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-10000 | Divisor of One of Coprime Numbers is Coprime to Other | Let $a, b \in \N$ be numbers such that $a$ and $b$ are coprime:
:$a \perp b$
Let $c > 1$ be a divisor of $a$:
:$c \divides a$
Then $c$ and $b$ are coprime:
:$c \perp b$
{{:Euclid:Proposition/VII/23}} | Let $a \perp b$ and $c > 1: c \divides a$.
{{AimForCont}} $c \not \perp b$.
So by definition of not coprime:
:$\exists d > 1: d \divides c, d \divides b$
But from Divisor Relation is Transitive:
:$d \divides c, c \divides a \implies d \divides a$
So $d$ is a common divisor of both $a$ and $b$.
So, by definition, $a$ an... | Let $a, b \in \N$ be [[Definition:Natural Number|numbers]] such that $a$ and $b$ are [[Definition:Coprime Integers|coprime]]:
:$a \perp b$
Let $c > 1$ be a [[Definition:Divisor of Integer|divisor]] of $a$:
:$c \divides a$
Then $c$ and $b$ are coprime:
:$c \perp b$
{{:Euclid:Proposition/VII/23}} | Let $a \perp b$ and $c > 1: c \divides a$.
{{AimForCont}} $c \not \perp b$.
So by definition of [[Definition:Coprime Integers|not coprime]]:
:$\exists d > 1: d \divides c, d \divides b$
But from [[Divisor Relation is Transitive]]:
:$d \divides c, c \divides a \implies d \divides a$
So $d$ is a [[Definition:Common ... | Divisor of One of Coprime Numbers is Coprime to Other/Proof 1 | https://proofwiki.org/wiki/Divisor_of_One_of_Coprime_Numbers_is_Coprime_to_Other | https://proofwiki.org/wiki/Divisor_of_One_of_Coprime_Numbers_is_Coprime_to_Other/Proof_1 | [
"Coprime Integers",
"Divisor of One of Coprime Numbers is Coprime to Other"
] | [
"Definition:Natural Numbers",
"Definition:Coprime/Integers",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Coprime/Integers",
"Divisor Relation is Transitive",
"Definition:Common Divisor/Integers",
"Definition:Coprime/Integers",
"Proof by Contradiction"
] |
proofwiki-10001 | Divisor of One of Coprime Numbers is Coprime to Other | Let $a, b \in \N$ be numbers such that $a$ and $b$ are coprime:
:$a \perp b$
Let $c > 1$ be a divisor of $a$:
:$c \divides a$
Then $c$ and $b$ are coprime:
:$c \perp b$
{{:Euclid:Proposition/VII/23}} | Let $A, B$ be two numbers which are prime to one another.
Let $C$ be any number greater than $1$ which measures $A$.
:250px
Suppose $C$ and $B$ are not prime to one another.
Then some number $D$ will measure them both.
We have that $D$ measures $C$ and $C$ measures $A$.
So $D$ measures $A$.
But $D$ also measures $B$.
S... | Let $a, b \in \N$ be [[Definition:Natural Number|numbers]] such that $a$ and $b$ are [[Definition:Coprime Integers|coprime]]:
:$a \perp b$
Let $c > 1$ be a [[Definition:Divisor of Integer|divisor]] of $a$:
:$c \divides a$
Then $c$ and $b$ are coprime:
:$c \perp b$
{{:Euclid:Proposition/VII/23}} | Let $A, B$ be two [[Definition:Natural Number|numbers]] which are [[Definition:Coprime Integers|prime to one another]].
Let $C$ be any [[Definition:Natural Number|number]] greater than $1$ which [[Definition:Divisor of Integer|measures]] $A$.
:[[File:Euclid-VII-23.png|250px]]
Suppose $C$ and $B$ are not [[Definition... | Divisor of One of Coprime Numbers is Coprime to Other/Proof 2 | https://proofwiki.org/wiki/Divisor_of_One_of_Coprime_Numbers_is_Coprime_to_Other | https://proofwiki.org/wiki/Divisor_of_One_of_Coprime_Numbers_is_Coprime_to_Other/Proof_2 | [
"Coprime Integers",
"Divisor of One of Coprime Numbers is Coprime to Other"
] | [
"Definition:Natural Numbers",
"Definition:Coprime/Integers",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Natural Numbers",
"Definition:Coprime/Integers",
"Definition:Natural Numbers",
"Definition:Divisor (Algebra)/Integer",
"File:Euclid-VII-23.png",
"Definition:Coprime/Integers",
"Definition:Natural Numbers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
... |
proofwiki-10002 | Integer Coprime to Factors is Coprime to Whole | Let $a, b, c \in \Z$ be integers.
Let:
:$a \perp b$
:$a \perp c$
where $\perp$ denotes coprimality.
Then:
:$a \perp b c$ | ''This proof follows the structure of {{AuthorRef|Euclid}}'s proof, if not its detail.''
Let $a, b, c \in \Z$ such that $c$ is coprime to each of $a$ and $b$.
Let $d = a b$.
{{AimForCont}} $c$ and $d$ are not coprime.
Then:
:$\exists e \in \Z_{>1}: e \divides c, e \divides d$
We have that $c \perp a$ and $e \divides c$... | Let $a, b, c \in \Z$ be [[Definition:Integer|integers]].
Let:
:$a \perp b$
:$a \perp c$
where $\perp$ denotes [[Definition:Coprime Integers|coprimality]].
Then:
:$a \perp b c$ | ''This proof follows the structure of {{AuthorRef|Euclid}}'s proof, if not its detail.''
Let $a, b, c \in \Z$ such that $c$ is [[Definition:Coprime Integers|coprime]] to each of $a$ and $b$.
Let $d = a b$.
{{AimForCont}} $c$ and $d$ are not [[Definition:Coprime Integers|coprime]].
Then:
:$\exists e \in \Z_{>1}: e \... | Integer Coprime to Factors is Coprime to Whole/Proof 1 | https://proofwiki.org/wiki/Integer_Coprime_to_Factors_is_Coprime_to_Whole | https://proofwiki.org/wiki/Integer_Coprime_to_Factors_is_Coprime_to_Whole/Proof_1 | [
"Integer Coprime to Factors is Coprime to Whole",
"Coprime Integers"
] | [
"Definition:Integer",
"Definition:Coprime/Integers"
] | [
"Definition:Coprime/Integers",
"Definition:Coprime/Integers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Rational Number/Canonical Form",
"Definition:Divisor (Algebra)/Integer",
"Proof by Contradiction",
"Definition:Coprime/Integers"
] |
proofwiki-10003 | Integer Coprime to Factors is Coprime to Whole | Let $a, b, c \in \Z$ be integers.
Let:
:$a \perp b$
:$a \perp c$
where $\perp$ denotes coprimality.
Then:
:$a \perp b c$ | Let $a, b, c \in \Z$ such that $a$ is coprime to each of $b$ and $c$.
We have:
{{begin-eqn}}
{{eqn | l = a
| o = \perp
| r = b
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| q = \exists x, y \in \Z
| l = 1
| r = a x + b y
| c = Integer Combination of Coprime Integers
}}
{{eqn |... | Let $a, b, c \in \Z$ be [[Definition:Integer|integers]].
Let:
:$a \perp b$
:$a \perp c$
where $\perp$ denotes [[Definition:Coprime Integers|coprimality]].
Then:
:$a \perp b c$ | Let $a, b, c \in \Z$ such that $a$ is [[Definition:Coprime Integers|coprime]] to each of $b$ and $c$.
We have:
{{begin-eqn}}
{{eqn | l = a
| o = \perp
| r = b
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| q = \exists x, y \in \Z
| l = 1
| r = a x + b y
| c = [[Integer Combin... | Integer Coprime to Factors is Coprime to Whole/Proof 2 | https://proofwiki.org/wiki/Integer_Coprime_to_Factors_is_Coprime_to_Whole | https://proofwiki.org/wiki/Integer_Coprime_to_Factors_is_Coprime_to_Whole/Proof_2 | [
"Integer Coprime to Factors is Coprime to Whole",
"Coprime Integers"
] | [
"Definition:Integer",
"Definition:Coprime/Integers"
] | [
"Definition:Coprime/Integers",
"Integer Combination of Coprime Integers",
"Integer Combination of Coprime Integers",
"Integer Combination of Coprime Integers"
] |
proofwiki-10004 | Square of Coprime Number is Coprime | Let $a$ and $b$ be coprime integers:
:$a, b \in \Z: a \perp b$
Then:
:$a^2 \perp b$
{{:Euclid:Proposition/VII/25}} | Let $a \perp b$.
Let $a^2 = c$.
Let $d = a$.
As $a \perp b$ it follows that $d \perp b$.
From {{EuclidPropLink|book = VII|prop = 24|title = Integer Coprime to Factors is Coprime to Whole}}:
:$a d \perp b$
But $a d = c = a^2$.
Hence the result.
{{qed}}
{{Euclid Note|25|VII}} | Let $a$ and $b$ be [[Definition:Coprime Integers|coprime integers]]:
:$a, b \in \Z: a \perp b$
Then:
:$a^2 \perp b$
{{:Euclid:Proposition/VII/25}} | Let $a \perp b$.
Let $a^2 = c$.
Let $d = a$.
As $a \perp b$ it follows that $d \perp b$.
From {{EuclidPropLink|book = VII|prop = 24|title = Integer Coprime to Factors is Coprime to Whole}}:
:$a d \perp b$
But $a d = c = a^2$.
Hence the result.
