Recursive",
":Category:Primitive Recursive Function... |
proofwiki-2191 | URM Computable Function is Recursive | Every URM computable function is recursive. | Let $f: \N^k \to \N$ be a URM computable function.
Then by hypothesis there is a URM program that computes $f$.
Let $P$ be the URM program with the smallest code number that computes $f$.
Let $e = \map \gamma P$ be the code number of $P$.
Consider the function $g: \N^k \to \N$ given by:
:$\map g {n_1, n_2, \ldots, n_k}... | Every [[Definition:URM Computability#Function|URM computable function]] is [[Definition:Recursive Function|recursive]]. | Let $f: \N^k \to \N$ be a [[Definition:URM Computability#Function|URM computable function]].
Then by hypothesis there is a [[Definition:URM Program|URM program]] that computes $f$.
Let $P$ be the [[Definition:URM Program|URM program]] with the smallest [[Unique Code for URM Program|code number]] that computes $f$.
L... | URM Computable Function is Recursive | https://proofwiki.org/wiki/URM_Computable_Function_is_Recursive | https://proofwiki.org/wiki/URM_Computable_Function_is_Recursive | [
"Recursive Functions",
"URM Programs"
] | [
"Definition:URM Computability",
"Definition:Recursive/Function"
] | [
"Definition:URM Computability",
"Definition:Unlimited Register Machine/Program",
"Definition:Unlimited Register Machine/Program",
"Unique Code for URM Program",
"Unique Code for URM Program",
"Definition:Function",
"Definition:Length of Integer",
"Definition:Minimization",
"Definition:Partial Functi... |
proofwiki-2192 | Kleene's Normal Form Theorem | For each integer $k \ge 1$, there exists:
:a primitive recursive $k + 1$-ary relation $T_k$
:a primitive recursive function $U: \N \to \N$
such that a partial function $f: \N^k \to \N$ is recursive {{iff}}, for some $e \in \N$ and all $\tuple {n_1, n_2, \ldots, n_k} \in \N^k$:
:$\map f {n_1, n_2, \ldots, n_k} \approx \... | See the proof of URM Computable Function is Recursive for an explanation of the symbols used here. | For each [[Definition:Integer|integer]] $k \ge 1$, there exists:
:a [[Definition:Primitive Recursive Relation|primitive recursive $k + 1$-ary relation]] $T_k$
:a [[Definition:Primitive Recursive Function|primitive recursive function]] $U: \N \to \N$
such that a [[Definition:Partial Function|partial function]] $f: \N^k ... | See the proof of [[URM Computable Function is Recursive]] for an explanation of the symbols used here. | Kleene's Normal Form Theorem | https://proofwiki.org/wiki/Kleene's_Normal_Form_Theorem | https://proofwiki.org/wiki/Kleene's_Normal_Form_Theorem | [
"Recursion Theory"
] | [
"Definition:Integer",
"Definition:Primitive Recursive/Relation",
"Definition:Primitive Recursive/Function",
"Definition:Partial Function",
"Definition:Recursive/Function"
] | [
"URM Computable Function is Recursive"
] |
proofwiki-2193 | Recursive Function uses One Minimization | Every recursive function can be obtained from the basic primitive recursive functions using:
* substitution
* primitive recursion
* at most one minimization on a function. | Let $f: \N^k \to \N$ be any recursive function.
Consider the minimization operation on the $k + 2$-ary relation $\map \RR {n_1, n_2, \ldots, n_k, y}$:
:$\mu y \mathrel \RR \tuple {n_1, n_2, \ldots, n_k, y}$
From Minimization on Relation Equivalent to Minimization on Function, this is equivalent to:
:$\map {\mu y} {\map... | Every [[Definition:Recursive Function|recursive function]] can be obtained from the [[Definition:Basic Primitive Recursive Function|basic primitive recursive functions]] using:
* [[Definition:Substitution (Mathematical Logic)|substitution]]
* [[Definition:Primitive Recursion|primitive recursion]]
* at most one [[Defini... | Let $f: \N^k \to \N$ be any [[Definition:Recursive Function|recursive function]].