{{qed}}
{{Euclid Note|25|VII}} | Square of Coprime Number is Coprime | https://proofwiki.org/wiki/Square_of_Coprime_Number_is_Coprime | https://proofwiki.org/wiki/Square_of_Coprime_Number_is_Coprime | [
"Coprime Integers"
] | [
"Definition:Coprime/Integers"
] | [] |
proofwiki-10005 | Product of Coprime Pairs is Coprime | Let $a, b, c, d$ be integers.
Let:
:$a \perp c, b \perp c, a \perp d, b \perp d$
where $a \perp c$ denotes that $a$ and $c$ are coprime.
Then:
:$a b \perp c d$
{{:Euclid:Proposition/VII/26}} | Let $e = a b, f = c d$.
{{begin-eqn}}
{{eqn | l = a
| o = \perp
| r = c
| c =
}}
{{eqn | ll= \land
| l = b
| o = \perp
| r = c
| c =
}}
{{eqn | n = 1
| lll= \implies
| l = a b
| o = \perp
| r = c
| c = {{EuclidPropLink|book = VII|prop = 24|title ... | Let $a, b, c, d$ be [[Definition:Integer|integers]].
Let:
:$a \perp c, b \perp c, a \perp d, b \perp d$
where $a \perp c$ denotes that $a$ and $c$ are [[Definition:Coprime Integers|coprime]].
Then:
:$a b \perp c d$
{{:Euclid:Proposition/VII/26}} | Let $e = a b, f = c d$.
{{begin-eqn}}
{{eqn | l = a
| o = \perp
| r = c
| c =
}}
{{eqn | ll= \land
| l = b
| o = \perp
| r = c
| c =
}}
{{eqn | n = 1
| lll= \implies
| l = a b
| o = \perp
| r = c
| c = {{EuclidPropLink|book = VII|prop = 24|title... | Product of Coprime Pairs is Coprime | https://proofwiki.org/wiki/Product_of_Coprime_Pairs_is_Coprime | https://proofwiki.org/wiki/Product_of_Coprime_Pairs_is_Coprime | [
"Coprime Integers"
] | [
"Definition:Integer",
"Definition:Coprime/Integers"
] | [] |
proofwiki-10006 | Powers of Coprime Numbers are Coprime | Let $a, b$ be coprime integers:
:$a \perp b$
Then:
:$\forall n \in \N_{>0}: a^n \perp b^n$
{{:Euclid:Proposition/VII/27}} | Proof by induction:
Let $a \perp b$.
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:$a^n \perp b^n$
$\map P 1$ is true, as this just says:
:$a \perp b$ | Let $a, b$ be [[Definition:Coprime Integers|coprime integers]]:
:$a \perp b$
Then:
:$\forall n \in \N_{>0}: a^n \perp b^n$
{{:Euclid:Proposition/VII/27}} | Proof by [[Principle of Mathematical Induction|induction]]:
Let $a \perp b$.
For all $n \in \N_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$a^n \perp b^n$
$\map P 1$ is true, as this just says:
:$a \perp b$ | Powers of Coprime Numbers are Coprime | https://proofwiki.org/wiki/Powers_of_Coprime_Numbers_are_Coprime | https://proofwiki.org/wiki/Powers_of_Coprime_Numbers_are_Coprime | [
"Coprime Integers"
] | [
"Definition:Coprime/Integers"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-10007 | Numbers are Coprime iff Sum is Coprime to Both | Let $a, b$ be integers.
Then:
:$a \perp b \iff a \perp \paren {a + b}$
where $a \perp b$ denotes that $a$ and $b$ are coprime.
{{:Euclid:Proposition/VII/28}} | === Necessary Condition ===
Let $a \perp b$.
Suppose $a + b$ is not coprime to $a$.
Then:
:$\exists d \in \Z_{>1}: d \divides a, d \divides \paren {a + b}$
But then:
:$d \divides \paren {\paren {a + b} - a}$
and so:
:$d \divides b$
and so $a$ and $b$ are not coprime.
From this contradiction it follows that $a + b$ is c... | Let $a, b$ be [[Definition:Integer|integers]].
Then:
:$a \perp b \iff a \perp \paren {a + b}$
where $a \perp b$ denotes that $a$ and $b$ are [[Definition:Coprime Integers|coprime]].
{{:Euclid:Proposition/VII/28}} | === Necessary Condition ===
Let $a \perp b$.
Suppose $a + b$ is not [[Definition:Coprime Integers|coprime]] to $a$.
Then:
:$\exists d \in \Z_{>1}: d \divides a, d \divides \paren {a + b}$
But then:
:$d \divides \paren {\paren {a + b} - a}$
and so:
:$d \divides b$
and so $a$ and $b$ are not [[Definition:Coprime Int... | Numbers are Coprime iff Sum is Coprime to Both | https://proofwiki.org/wiki/Numbers_are_Coprime_iff_Sum_is_Coprime_to_Both | https://proofwiki.org/wiki/Numbers_are_Coprime_iff_Sum_is_Coprime_to_Both | [
"Coprime Integers"
] | [
"Definition:Integer",
"Definition:Coprime/Integers"
] | [
"Definition:Coprime/Integers",
"Definition:Coprime/Integers",
"Proof by Contradiction",
"Definition:Coprime/Integers",
"Definition:Coprime/Integers",
"Definition:Coprime/Integers",
"Definition:Coprime/Integers",
"Proof by Contradiction",
"Definition:Coprime/Integers"
] |
proofwiki-10008 | Composite Number has Prime Factor | Let $a$ be a composite number.
Then there exists a prime number $p$ such that:
:$p \divides a$
where $\divides$ means '''is a divisor of'''.
{{:Euclid:Proposition/VII/31}} | By definition of composite number:
:$\exists b \in \Z: b \divides a$
If $b$ is a prime number then the proof is complete.
Otherwise $b$ is composite.
By definition of composite number:
:$\exists c \in \Z: c \divides b$
and so:
:$c \divides a$
Again, if $c$ is a prime number then the proof is complete.
Continuing in thi... | Let $a$ be a [[Definition:Composite Number|composite number]].
Then there exists a [[Definition:Prime Number|prime number]] $p$ such that:
:$p \divides a$
where $\divides$ means '''is a [[Definition:Divisor of Integer|divisor]] of'''.
{{:Euclid:Proposition/VII/31}} | By definition of [[Definition:Composite Number|composite number]]:
:$\exists b \in \Z: b \divides a$
If $b$ is a [[Definition:Prime Number|prime number]] then the proof is complete.
Otherwise $b$ is [[Definition:Composite Number|composite]].
By definition of [[Definition:Composite Number|composite number]]:
:$\ex... | Composite Number has Prime Factor | https://proofwiki.org/wiki/Composite_Number_has_Prime_Factor | https://proofwiki.org/wiki/Composite_Number_has_Prime_Factor | [
"Prime Numbers"
] | [
"Definition:Composite Number",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Composite Number",
"Definition:Prime Number",
"Definition:Composite Number",
"Definition:Composite Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Natural Numbers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Se... |
proofwiki-10009 | Natural Number is Prime or has Prime Factor | Let $a$ be a natural number greater than $1$.
Then either:
:$a$ is a prime number
or:
:there exists a prime number $p \ne a$ such that $p \divides a$
where $\divides$ denotes '''is a divisor of'''.
{{:Euclid:Proposition/VII/32}} | By definition of composite number $a$ is either prime or composite.
Let $a$ be prime.
Then the statement of the result is fulfilled.
Let $a$ be composite.
Then by {{EuclidPropLink|book = VII|prop = 31|title = Composite Number has Prime Factor}}:
:$\exists p: p \divides a$
where $p$ is a prime number.
The result follows... | Let $a$ be a [[Definition:Natural Number|natural number]] greater than $1$.
Then either:
:$a$ is a [[Definition:Prime Number|prime number]]
or:
:there exists a [[Definition:Prime Number|prime number]] $p \ne a$ such that $p \divides a$
where $\divides$ denotes '''is a [[Definition:Divisor of Integer|divisor]] of'''.
... | By definition of [[Definition:Composite Number|composite number]] $a$ is either [[Definition:Prime Number|prime]] or [[Definition:Composite Number|composite]].
Let $a$ be [[Definition:Prime Number|prime]].
Then the statement of the result is fulfilled.
Let $a$ be [[Definition:Composite Number|composite]].
Then by... | Natural Number is Prime or has Prime Factor | https://proofwiki.org/wiki/Natural_Number_is_Prime_or_has_Prime_Factor | https://proofwiki.org/wiki/Natural_Number_is_Prime_or_has_Prime_Factor | [
"Prime Numbers"
] | [
"Definition:Natural Numbers",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Composite Number",
"Definition:Prime Number",
"Definition:Composite Number",
"Definition:Prime Number",
"Definition:Composite Number",
"Definition:Prime Number",
"Proof by Cases"
] |
proofwiki-10010 | LCM of Three Numbers | Let $a, b, c \in \Z: a b c \ne 0$.
The lowest common multiple of $a, b, c$, denoted $\lcm \set {a, b, c}$, can always be found.
{{:Euclid:Proposition/VII/36}} | Let $d = \lcm \set {a, b}$.
This exists from {{EuclidPropLink|book = VII|prop = 34|title = Existence of Lowest Common Multiple}}.
Either $c \divides d$ or not, where $\divides$ denotes divisibility.
Suppose $c \divides d$.
But by definition of lowest common multiple, $a \divides d$ and $b \divides d$ also.
Suppose $a, ... | Let $a, b, c \in \Z: a b c \ne 0$.
The [[Definition:Lowest Common Multiple of Integers|lowest common multiple]] of $a, b, c$, denoted $\lcm \set {a, b, c}$, can always be found.
{{:Euclid:Proposition/VII/36}} | Let $d = \lcm \set {a, b}$.
This exists from {{EuclidPropLink|book = VII|prop = 34|title = Existence of Lowest Common Multiple}}.
Either $c \divides d$ or not, where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
Suppose $c \divides d$.
But by definition of [[Definition:Lowest Common Multiple o... | LCM of Three Numbers | https://proofwiki.org/wiki/LCM_of_Three_Numbers | https://proofwiki.org/wiki/LCM_of_Three_Numbers | [
"Lowest Common Multiple"
] | [
"Definition:Lowest Common Multiple/Integers"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Lowest Common Multiple/Integers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Common Divisor/Integers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Diviso... |
proofwiki-10011 | Negative of Real Function that Increases Without Bound | Let $f: \R \to \R$ be a real function.
Then:
:$(1): \quad \ds \lim_{x \mathop \to +\infty} \map f x = +\infty \implies \lim_{x \mathop \to +\infty} -\map f x = -\infty$
:$(2): \quad \ds \lim_{x \mathop \to -\infty} \map f x = +\infty \implies \lim_{x \mathop \to -\infty} -\map f x = -\infty$ | Suppose $\ds \lim_{x \mathop \to +\infty} \map f x = +\infty$.
Then by the definition of infinite limits at infinity:
:$\forall M > 0: \exists N > 0: x > N \implies \map f x > M$
But $M > 0 \iff -M < 0$.
Likewise $\map f x > M \iff -\map f x < -M$.
Putting $M' = -M$:
:$\forall M' < 0: \exists N > 0: x > N \implies -\ma... | Let $f: \R \to \R$ be a [[Definition:Real Function|real function]].
Then:
:$(1): \quad \ds \lim_{x \mathop \to +\infty} \map f x = +\infty \implies \lim_{x \mathop \to +\infty} -\map f x = -\infty$
:$(2): \quad \ds \lim_{x \mathop \to -\infty} \map f x = +\infty \implies \lim_{x \mathop \to -\infty} -\map f x = -\i... | Suppose $\ds \lim_{x \mathop \to +\infty} \map f x = +\infty$.
Then by the definition of [[Definition:Infinite Limit at Infinity|infinite limits at infinity]]:
:$\forall M > 0: \exists N > 0: x > N \implies \map f x > M$
But $M > 0 \iff -M < 0$.
Likewise $\map f x > M \iff -\map f x < -M$.
Putting $M' = -M$:
:$... | Negative of Real Function that Increases Without Bound | https://proofwiki.org/wiki/Negative_of_Real_Function_that_Increases_Without_Bound | https://proofwiki.org/wiki/Negative_of_Real_Function_that_Increases_Without_Bound | [
"Unbounded Mappings"
] | [
"Definition:Real Function"
] | [
"Definition:Limit of Real Function/Limit at Infinity/Positive/Increasing Without Bound",
"Definition:Limit of Real Function/Limit at Infinity/Positive/Decreasing Without Bound"
] |
proofwiki-10012 | Integer Divided by Divisor is Integer | Let $a, b \in \N$.
Then:
:$b \divides a \implies \dfrac 1 b \times a \in \N$
where $\divides$ denotes divisibilty.
{{:Euclid:Proposition/VII/37}} | Let $b \divides a$.
By definition of divisibilty:
:$\exists c \in \N: c \times b = a$
Then also:
:$c \times 1 = c$
So by {{EuclidPropLink|book = VII|prop = 15|title = Alternate Ratios of Multiples}}:
:$1 : b = c : a$
Hence the result.
{{qed}}
{{Euclid Note|37|VII|}} | Let $a, b \in \N$.
Then:
:$b \divides a \implies \dfrac 1 b \times a \in \N$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibilty]].
{{:Euclid:Proposition/VII/37}} | Let $b \divides a$.
By definition of [[Definition:Divisor of Integer|divisibilty]]:
:$\exists c \in \N: c \times b = a$
Then also:
:$c \times 1 = c$
So by {{EuclidPropLink|book = VII|prop = 15|title = Alternate Ratios of Multiples}}:
:$1 : b = c : a$
Hence the result.
{{qed}}
{{Euclid Note|37|VII|}} | Integer Divided by Divisor is Integer | https://proofwiki.org/wiki/Integer_Divided_by_Divisor_is_Integer | https://proofwiki.org/wiki/Integer_Divided_by_Divisor_is_Integer | [
"Divisors"
] | [
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-10013 | Divisor is Reciprocal of Divisor of Integer | Let $a, b, c \in \Z_{>0}$.
Then:
:$b = \dfrac 1 c \times a \implies c \divides a$
where $\divides$ denotes divisibilty.
{{:Euclid:Proposition/VII/38}} | Let $a$ have an aliquot part $b$.
Let $c$ be an integer called by the same name as the aliquot part $b$.
Then:
:$1 = \dfrac 1 c \times c$
and so by {{EuclidPropLink|book = VII|prop = 15|title = Alternate Ratios of Multiples}}:
:$ 1 : c = b : a$
Hence the result.
{{qed}}
{{Euclid Note|38|VII|}} | Let $a, b, c \in \Z_{>0}$.
Then:
:$b = \dfrac 1 c \times a \implies c \divides a$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibilty]].
{{:Euclid:Proposition/VII/38}} | Let $a$ have an [[Definition:Aliquot Part|aliquot part]] $b$.
Let $c$ be an [[Definition:Integer|integer]] called by the same name as the [[Definition:Aliquot Part|aliquot part]] $b$.
Then:
:$1 = \dfrac 1 c \times c$
and so by {{EuclidPropLink|book = VII|prop = 15|title = Alternate Ratios of Multiples}}:
:$ 1 : c = b... | Divisor is Reciprocal of Divisor of Integer | https://proofwiki.org/wiki/Divisor_is_Reciprocal_of_Divisor_of_Integer | https://proofwiki.org/wiki/Divisor_is_Reciprocal_of_Divisor_of_Integer | [
"Divisors"
] | [
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Divisor (Algebra)/Integer/Aliquot Part",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer/Aliquot Part"
] |
proofwiki-10014 | Negative of Real Function that Decreases Without Bound | Let $f: \R \to \R$ be a real function.
Then:
:$(1): \quad \ds \lim_{x \mathop \to +\infty} \map f x = -\infty \implies \lim_{x \mathop \to +\infty} -\map f x = +\infty$
:$(2): \quad \ds \lim_{x \mathop \to -\infty} \map f x = -\infty \implies \lim_{x \mathop \to -\infty} -\map f x = +\infty$ | Suppose $\ds \lim_{x \mathop \to +\infty} \map f x = -\infty$.
Then by the definition of negative infinite limit at infinity:
:$\forall M < 0: \exists N > 0: x > N \implies \map f x < M$
But:
:$M < 0 \iff -M > 0$
Likewise:
:$\map f x < M \iff -\map f x > -M$
Putting $M' = -M$:
:$\forall M' > 0: \exists N > 0: x > N \im... | Let $f: \R \to \R$ be a [[Definition:Real Function|real function]].
Then:
:$(1): \quad \ds \lim_{x \mathop \to +\infty} \map f x = -\infty \implies \lim_{x \mathop \to +\infty} -\map f x = +\infty$
:$(2): \quad \ds \lim_{x \mathop \to -\infty} \map f x = -\infty \implies \lim_{x \mathop \to -\infty} -\map f x = +\i... | Suppose $\ds \lim_{x \mathop \to +\infty} \map f x = -\infty$.
Then by the definition of [[Definition:Negative Infinite Limit at Infinity|negative infinite limit at infinity]]:
:$\forall M < 0: \exists N > 0: x > N \implies \map f x < M$
But:
:$M < 0 \iff -M > 0$
Likewise:
:$\map f x < M \iff -\map f x > -M$
Put... | Negative of Real Function that Decreases Without Bound | https://proofwiki.org/wiki/Negative_of_Real_Function_that_Decreases_Without_Bound | https://proofwiki.org/wiki/Negative_of_Real_Function_that_Decreases_Without_Bound | [
"Unbounded Mappings"
] | [
"Definition:Real Function"
] | [
"Definition:Limit of Real Function/Limit at Infinity/Positive/Decreasing Without Bound",
"Definition:Limit of Real Function/Limit at Infinity/Positive/Increasing Without Bound"
] |
proofwiki-10015 | Construction of Geometric Sequence in Lowest Terms | It is possible to find a geometric sequence of integers $G_n$ of length $n + 1$ with a given common ratio such that $G_n$ is in its lowest terms.
{{:Euclid:Proposition/VIII/2}} | Let $r = \dfrac a b$ be the given common ratio.
Let the required geometric sequence have a length of $4$.
Let $a^2 = c$.
Let $a b = d$.
Let $b^2 = e$.
Let:
:$a c = f$
:$a d = g$
:$a e = h$
and let:
:$b e = k$
As:
:$a^2 = c$
:$a b = d$
it follows from {{EuclidPropLink|book = VII|prop = 17|title = Multiples of Ratios of ... | It is possible to find a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] $G_n$ of [[Definition:Length of Sequence|length]] $n + 1$ with a given [[Definition:Common Ratio of Geometric Sequence|common ratio]] such that $G_n$ is in its [[Definition:Geometric Sequence of Integers in Lowest Term... | Let $r = \dfrac a b$ be the given [[Definition:Common Ratio of Geometric Sequence|common ratio]].
Let the required [[Definition:Geometric Sequence|geometric sequence]] have a [[Definition:Length of Sequence|length]] of $4$.
Let $a^2 = c$.
Let $a b = d$.
Let $b^2 = e$.