Consider the [[Definition:Minimization/Relation|minimization operation]] on the [[Definition:Relation|$k + 2$-ary relation]] $\map \RR {n_1, n_2, \ldots, n_k, y}$:
:$\mu y \mathrel \RR \tuple {n_1, n_2, \ldots, n_k, y}$
From [[Minimizat... | Recursive Function uses One Minimization | https://proofwiki.org/wiki/Recursive_Function_uses_One_Minimization | https://proofwiki.org/wiki/Recursive_Function_uses_One_Minimization | [
"Recursive Functions"
] | [
"Definition:Recursive/Function",
"Definition:Basic Primitive Recursive Function",
"Definition:Substitution (Mathematical Logic)",
"Definition:Primitive Recursion",
"Definition:Minimization/Function"
] | [
"Definition:Recursive/Function",
"Definition:Minimization/Relation",
"Definition:Relation",
"Minimization on Relation Equivalent to Minimization on Function",
"Kleene's Normal Form Theorem",
"Definition:Primitive Recursive/Function",
"Definition:Characteristic Function (Set Theory)/Relation",
"Signum ... |
proofwiki-2194 | Universal URM Computable Functions | For each integer $k \ge 1$, there exists a URM computable function:
:$\Phi_k: \N^{k+1} \to \N$
such that for each URM computable function $f: \N^k \to \N$ there exists a natural number $e$ such that:
:$\forall \left({n_1, n_2, \ldots, n_k}\right) \in \N^k: f \left({n_1, n_2, \ldots, n_k}\right) \approx \Phi_k \left({e,... | Let $\Phi_k: \N^{k+1} \to \N$ be given by:
:$\Phi_k \left({e, n_1, n_2, \ldots, n_k}\right) = U \left({\mu z \ T_k \left({e, n_1, n_2, \ldots, n_k, z}\right)}\right)$
where $T_k$ and $U$ are as in Kleene's Normal Form Theorem.
Thus we have reinterpreted Kleene's Normal Form Theorem as being about URM computable functio... | For each [[Definition:Integer|integer]] $k \ge 1$, there exists a [[Definition:URM Computability#Function|URM computable function]]:
:$\Phi_k: \N^{k+1} \to \N$
such that for each [[Definition:URM Computability#Function|URM computable function]] $f: \N^k \to \N$ there exists a [[Definition:Natural Numbers|natural number... | Let $\Phi_k: \N^{k+1} \to \N$ be given by:
:$\Phi_k \left({e, n_1, n_2, \ldots, n_k}\right) = U \left({\mu z \ T_k \left({e, n_1, n_2, \ldots, n_k, z}\right)}\right)$
where $T_k$ and $U$ are as in [[Kleene's Normal Form Theorem]].
Thus we have reinterpreted [[Kleene's Normal Form Theorem]] as being about [[Definition:... | Universal URM Computable Functions | https://proofwiki.org/wiki/Universal_URM_Computable_Functions | https://proofwiki.org/wiki/Universal_URM_Computable_Functions | [
"URM Programs"
] | [
"Definition:Integer",
"Definition:URM Computability",
"Definition:URM Computability",
"Definition:Natural Numbers"
] | [
"Kleene's Normal Form Theorem",
"Kleene's Normal Form Theorem",
"Definition:URM Computability",
"URM Computable Function is Recursive",
"Recursive Function is URM Computable"
] |
proofwiki-2195 | Universal URM Programs | For each integer $k \ge 1$, there exists a URM program $P_k$ such that:
For each URM program $P$ there exists a natural number $e$ such that:
For all $\left({n_1, n_2, \ldots, n_k}\right) \in \N^k$, the computation using the program $P_k$ with input $\left({e, n_1, n_2, \ldots, n_k}\right)$
has the same output as the c... | This follows directly from:
* Kleene's Normal Form Theorem;
* Universal URM Computable Functions.
{{qed}}
Category:URM Programs
557hxwrpy6pu9y2am1yzmuwcl59xj3w | For each [[Definition:Integer|integer]] $k \ge 1$, there exists a [[Definition:URM Program|URM program]] $P_k$ such that:
For each [[Definition:URM Program|URM program]] $P$ there exists a [[Definition:Natural Numbers|natural number]] $e$ such that:
For all $\left({n_1, n_2, \ldots, n_k}\right) \in \N^k$, the computa... | This follows directly from:
* [[Kleene's Normal Form Theorem]];
* [[Universal URM Computable Functions]].