Let:
:$a c = f$
:$a d = g$
:$a e = h$
and let... | Construction of Geometric Sequence in Lowest Terms/Proof 1 | https://proofwiki.org/wiki/Construction_of_Geometric_Sequence_in_Lowest_Terms | https://proofwiki.org/wiki/Construction_of_Geometric_Sequence_in_Lowest_Terms/Proof_1 | [
"Geometric Sequences of Integers",
"Construction of Geometric Sequence in Lowest Terms"
] | [
"Definition:Geometric Sequence/Integers",
"Definition:Length of Sequence",
"Definition:Geometric Sequence/Common Ratio",
"Definition:Geometric Sequence of Integers in Lowest Terms"
] | [
"Definition:Geometric Sequence/Common Ratio",
"Definition:Geometric Sequence",
"Definition:Length of Sequence",
"Definition:Geometric Sequence",
"Definition:Geometric Sequence/Common Ratio",
"Definition:Geometric Sequence",
"Definition:Geometric Sequence/Common Ratio",
"Definition:Ratio",
"Definitio... |
proofwiki-10016 | Construction of Geometric Sequence in Lowest Terms | It is possible to find a geometric sequence of integers $G_n$ of length $n + 1$ with a given common ratio such that $G_n$ is in its lowest terms.
{{:Euclid:Proposition/VIII/2}} | Let the required length of the geometric sequence $P$ be $n$.
Let $r$ be the given common ratio.
From Common Ratio in Integer Geometric Sequence is Rational, $r$ is a rational number.
Let $r = \dfrac p q$ be in canonical form.
Thus, by definition:
:$p \perp q$
Let $a$ be the first term of $P$.
Then the sequence $P$ is:... | It is possible to find a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] $G_n$ of [[Definition:Length of Sequence|length]] $n + 1$ with a given [[Definition:Common Ratio of Geometric Sequence|common ratio]] such that $G_n$ is in its [[Definition:Geometric Sequence of Integers in Lowest Term... | Let the required [[Definition:Length of Sequence|length]] of the [[Definition:Geometric Sequence|geometric sequence]] $P$ be $n$.
Let $r$ be the given [[Definition:Common Ratio of Geometric Sequence|common ratio]].
From [[Common Ratio in Integer Geometric Sequence is Rational]], $r$ is a [[Definition:Rational Number|... | Construction of Geometric Sequence in Lowest Terms/Proof 2 | https://proofwiki.org/wiki/Construction_of_Geometric_Sequence_in_Lowest_Terms | https://proofwiki.org/wiki/Construction_of_Geometric_Sequence_in_Lowest_Terms/Proof_2 | [
"Geometric Sequences of Integers",
"Construction of Geometric Sequence in Lowest Terms"
] | [
"Definition:Geometric Sequence/Integers",
"Definition:Length of Sequence",
"Definition:Geometric Sequence/Common Ratio",
"Definition:Geometric Sequence of Integers in Lowest Terms"
] | [
"Definition:Length of Sequence",
"Definition:Geometric Sequence",
"Definition:Geometric Sequence/Common Ratio",
"Common Ratio in Integer Geometric Sequence is Rational",
"Definition:Rational Number",
"Definition:Rational Number/Canonical Form",
"Definition:Geometric Sequence/Term",
"Definition:Sequenc... |
proofwiki-10017 | Geometric Sequence in Lowest Terms has Coprime Extremes | A geometric sequence of integers in lowest terms has extremes which are coprime.
{{:Euclid:Proposition/VIII/3}} | Let $a_0, a_1, a_2, \ldots, a_n$ be natural numbers.
Let $\sequence {a_k}_{0 \mathop \le k \mathop \le n}$ be a geometric sequence with common ratio $r$.
Let $a_0, a_1, \ldots, a_n$ be the smallest such natural numbers.
From {{EuclidPropLink|book = VII|prop = 33|title = Least Ratio of Numbers}}, let $d_0, d_1$ be the s... | A [[Definition:Geometric Sequence of Integers in Lowest Terms|geometric sequence of integers in lowest terms]] has [[Definition:Extremes of Finite Geometric Sequence|extremes]] which are [[Definition:Coprime Integers|coprime]].
{{:Euclid:Proposition/VIII/3}} | Let $a_0, a_1, a_2, \ldots, a_n$ be [[Definition:Natural Number|natural numbers]].
Let $\sequence {a_k}_{0 \mathop \le k \mathop \le n}$ be a [[Definition:Geometric Sequence|geometric sequence]] with [[Definition:Common Ratio|common ratio]] $r$.
Let $a_0, a_1, \ldots, a_n$ be the smallest such [[Definition:Natural Nu... | Geometric Sequence in Lowest Terms has Coprime Extremes/Proof 1 | https://proofwiki.org/wiki/Geometric_Sequence_in_Lowest_Terms_has_Coprime_Extremes | https://proofwiki.org/wiki/Geometric_Sequence_in_Lowest_Terms_has_Coprime_Extremes/Proof_1 | [
"Geometric Sequences of Integers",
"Geometric Sequence in Lowest Terms has Coprime Extremes"
] | [
"Definition:Geometric Sequence of Integers in Lowest Terms",
"Definition:Geometric Sequence/Finite/Extremes",
"Definition:Coprime/Integers"
] | [
"Definition:Natural Numbers",
"Definition:Geometric Sequence",
"Definition:Geometric Sequence/Common Ratio",
"Definition:Natural Numbers",
"Definition:Natural Numbers",
"Definition:Geometric Sequence",
"Definition:Coprime/Integers",
"Definition:Geometric Sequence",
"Definition:Power (Algebra)/Intege... |
proofwiki-10018 | Geometric Sequence in Lowest Terms has Coprime Extremes | A geometric sequence of integers in lowest terms has extremes which are coprime.
{{:Euclid:Proposition/VIII/3}} | Let $P$ be a geometric sequence of natural numbers of length $n$.
Let the common ratio of $P$ be expressed in canonical form as $\dfrac p q$.
From Construction of Geometric Sequence in Lowest Terms:
:$P = \paren {q^n, p q^{n - 1}, p^2 q^{n - 2}, \ldots, p^{n - 1} q, p^n}$
By definition of canonical form:
:$p \perp q$
I... | A [[Definition:Geometric Sequence of Integers in Lowest Terms|geometric sequence of integers in lowest terms]] has [[Definition:Extremes of Finite Geometric Sequence|extremes]] which are [[Definition:Coprime Integers|coprime]].
{{:Euclid:Proposition/VIII/3}} | Let $P$ be a [[Definition:Geometric Sequence|geometric sequence]] of [[Definition:Natural Number|natural numbers]] of [[Definition:Length of Sequence|length]] $n$.
Let the [[Definition:Common Ratio|common ratio]] of $P$ be expressed in [[Definition:Canonical Form of Rational Number|canonical form]] as $\dfrac p q$.
F... | Geometric Sequence in Lowest Terms has Coprime Extremes/Proof 2 | https://proofwiki.org/wiki/Geometric_Sequence_in_Lowest_Terms_has_Coprime_Extremes | https://proofwiki.org/wiki/Geometric_Sequence_in_Lowest_Terms_has_Coprime_Extremes/Proof_2 | [
"Geometric Sequences of Integers",
"Geometric Sequence in Lowest Terms has Coprime Extremes"
] | [
"Definition:Geometric Sequence of Integers in Lowest Terms",
"Definition:Geometric Sequence/Finite/Extremes",
"Definition:Coprime/Integers"
] | [
"Definition:Geometric Sequence",
"Definition:Natural Numbers",
"Definition:Length of Sequence",
"Definition:Geometric Sequence/Common Ratio",
"Definition:Rational Number/Canonical Form",
"Construction of Geometric Sequence in Lowest Terms",
"Definition:Rational Number/Canonical Form",
"Powers of Copri... |
proofwiki-10019 | General Associativity Theorem/Formulation 3 | Let $\struct {S, \circ}$ be a semigroup.
Let $a_i$ denote elements of $S$.
Let $\circ$ be associative.
Let $n \in \Z$ be a positive integer such that $n \ge 3$.
Then all possible parenthesizations of the expression:
:$a_1 \circ a_2 \circ \cdots \circ a_n$
are equivalent. | Let $\circ$ be associative.
It will be shown that any parenthesization of $a_1 \circ a_2 \circ \dots \circ a_n$ is equal to the '''left-associated expression''':
:$\paren {\paren {\paren {\cdots \paren {a_1 \circ a_2} \circ a_3} \circ \cdots} \circ a_n}$
The proof proceeds by induction on $n$. | Let $\struct {S, \circ}$ be a [[Definition:Semigroup|semigroup]].
Let $a_i$ denote [[Definition:Element|elements]] of $S$.
Let $\circ$ be [[Definition:Associative Operation|associative]].
Let $n \in \Z$ be a [[Definition:Positive Integer|positive integer]] such that $n \ge 3$.
Then all possible [[Definition:Parenth... | Let $\circ$ be [[Definition:Associative Operation|associative]].
It will be shown that any [[Definition:Parenthesization|parenthesization]] of $a_1 \circ a_2 \circ \dots \circ a_n$ is equal to the '''left-associated expression''':
:$\paren {\paren {\paren {\cdots \paren {a_1 \circ a_2} \circ a_3} \circ \cdots} \circ a... | General Associativity Theorem/Formulation 3 | https://proofwiki.org/wiki/General_Associativity_Theorem/Formulation_3 | https://proofwiki.org/wiki/General_Associativity_Theorem/Formulation_3 | [
"General Associativity Theorem"
] | [
"Definition:Semigroup",
"Definition:Element",
"Definition:Associative Operation",
"Definition:Positive/Integer",
"Definition:Parenthesization"
] | [
"Definition:Associative Operation",
"Definition:Parenthesization",
"Principle of Mathematical Induction",
"Definition:Parenthesization",
"Definition:Parenthesization",
"Definition:Parenthesization",
"Definition:Parenthesization",
"Definition:Parenthesization",
"Definition:Parenthesization",
"Defin... |
proofwiki-10020 | First Element of Geometric Sequence not dividing Second | Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers of length $n$.