{{qed}}
[[Category:URM Programs]]
557hxwrpy6pu9y2am1yzmuwcl59xj3w | Universal URM Programs | https://proofwiki.org/wiki/Universal_URM_Programs | https://proofwiki.org/wiki/Universal_URM_Programs | [
"URM Programs"
] | [
"Definition:Integer",
"Definition:Unlimited Register Machine/Program",
"Definition:Unlimited Register Machine/Program",
"Definition:Natural Numbers",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine",
"Definition:Unlimited Register Machine"
] | [
"Kleene's Normal Form Theorem",
"Universal URM Computable Functions",
"Category:URM Programs"
] |
proofwiki-2196 | Combination of Recursive Functions | Let $f: \N^k \to \N$ and $g: \N^k \to \N$ be recursive functions (not necessarily total), where $k \ge 1$.
Let $\RR$ be a $k$-ary relation such that:
:if $\map \RR {n_1, n_2, \ldots, n_k}$ holds, then $\map f {n_1, n_2, \ldots, n_k}$ is defined
:if $\map \RR {n_1, n_2, \ldots, n_k}$ does not hold, then $\map g {n_1, n_... | Let $P_f, P_g, P_\RR$ be the URM programs computing, respectively, the functions $f$ and $g$ and the characteristic function $\chi_\RR$.
From Recursive Function is URM Computable, these programs are guaranteed to exist.
An informal algorithm for computing $h$ is as follows.
# Input $\tuple {n_1, n_2, \ldots, n_k}$.
# U... | Let $f: \N^k \to \N$ and $g: \N^k \to \N$ be [[Definition:Recursive Function|recursive functions]] (not necessarily [[Definition:Total Function|total]]), where $k \ge 1$.
Let $\RR$ be a [[Definition:Relation|$k$-ary relation]] such that:
:if $\map \RR {n_1, n_2, \ldots, n_k}$ holds, then $\map f {n_1, n_2, \ldots, n_k... | Let $P_f, P_g, P_\RR$ be the [[Definition:URM Program|URM programs]] computing, respectively, the functions $f$ and $g$ and the [[Definition:Characteristic Function of Relation|characteristic function]] $\chi_\RR$.
From [[Recursive Function is URM Computable]], these programs are guaranteed to exist.
An informal algo... | Combination of Recursive Functions | https://proofwiki.org/wiki/Combination_of_Recursive_Functions | https://proofwiki.org/wiki/Combination_of_Recursive_Functions | [
"Recursive Functions"
] | [
"Definition:Recursive/Function",
"Definition:Total Function",
"Definition:Relation",
"Definition:Function",
"Definition:Total Function",
"Definition:Recursive/Function"
] | [
"Definition:Unlimited Register Machine/Program",
"Definition:Characteristic Function (Set Theory)/Relation",
"Recursive Function is URM Computable",
"URM Computable Function is Recursive"
] |
proofwiki-2197 | Not All URM Computable Functions are Primitive Recursive | There exist URM computable functions which are not primitive recursive. | Consider the basic primitive recursive functions.
To each basic primitive recursive function $f$ let us assign a code number $\map \delta f$, as follows:
* $\map \delta {\operatorname{zero} } = 3$
* $\map \delta {\operatorname{succ} } = 9$
* $\forall k, m \in \N^*: m \le k: \map \delta {\pr^k_m} = 2^k 3^m$
Suppose the ... | There exist [[Definition:URM Computability#Function|URM computable functions]] which are not [[Definition:Primitive Recursive Function|primitive recursive]]. | Consider the [[Definition:Basic Primitive Recursive Function|basic primitive recursive functions]].
To each [[Definition:Basic Primitive Recursive Function|basic primitive recursive function]] $f$ let us assign a code number $\map \delta f$, as follows:
* $\map \delta {\operatorname{zero} } = 3$
* $\map \delta {\oper... | Not All URM Computable Functions are Primitive Recursive | https://proofwiki.org/wiki/Not_All_URM_Computable_Functions_are_Primitive_Recursive | https://proofwiki.org/wiki/Not_All_URM_Computable_Functions_are_Primitive_Recursive | [
"URM Programs",
"Primitive Recursive Functions"
] | [
"Definition:URM Computability",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Basic Primitive Recursive Function",
"Definition:Basic Primitive Recursive Function",
"Definition:Function",
"Definition:Substitution (Mathematical Logic)",
"Definition:Function",
"Definition:Primitive Recursion",
"Definition:Primitive Recursive/Function",
"Definition:Natural Numbers",
"... |
proofwiki-2198 | Cantor's Diagonal Argument | Let $S$ be a set such that $\card S > 1$, that is, such that $S$ is not a singleton.