Let $a_0$ not be a divisor of $a_1$.
Then:
:$\forall j, k \in \set {0, 1, \ldots, n}, j \ne k: a_j \nmid a_k$
That is, if the initial term of $P$ does not divide the second, no term of $P$ divides any other term... | Let $P_a = \tuple {a_0, a_1, \ldots, a_n}$ be a geometric sequence of natural numbers such that $a_0 \nmid a_1$.
{{AimForCont}} $a_0 \divides a_k$ for some $k: 2 \le k \le n$.
Let $b_0, b_1, \ldots, b_k$ be the least natural numbers which have the same common ratio as $a_0, a_1, \ldots, a_k$.
These can be found by mean... | Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] of [[Definition:Length of Sequence|length]] $n$.
Let $a_0$ not be a [[Definition:Divisor of Integer|divisor]] of $a_1$.
Then:
:$\forall j, k \in \set {0, 1, \ldots, n}, j \ne k:... | Let $P_a = \tuple {a_0, a_1, \ldots, a_n}$ be a [[Definition:Geometric Sequence|geometric sequence]] of [[Definition:Natural Number|natural numbers]] such that $a_0 \nmid a_1$.
{{AimForCont}} $a_0 \divides a_k$ for some $k: 2 \le k \le n$.
Let $b_0, b_1, \ldots, b_k$ be the least [[Definition:Natural Number|natural n... | First Element of Geometric Sequence not dividing Second/Proof 1 | https://proofwiki.org/wiki/First_Element_of_Geometric_Sequence_not_dividing_Second | https://proofwiki.org/wiki/First_Element_of_Geometric_Sequence_not_dividing_Second/Proof_1 | [
"Geometric Sequences of Integers",
"First Element of Geometric Sequence not dividing Second"
] | [
"Definition:Geometric Sequence/Integers",
"Definition:Length of Sequence",
"Definition:Divisor (Algebra)/Integer",
"Definition:Geometric Sequence/Initial Term",
"Definition:Divisor (Algebra)/Integer",
"Definition:Geometric Sequence/Term",
"Definition:Divisor (Algebra)/Integer",
"Definition:Geometric S... | [
"Definition:Geometric Sequence",
"Definition:Natural Numbers",
"Definition:Natural Numbers",
"Definition:Geometric Sequence/Common Ratio",
"Integer Divisor Results/One Divides all Integers",
"Definition:Natural Numbers",
"Definition:Geometric Sequence/Common Ratio"
] |
proofwiki-10021 | First Element of Geometric Sequence not dividing Second | Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers of length $n$.
Let $a_0$ not be a divisor of $a_1$.
Then:
:$\forall j, k \in \set {0, 1, \ldots, n}, j \ne k: a_j \nmid a_k$
That is, if the initial term of $P$ does not divide the second, no term of $P$ divides any other term... | From Form of Geometric Sequence of Integers, the terms of $P$ are in the form:
:$\paren 1: \quad: a_j = k q^j p^{n - j}$
where $p \perp q$.
Let $a_0 \nmid a_1$.
That is:
{{begin-eqn}}
{{eqn | l = a_0
| o = \nmid
| r = a_1
| c =
}}
{{eqn | ll= \leadsto
| l = k q^n
| o = \nmid
| r = k... | Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] of [[Definition:Length of Sequence|length]] $n$.
Let $a_0$ not be a [[Definition:Divisor of Integer|divisor]] of $a_1$.
Then:
:$\forall j, k \in \set {0, 1, \ldots, n}, j \ne k:... | From [[Form of Geometric Sequence of Integers]], the [[Definition:Term of Geometric Sequence|terms]] of $P$ are in the form:
:$\paren 1: \quad: a_j = k q^j p^{n - j}$
where $p \perp q$.
Let $a_0 \nmid a_1$.
That is:
{{begin-eqn}}
{{eqn | l = a_0
| o = \nmid
| r = a_1
| c =
}}
{{eqn | ll= \leadsto
... | First Element of Geometric Sequence not dividing Second/Proof 2 | https://proofwiki.org/wiki/First_Element_of_Geometric_Sequence_not_dividing_Second | https://proofwiki.org/wiki/First_Element_of_Geometric_Sequence_not_dividing_Second/Proof_2 | [
"Geometric Sequences of Integers",
"First Element of Geometric Sequence not dividing Second"
] | [
"Definition:Geometric Sequence/Integers",
"Definition:Length of Sequence",
"Definition:Divisor (Algebra)/Integer",
"Definition:Geometric Sequence/Initial Term",
"Definition:Divisor (Algebra)/Integer",
"Definition:Geometric Sequence/Term",
"Definition:Divisor (Algebra)/Integer",
"Definition:Geometric S... | [
"Form of Geometric Sequence of Integers",
"Definition:Geometric Sequence/Term",
"Definition:Contrapositive Statement",
"Multiple of Divisor Divides Multiple",
"Definition:Contrapositive Statement",
"Multiple of Divisor Divides Multiple",
"Powers of Coprime Numbers are Coprime"
] |
proofwiki-10022 | First Element of Geometric Sequence that divides Last also divides Second | Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers of length $n$.
Let $a_0$ be a divisor of $a_n$.
Then $a_0$ is a divisor of $a_2$.
{{:Euclid:Proposition/VIII/7}} | By hypothesis, let $a_0$ be a divisor of $a_n$.
{{AimForCont}} $a_0$ is not a divisor of $a_2$.
From First Element of Geometric Sequence not dividing Second it would follow that $a_0$ does not divide $a_n$.
From this contradiction follows the result.
{{qed}}
{{Euclid Note|7|VIII}} | Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] of [[Definition:Length of Sequence|length]] $n$.
Let $a_0$ be a [[Definition:Divisor of Integer|divisor]] of $a_n$.
Then $a_0$ is a [[Definition:Divisor of Integer|divisor]] of ... | [[Definition:By Hypothesis|By hypothesis]], let $a_0$ be a [[Definition:Divisor of Integer|divisor]] of $a_n$.
{{AimForCont}} $a_0$ is not a [[Definition:Divisor of Integer|divisor]] of $a_2$.
From [[First Element of Geometric Sequence not dividing Second]] it would follow that $a_0$ does not divide $a_n$.
From this... | First Element of Geometric Sequence that divides Last also divides Second | https://proofwiki.org/wiki/First_Element_of_Geometric_Sequence_that_divides_Last_also_divides_Second | https://proofwiki.org/wiki/First_Element_of_Geometric_Sequence_that_divides_Last_also_divides_Second | [
"Geometric Sequences of Integers"
] | [
"Definition:Geometric Sequence/Integers",
"Definition:Length of Sequence",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:By Hypothesis",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"First Element of Geometric Sequence not dividing Second",
"Proof by Contradiction"
] |
proofwiki-10023 | Geometric Sequences in Proportion have Same Number of Elements | Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers of length $n$.
Let $r$ be the common ratio of $P$.
Let $Q = \sequence {b_j}_{0 \mathop \le j \mathop \le m}$ be a geometric sequence of integers of length $m$.
Let $r$ be the common ratio of $Q$.
Let $b_0$ and $b_m$ be such th... | Let $S = \sequence {c_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers of length $n$ such that:
:$r$ is the common ratio of $S$
:$S$ is in its lowest terms.
From Geometric Sequence in Lowest Terms has Coprime Extremes, $c_0$ is coprime to $c_n$.
Then:
:$\dfrac {c_0} {c_n} = \dfrac {a_0} {a_n} = \... | Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] of [[Definition:Length of Sequence|length]] $n$.
Let $r$ be the [[Definition:Common Ratio|common ratio]] of $P$.
Let $Q = \sequence {b_j}_{0 \mathop \le j \mathop \le m}$ be a [... | Let $S = \sequence {c_j}_{0 \mathop \le j \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] of [[Definition:Length of Sequence|length]] $n$ such that:
:$r$ is the [[Definition:Common Ratio|common ratio]] of $S$
:$S$ is in its [[Definition:Geometric Sequence of Integers in... | Geometric Sequences in Proportion have Same Number of Elements | https://proofwiki.org/wiki/Geometric_Sequences_in_Proportion_have_Same_Number_of_Elements | https://proofwiki.org/wiki/Geometric_Sequences_in_Proportion_have_Same_Number_of_Elements | [
"Geometric Sequences of Integers"
] | [
"Definition:Geometric Sequence/Integers",
"Definition:Length of Sequence",
"Definition:Geometric Sequence/Common Ratio",
"Definition:Geometric Sequence/Integers",
"Definition:Length of Sequence",
"Definition:Geometric Sequence/Common Ratio"
] | [
"Definition:Geometric Sequence/Integers",
"Definition:Length of Sequence",
"Definition:Geometric Sequence/Common Ratio",
"Definition:Geometric Sequence of Integers in Lowest Terms",
"Geometric Sequence in Lowest Terms has Coprime Extremes",
"Definition:Coprime/Integers"
] |
proofwiki-10024 | Elements of Geometric Sequence between Coprime Numbers | Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers of length $n + 1$.
Let $a_0$ be coprime to $a_n$.
Then there exist geometric sequences of integers $Q_1$ and $Q_2$ of length $n + 1$ such that:
:the initial term of both $Q_1$ and $Q_2$ is $1$
:the final term of $Q_1$ is $a_0$... | Let the common ratio of $P$ be $r$.
By Form of Geometric Sequence of Integers with Coprime Extremes, the $j$th term of $P$ is given by:
:$a_j = q^j p^{n - j}$
such that:
:$a_0 = p^n$
:$a_n = q^n$
For $j \in \set {0, 1, \ldots, n}$, let $r_j = q^j$.
Let the finite sequence $Q_2 = \sequence {t_j}_{0 \mathop \le j \mathop... | Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] of [[Definition:Length of Sequence|length]] $n + 1$.
Let $a_0$ be [[Definition:Coprime Integers|coprime]] to $a_n$.
Then there exist [[Definition:Geometric Sequence of Integers|g... | Let the [[Definition:Common Ratio|common ratio]] of $P$ be $r$.
By [[Form of Geometric Sequence of Integers with Coprime Extremes]], the $j$th [[Definition:Term of Geometric Sequence|term]] of $P$ is given by:
:$a_j = q^j p^{n - j}$
such that:
:$a_0 = p^n$
:$a_n = q^n$
For $j \in \set {0, 1, \ldots, n}$, let $r_j = q... | Elements of Geometric Sequence between Coprime Numbers | https://proofwiki.org/wiki/Elements_of_Geometric_Sequence_between_Coprime_Numbers | https://proofwiki.org/wiki/Elements_of_Geometric_Sequence_between_Coprime_Numbers | [
"Geometric Sequences of Integers"
] | [
"Definition:Geometric Sequence/Integers",
"Definition:Length of Sequence",
"Definition:Coprime/Integers",
"Definition:Geometric Sequence/Integers",
"Definition:Length of Sequence",
"Definition:Geometric Sequence/Initial Term",
"Definition:Geometric Sequence/Finite/Final Term",
"Definition:Geometric Se... | [
"Definition:Geometric Sequence/Common Ratio",
"Form of Geometric Sequence of Integers with Coprime Extremes",
"Definition:Geometric Sequence/Term",
"Definition:Finite Sequence",
"Definition:Geometric Sequence/Integers",
"Definition:Length of Sequence",
"Definition:Finite Sequence",
"Definition:Geometr... |
proofwiki-10025 | Product of Geometric Sequences from One | Let $Q_1 = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ and $Q_2 = \sequence {b_j}_{0 \mathop \le j \mathop \le n}$ be geometric sequences of integers of length $n + 1$.
Let $a_0 = b_0 = 1$.
Then the sequence $P = \sequence {p_j}_{0 \mathop \le j \mathop \le n}$ defined as:
:$\forall j \in \set {0, \ldots, n}: p_j ... | By Form of Geometric Sequence of Integers with Coprime Extremes, the $j$th term of $Q_1$ is given by:
:$a_j = a^j$
and of $Q_1$ is given by:
:$b_j = b^j$
Let the terms of $P$ be defined as:
:$\forall j \in \left\{{0, 1, \ldots, n}\right\}: p_j = b^j a^{n - j}$
Then from Form of Geometric Sequence of Integers it follows... | Let $Q_1 = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ and $Q_2 = \sequence {b_j}_{0 \mathop \le j \mathop \le n}$ be [[Definition:Geometric Sequence of Integers|geometric sequences of integers]] of [[Definition:Length of Sequence|length]] $n + 1$.
Let $a_0 = b_0 = 1$.
Then the [[Definition:Finite Sequence|seque... | By [[Form of Geometric Sequence of Integers with Coprime Extremes]], the $j$th [[Definition:Term of Geometric Sequence|term]] of $Q_1$ is given by:
:$a_j = a^j$
and of $Q_1$ is given by:
:$b_j = b^j$
Let the [[Definition:Term of Geometric Sequence|terms]] of $P$ be defined as:
:$\forall j \in \left\{{0, 1, \ldots, n}... | Product of Geometric Sequences from One | https://proofwiki.org/wiki/Product_of_Geometric_Sequences_from_One | https://proofwiki.org/wiki/Product_of_Geometric_Sequences_from_One | [
"Geometric Sequences of Integers"
] | [
"Definition:Geometric Sequence/Integers",
"Definition:Length of Sequence",
"Definition:Finite Sequence",
"Definition:Geometric Sequence"
] | [
"Form of Geometric Sequence of Integers with Coprime Extremes",
"Definition:Geometric Sequence/Term",
"Definition:Geometric Sequence/Term",
"Form of Geometric Sequence of Integers",
"Definition:Geometric Sequence"
] |
proofwiki-10026 | Powers of Elements of Geometric Sequence are in Geometric Sequence | Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers.
Then the sequence $Q = \sequence {b_j}_{0 \mathop \le j \mathop \le n}$ defined as:
:$\forall j \in \set {0, 1, \ldots, n}: b_j = a_j^m$
where $m \in \Z_{>0}$, is a geometric sequence.
{{:Euclid:Proposition/VIII/13}} | From Form of Geometric Sequence of Integers, the $j$th term of $P$ is given by:
:$a_j = k q^j p^{n - j}$
Thus the $j$th term of $Q$ is given by:
:$b_j = k^m \paren {q^m}^j \paren {p^m}^{n - j}$
From Form of Geometric Sequence of Integers, this is a geometric sequence.
{{qed}}
{{Euclid Note|13|VIII}} | Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]].
Then the [[Definition:Finite Sequence|sequence]] $Q = \sequence {b_j}_{0 \mathop \le j \mathop \le n}$ defined as:
:$\forall j \in \set {0, 1, \ldots, n}: b_j = a_j^m$
where $m \... | From [[Form of Geometric Sequence of Integers]], the $j$th [[Definition:Term of Geometric Sequence|term]] of $P$ is given by:
:$a_j = k q^j p^{n - j}$
Thus the $j$th [[Definition:Term of Geometric Sequence|term]] of $Q$ is given by:
:$b_j = k^m \paren {q^m}^j \paren {p^m}^{n - j}$
From [[Form of Geometric Sequence of... | Powers of Elements of Geometric Sequence are in Geometric Sequence | https://proofwiki.org/wiki/Powers_of_Elements_of_Geometric_Sequence_are_in_Geometric_Sequence | https://proofwiki.org/wiki/Powers_of_Elements_of_Geometric_Sequence_are_in_Geometric_Sequence | [
"Geometric Sequences of Integers"
] | [
"Definition:Geometric Sequence/Integers",
"Definition:Finite Sequence",
"Definition:Geometric Sequence/Integers"
] | [
"Form of Geometric Sequence of Integers",
"Definition:Geometric Sequence/Term",
"Definition:Geometric Sequence/Term",
"Form of Geometric Sequence of Integers",
"Definition:Geometric Sequence/Integers"
] |
proofwiki-10027 | Number divides Number iff Square divides Square | Let $a, b \in \Z$.
Then:
:$a^2 \divides b^2 \iff a \divides b$
where $\divides$ denotes integer divisibility.
{{:Euclid:Proposition/VIII/14}} | From Between two Squares exists one Mean Proportional:
:$\tuple {a^2, ab, b^2}$
is a geometric sequence.
Let $a, b \in \Z$ such that $a^2 \divides b^2$.
Then from First Element of Geometric Sequence that divides Last also divides Second:
:$a^2 \divides a b$
Thus:
{{begin-eqn}}
{{eqn | l = a^2
| o = \divides
... | Let $a, b \in \Z$.
Then:
:$a^2 \divides b^2 \iff a \divides b$
where $\divides$ denotes [[Definition:Divisor of Integer|integer divisibility]].
{{:Euclid:Proposition/VIII/14}} | From [[Between two Squares exists one Mean Proportional]]:
:$\tuple {a^2, ab, b^2}$
is a [[Definition:Geometric Sequence of Integers|geometric sequence]].
Let $a, b \in \Z$ such that $a^2 \divides b^2$.
Then from [[First Element of Geometric Sequence that divides Last also divides Second]]:
:$a^2 \divides a b$
Thus... | Number divides Number iff Square divides Square | https://proofwiki.org/wiki/Number_divides_Number_iff_Square_divides_Square | https://proofwiki.org/wiki/Number_divides_Number_iff_Square_divides_Square | [
"Divisors",
"Square Numbers"
] | [
"Definition:Divisor (Algebra)/Integer"
] | [
"Between two Squares exists one Mean Proportional",
"Definition:Geometric Sequence/Integers",
"First Element of Geometric Sequence that divides Last also divides Second"
] |
proofwiki-10028 | Number divides Number iff Cube divides Cube | Let $a, b \in \Z$.
Then:
:$a^3 \divides b^3 \iff a \divides b$
where $\divides$ denotes integer divisibility.
{{:Euclid:Proposition/VIII/15}} | Let $a^3$ and $b^3$ be cube numbers.
From {{Corollary|Form of Geometric Sequence of Integers}}:
:$\tuple {a^3, a^2 b, a b^2, b^3}$
is a geometric sequence.
Let $a, b \in \Z$ such that $a^2 \divides b^2$.
Then from First Element of Geometric Sequence that divides Last also divides Second:
:$a^3 \divides a^2 b$
Thus:
{{b... | Let $a, b \in \Z$.
Then:
:$a^3 \divides b^3 \iff a \divides b$
where $\divides$ denotes [[Definition:Divisor of Integer|integer divisibility]].
{{:Euclid:Proposition/VIII/15}} | Let $a^3$ and $b^3$ be [[Definition:Cube Number|cube numbers]].
From {{Corollary|Form of Geometric Sequence of Integers}}:
:$\tuple {a^3, a^2 b, a b^2, b^3}$
is a [[Definition:Geometric Sequence of Integers|geometric sequence]].
Let $a, b \in \Z$ such that $a^2 \divides b^2$.