Let $\mathbb F$ be the set of all mappings from the natural numbers $\N$ to $S$:
:$\mathbb F = \set {f: \N \to S}$
Then $\mathbb F$ is uncountably infinite. | First we note that as $\card S > 1$, there are at least two elements of $S$ which are distinct.
Call these distinct elements $a$ and $b$.
That is:
:$\exists a, b \in S: a \ne b$
First we show that $\mathbb F$ is infinite, as follows.
For each $m \in \N$, let $\phi_m$ be the mapping defined as:
:<nowiki>$\map {\phi_m} n... | Let $S$ be a [[Definition:Set|set]] such that $\card S > 1$, that is, such that $S$ is not a [[Definition:Singleton|singleton]].
Let $\mathbb F$ be the [[Definition:Set|set]] of all [[Definition:Mapping|mappings]] from the [[Definition:Natural Numbers|natural numbers]] $\N$ to $S$:
:$\mathbb F = \set {f: \N \to S}$
T... | First we note that as $\card S > 1$, there are at least two [[Definition:Element|elements]] of $S$ which are [[Definition:Distinct Elements|distinct]].
Call these [[Definition:Distinct Elements|distinct elements]] $a$ and $b$.
That is:
:$\exists a, b \in S: a \ne b$
First we show that $\mathbb F$ is [[Definition:In... | Cantor's Diagonal Argument | https://proofwiki.org/wiki/Cantor's_Diagonal_Argument | https://proofwiki.org/wiki/Cantor's_Diagonal_Argument | [
"Cantor's Diagonal Argument",
"Diagonal Arguments",
"Uncountable Sets",
"Mapping Theory",
"Proof Techniques"
] | [
"Definition:Set",
"Definition:Singleton",
"Definition:Set",
"Definition:Mapping",
"Definition:Natural Numbers",
"Definition:Uncountable/Set"
] | [
"Definition:Element",
"Definition:Distinct/Plural",
"Definition:Distinct/Plural",
"Definition:Infinite Set",
"Definition:Mapping",
"Definition:Mapping",
"Definition:Injection",
"Definition:Infinite Set",
"Infinite if Injection from Natural Numbers",
"Definition:Uncountable/Set",
"Definition:Mapp... |
proofwiki-2199 | Infinite if Injection from Natural Numbers | Let $S$ be a set.
Let there exist an injection $\phi: \N \to S$ from the natural numbers to $S$.
Then $S$ is infinite. | {{AimForCont}} that $S$ is finite.
Let $k \in \N$ be such that there exists a bijection $\psi: S \to \N_k$.
Note that $\N_k \subset \N$ since $k \in \N$, $k \notin \N_k$.
Consider the restriction $\phi \restriction \N_k$ of $\phi$ to $\N_k$.
Then $\phi \restriction \N_k: \N_k \to S$ is an injection by Restriction of In... | Let $S$ be a [[Definition:Set|set]].
Let there exist an [[Definition:Injection|injection]] $\phi: \N \to S$ from the [[Definition:Natural Numbers|natural numbers]] to $S$.
Then $S$ is [[Definition:Infinite Set|infinite]]. | {{AimForCont}} that $S$ is [[Definition:Finite Set|finite]].
Let $k \in \N$ be such that there exists a [[Definition:Bijection|bijection]] $\psi: S \to \N_k$.
Note that $\N_k \subset \N$ since $k \in \N$, $k \notin \N_k$.
Consider the [[Definition:Restriction of Mapping|restriction]] $\phi \restriction \N_k$ of $\p... | Infinite if Injection from Natural Numbers | https://proofwiki.org/wiki/Infinite_if_Injection_from_Natural_Numbers | https://proofwiki.org/wiki/Infinite_if_Injection_from_Natural_Numbers | [
"Injections",
"Infinite Sets"
] | [
"Definition:Set",
"Definition:Injection",
"Definition:Natural Numbers",
"Definition:Infinite Set"
] | [
"Definition:Finite Set",
"Definition:Bijection",
"Definition:Restriction/Mapping",
"Definition:Injection",
"Restriction of Injection is Injection",
"Equivalence of Mappings between Finite Sets of Same Cardinality",
"Definition:Bijection",
"Definition:Surjection",
"Definition:Injection",
"Definitio... |
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