Then from [[First Element of Geometric S... | Number divides Number iff Cube divides Cube | https://proofwiki.org/wiki/Number_divides_Number_iff_Cube_divides_Cube | https://proofwiki.org/wiki/Number_divides_Number_iff_Cube_divides_Cube | [
"Divisors",
"Cube Numbers"
] | [
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Cube Number",
"Definition:Geometric Sequence/Integers",
"First Element of Geometric Sequence that divides Last also divides Second",
"Divisibility of Elements in Geometric Sequence of Integers"
] |
proofwiki-10029 | Number does not divide Number iff Square does not divide Square | Let $a, b \in \Z$ be integers.
Then:
:$a \nmid b \iff a^2 \nmid b^2$
where $a \nmid b$ denotes that $a$ is not a divisor of $b$.
{{:Euclid:Proposition/VIII/16}} | Let $a \nmid b$.
{{AimForCont}}:
:$a^2 \divides b^2$
where $\divides$ denotes divisibility.
Then by Number divides Number iff Square divides Square:
:$a \divides b$
From Proof by Contradiction it follows that $a^2 \divides b^2$ is false.
Thus $a^2 \nmid b^2$.
{{qed|lemma}}
Let $a^2 \nmid b^2$.
{{AimForCont}}:
:$a \divi... | Let $a, b \in \Z$ be [[Definition:Integer|integers]].
Then:
:$a \nmid b \iff a^2 \nmid b^2$
where $a \nmid b$ denotes that $a$ is not a [[Definition:Divisor of Integer|divisor]] of $b$.
{{:Euclid:Proposition/VIII/16}} | Let $a \nmid b$.
{{AimForCont}}:
:$a^2 \divides b^2$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
Then by [[Number divides Number iff Square divides Square]]:
:$a \divides b$
From [[Proof by Contradiction]] it follows that $a^2 \divides b^2$ is [[Definition:False|false]].
Thus $a^2 \nmid... | Number does not divide Number iff Square does not divide Square | https://proofwiki.org/wiki/Number_does_not_divide_Number_iff_Square_does_not_divide_Square | https://proofwiki.org/wiki/Number_does_not_divide_Number_iff_Square_does_not_divide_Square | [
"Divisors",
"Square Numbers"
] | [
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Divisor (Algebra)/Integer",
"Number divides Number iff Square divides Square",
"Proof by Contradiction",
"Definition:False",
"Number divides Number iff Square divides Square",
"Proof by Contradiction",
"Definition:False"
] |
proofwiki-10030 | Number does not divide Number iff Cube does not divide Cube | Let $a, b \in \Z$ be integers.
Then:
:$a \nmid b \iff a^3 \nmid b^3$
where $a \nmid b$ denotes that $a$ is not a divisor of $b$.
{{:Euclid:Proposition/VIII/17}} | Let $a \nmid b$.
{{AimForCont}}:
:$a^3 \divides b^3$
where $\divides$ denotes divisibility.
Then by Number divides Number iff Cube divides Cube:
:$a \divides b$
From Proof by Contradiction it follows that $a^2 \divides b^2$ is false.
Thus $a^3 \nmid b^3$.
{{qed|lemma}}
Let $a^3 \nmid b^3$.
{{AimForCont}}
:$a \divides b... | Let $a, b \in \Z$ be [[Definition:Integer|integers]].
Then:
:$a \nmid b \iff a^3 \nmid b^3$
where $a \nmid b$ denotes that $a$ is not a [[Definition:Divisor of Integer|divisor]] of $b$.
{{:Euclid:Proposition/VIII/17}} | Let $a \nmid b$.
{{AimForCont}}:
:$a^3 \divides b^3$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
Then by [[Number divides Number iff Cube divides Cube]]:
:$a \divides b$
From [[Proof by Contradiction]] it follows that $a^2 \divides b^2$ is [[Definition:False|false]].
Thus $a^3 \nmid b^3... | Number does not divide Number iff Cube does not divide Cube | https://proofwiki.org/wiki/Number_does_not_divide_Number_iff_Cube_does_not_divide_Cube | https://proofwiki.org/wiki/Number_does_not_divide_Number_iff_Cube_does_not_divide_Cube | [
"Divisors",
"Cube Numbers"
] | [
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Divisor (Algebra)/Integer",
"Number divides Number iff Cube divides Cube",
"Proof by Contradiction",
"Definition:False",
"Number divides Number iff Cube divides Cube",
"Proof by Contradiction",
"Definition:False"
] |
proofwiki-10031 | Numbers between which exists one Mean Proportional are Similar Plane | Let $a, b \in \Z$ such that the geometric mean is an integer.
Then $a$ and $b$ are similar plane numbers.
{{:Euclid:Proposition/VIII/20}} | Let the geometric mean of $a$ and $b$ be an integer $m$.
Then, in the language of {{AuthorRef|Euclid}}, $m$ is a mean proportional of $a$ and $b$.
Thus:
:$\left({a, m, b}\right)$
is a geometric sequence.
From Form of Geometric Sequence of Integers:
:$\exists k, p, q \in \Z: a = k p^2, b = k q^2$
So $a$ and $b$ are plan... | Let $a, b \in \Z$ such that the [[Definition:Geometric Mean|geometric mean]] is an [[Definition:Integer|integer]].
Then $a$ and $b$ are [[Definition:Similar Plane Numbers|similar plane numbers]].
{{:Euclid:Proposition/VIII/20}} | Let the [[Definition:Geometric Mean|geometric mean]] of $a$ and $b$ be an [[Definition:Integer|integer]] $m$.
Then, in the language of {{AuthorRef|Euclid}}, $m$ is a [[Definition:Mean Proportional|mean proportional]] of $a$ and $b$.
Thus:
:$\left({a, m, b}\right)$
is a [[Definition:Geometric Sequence of Integers|geom... | Numbers between which exists one Mean Proportional are Similar Plane | https://proofwiki.org/wiki/Numbers_between_which_exists_one_Mean_Proportional_are_Similar_Plane | https://proofwiki.org/wiki/Numbers_between_which_exists_one_Mean_Proportional_are_Similar_Plane | [
"Euclidean Number Theory"
] | [
"Definition:Geometric Mean",
"Definition:Integer",
"Definition:Plane Number/Similar Numbers"
] | [
"Definition:Geometric Mean",
"Definition:Integer",
"Definition:Geometric Mean/Mean Proportional",
"Definition:Geometric Sequence/Integers",
"Form of Geometric Sequence of Integers",
"Definition:Plane Number",
"Definition:Plane Number/Similar Numbers"
] |
proofwiki-10032 | Numbers between which exist two Mean Proportionals are Similar Solid | Let $a, b \in \Z$ be the extremes of a geometric sequence of integers whose length is $4$:
:$\tuple {a, m_1, m_2, b}$
That is, such that $a$ and $b$ have $2$ mean proportionals.
Then $a$ and $b$ are similar solid numbers.
{{:Euclid:Proposition/VIII/21}} | From Form of Geometric Sequence of Integers:
:$\exists k, p, q \in \Z: a = k p^3, b = k q^3$
So $a$ and $b$ are solid numbers whose sides are:
:$k p$, $p$ and $p$
and
:$k q$, $q$ and $q$
respectively.
Then:
:$\dfrac {k p} {k q} = \dfrac p q$
demonstrating that $a$ and $b$ are similar solid numbers by definition.
{{qed}... | Let $a, b \in \Z$ be the [[Definition:Extremes of Finite Geometric Sequence|extremes]] of a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] whose [[Definition:Length of Sequence|length]] is $4$:
:$\tuple {a, m_1, m_2, b}$
That is, such that $a$ and $b$ have $2$ [[Definition:General Mean Pr... | From [[Form of Geometric Sequence of Integers]]:
:$\exists k, p, q \in \Z: a = k p^3, b = k q^3$
So $a$ and $b$ are [[Definition:Solid Number|solid numbers]] whose sides are:
:$k p$, $p$ and $p$
and
:$k q$, $q$ and $q$
respectively.
Then:
:$\dfrac {k p} {k q} = \dfrac p q$
demonstrating that $a$ and $b$ are [[Defini... | Numbers between which exist two Mean Proportionals are Similar Solid | https://proofwiki.org/wiki/Numbers_between_which_exist_two_Mean_Proportionals_are_Similar_Solid | https://proofwiki.org/wiki/Numbers_between_which_exist_two_Mean_Proportionals_are_Similar_Solid | [
"Euclidean Number Theory"
] | [
"Definition:Geometric Sequence/Finite/Extremes",
"Definition:Geometric Sequence/Integers",
"Definition:Length of Sequence",
"Definition:Geometric Mean/Mean Proportional/General Definition",
"Definition:Solid Number/Similar Numbers"
] | [
"Form of Geometric Sequence of Integers",
"Definition:Solid Number",
"Definition:Solid Number/Similar Numbers"
] |
proofwiki-10033 | If First of Three Numbers in Geometric Sequence is Square then Third is Square | Let $P = \tuple {a, b, c}$ be a geometric sequence of integers.
Let $a$ be a square number.
Then $c$ is also a square number.
{{:Euclid:Proposition/VIII/22}} | From Form of Geometric Sequence of Integers:
:$P = \tuple {k p^2, k p q, k q^2}$
for some $k, p, q \in \Z$.
If $a = k p^2$ is a square number it follows that $k$ is a square number: $k = r^2$, say.
So:
:$P = \tuple {r^2 p^2, r^2 p q, r^2 q^2}$
and so $c = r^2 q^2 = \paren {r q}^2$.
{{qed}}
{{Euclid Note|22|VIII}} | Let $P = \tuple {a, b, c}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]].
Let $a$ be a [[Definition:Square Number|square number]].
Then $c$ is also a [[Definition:Square Number|square number]].
{{:Euclid:Proposition/VIII/22}} | From [[Form of Geometric Sequence of Integers]]:
:$P = \tuple {k p^2, k p q, k q^2}$
for some $k, p, q \in \Z$.
If $a = k p^2$ is a [[Definition:Square Number|square number]] it follows that $k$ is a [[Definition:Square Number|square number]]: $k = r^2$, say.
So:
:$P = \tuple {r^2 p^2, r^2 p q, r^2 q^2}$
and so $c = ... | If First of Three Numbers in Geometric Sequence is Square then Third is Square | https://proofwiki.org/wiki/If_First_of_Three_Numbers_in_Geometric_Sequence_is_Square_then_Third_is_Square | https://proofwiki.org/wiki/If_First_of_Three_Numbers_in_Geometric_Sequence_is_Square_then_Third_is_Square | [
"Geometric Sequences of Integers"
] | [
"Definition:Geometric Sequence/Integers",
"Definition:Square Number",
"Definition:Square Number"
] | [
"Form of Geometric Sequence of Integers",
"Definition:Square Number",
"Definition:Square Number"
] |
proofwiki-10034 | If First of Four Numbers in Geometric Sequence is Cube then Fourth is Cube | Let $P = \tuple {a, b, c, d}$ be a geometric sequence of integers.
Let $a$ be a cube number.
Then $d$ is also a cube number.
{{:Euclid:Proposition/VIII/23}} | From Form of Geometric Sequence of Integers:
:$P = \tuple {k p^3, k p^2 q, k p q^2, k q^3}$
for some $k, p, q \in \Z$.
If $a = k p^3$ is a cube number it follows that $k$ is a cube number: $k = r^3$, say.
So:
:$P = \tuple {r^3 p^3, r^3 p^2 q, r^3 p q^2, r^3 q^3}$
and so $d = r^3 q^3 = \paren {r q}^3$.
{{qed}}
{{Euclid ... | Let $P = \tuple {a, b, c, d}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]].
Let $a$ be a [[Definition:Cube Number|cube number]].
Then $d$ is also a [[Definition:Cube Number|cube number]].
{{:Euclid:Proposition/VIII/23}} | From [[Form of Geometric Sequence of Integers]]:
:$P = \tuple {k p^3, k p^2 q, k p q^2, k q^3}$
for some $k, p, q \in \Z$.
If $a = k p^3$ is a [[Definition:Cube Number|cube number]] it follows that $k$ is a [[Definition:Cube Number|cube number]]: $k = r^3$, say.
So:
:$P = \tuple {r^3 p^3, r^3 p^2 q, r^3 p q^2, r^3 q^... | If First of Four Numbers in Geometric Sequence is Cube then Fourth is Cube | https://proofwiki.org/wiki/If_First_of_Four_Numbers_in_Geometric_Sequence_is_Cube_then_Fourth_is_Cube | https://proofwiki.org/wiki/If_First_of_Four_Numbers_in_Geometric_Sequence_is_Cube_then_Fourth_is_Cube | [
"Geometric Sequences of Integers",
"Cube Numbers"
] | [
"Definition:Geometric Sequence/Integers",
"Definition:Cube Number",
"Definition:Cube Number"
] | [
"Form of Geometric Sequence of Integers",
"Definition:Cube Number",
"Definition:Cube Number"
] |
proofwiki-10035 | If Ratio of Square to Number is as between Two Squares then Number is Square | Let $a, b, c, d \in \Z$ be integers such that:
:$\dfrac a b = \dfrac {c^2} {d^2}$
Let $a$ be a square number.
Then $b$ is also a square number.
{{:Euclid:Proposition/VIII/24}} | From {{EuclidPropLink|title = Between two Similar Plane Numbers exists one Mean Proportional|book = VIII|prop = 18}}:
:$\tuple {c^2, c d, d^2}$
is a geometric sequence.
From {{EuclidPropLink|title = Geometric Sequences in Proportion have Same Number of Elements|book = VIII|prop = 8}}:
:$\tuple {a, m, b}$
is a geometric... | Let $a, b, c, d \in \Z$ be [[Definition:Integer|integers]] such that:
:$\dfrac a b = \dfrac {c^2} {d^2}$
Let $a$ be a [[Definition:Square Number|square number]].
Then $b$ is also a [[Definition:Square Number|square number]].
{{:Euclid:Proposition/VIII/24}} | From {{EuclidPropLink|title = Between two Similar Plane Numbers exists one Mean Proportional|book = VIII|prop = 18}}:
:$\tuple {c^2, c d, d^2}$
is a [[Definition:Geometric Sequence of Integers|geometric sequence]].
From {{EuclidPropLink|title = Geometric Sequences in Proportion have Same Number of Elements|book = VIII... | If Ratio of Square to Number is as between Two Squares then Number is Square | https://proofwiki.org/wiki/If_Ratio_of_Square_to_Number_is_as_between_Two_Squares_then_Number_is_Square | https://proofwiki.org/wiki/If_Ratio_of_Square_to_Number_is_as_between_Two_Squares_then_Number_is_Square | [
"Ratios",
"Square Numbers"
] | [
"Definition:Integer",
"Definition:Square Number",
"Definition:Square Number"
] | [
"Definition:Geometric Sequence/Integers",
"Definition:Geometric Sequence/Integers",
"Definition:Square Number",
"Definition:Square Number"
] |
proofwiki-10036 | If Ratio of Cube to Number is as between Two Cubes then Number is Cube | Let $a, b, c, d \in \Z$ be integers such that:
:$\dfrac a b = \dfrac {c^3} {d^3}$
Let $a$ be a cube number.
Then $b$ is also a cube number.
{{:Euclid:Proposition/VIII/25}} | From {{EuclidPropLink|title = Between two Similar Solid Numbers exist two Mean Proportionals|book = VIII|prop = 19}}:
:$\left({c^3, c^2 d, c d^2, d^3}\right)$
is a geometric sequence.
From {{EuclidPropLink|title = Geometric Sequences in Proportion have Same Number of Elements|book = VIII|prop = 8}}:
:$\left({a, m_1, m_... | Let $a, b, c, d \in \Z$ be [[Definition:Integer|integers]] such that:
:$\dfrac a b = \dfrac {c^3} {d^3}$
Let $a$ be a [[Definition:Cube Number|cube number]].
Then $b$ is also a [[Definition:Cube Number|cube number]].
{{:Euclid:Proposition/VIII/25}} | From {{EuclidPropLink|title = Between two Similar Solid Numbers exist two Mean Proportionals|book = VIII|prop = 19}}:
:$\left({c^3, c^2 d, c d^2, d^3}\right)$
is a [[Definition:Geometric Sequence of Integers|geometric sequence]].
From {{EuclidPropLink|title = Geometric Sequences in Proportion have Same Number of Eleme... | If Ratio of Cube to Number is as between Two Cubes then Number is Cube | https://proofwiki.org/wiki/If_Ratio_of_Cube_to_Number_is_as_between_Two_Cubes_then_Number_is_Cube | https://proofwiki.org/wiki/If_Ratio_of_Cube_to_Number_is_as_between_Two_Cubes_then_Number_is_Cube | [
"Ratios",
"Cube Numbers"
] | [
"Definition:Integer",
"Definition:Cube Number",
"Definition:Cube Number"
] | [
"Definition:Geometric Sequence/Integers",
"Definition:Geometric Sequence/Integers",
"Definition:Cube Number",
"Definition:Cube Number"
] |
proofwiki-10037 | Square of Cube Number is Cube | Let $a \in \N$ be a natural number.
Let $a$ be a cube number.
Then $a^2$ is also a cube number.
{{:Euclid:Proposition/IX/3}} | By the definition of cube number:
:$\exists k \in \N: k^3 = a$
Thus:
{{begin-eqn}}
{{eqn | l = a^2
| r = \paren {k^3}^2
| c =
}}
{{eqn | r = k^6
| c =
}}
{{eqn | r = \paren {k^2}^3
| c =
}}
{{end-eqn}}
Thus:
:$\exists r = k^2 \in \N: a = r^3$
Hence the result by definition of cube number.
{{q... | Let $a \in \N$ be a [[Definition:Natural Number|natural number]].
Let $a$ be a [[Definition:Cube Number|cube number]].
Then $a^2$ is also a [[Definition:Cube Number|cube number]].
{{:Euclid:Proposition/IX/3}} | By the definition of [[Definition:Cube Number|cube number]]:
:$\exists k \in \N: k^3 = a$
Thus:
{{begin-eqn}}
{{eqn | l = a^2
| r = \paren {k^3}^2
| c =
}}
{{eqn | r = k^6
| c =
}}
{{eqn | r = \paren {k^2}^3
| c =
}}
{{end-eqn}}
Thus:
:$\exists r = k^2 \in \N: a = r^3$
Hence the result by ... | Square of Cube Number is Cube/Proof 1 | https://proofwiki.org/wiki/Square_of_Cube_Number_is_Cube | https://proofwiki.org/wiki/Square_of_Cube_Number_is_Cube/Proof_1 | [
"Euclidean Number Theory",
"Square of Cube Number is Cube"
] | [
"Definition:Natural Numbers",
"Definition:Cube Number",
"Definition:Cube Number"
] | [
"Definition:Cube Number",
"Definition:Cube Number"
] |
proofwiki-10038 | Square of Cube Number is Cube | Let $a \in \N$ be a natural number.
Let $a$ be a cube number.
Then $a^2$ is also a cube number.
{{:Euclid:Proposition/IX/3}} | From Cube Number multiplied by Cube Number is Cube, if $a$ and $b$ are cube numbers then $a b$ is a cube number.
The result follows by setting $b = a$.
{{qed}} | Let $a \in \N$ be a [[Definition:Natural Number|natural number]].
Let $a$ be a [[Definition:Cube Number|cube number]].
Then $a^2$ is also a [[Definition:Cube Number|cube number]].
{{:Euclid:Proposition/IX/3}} | From [[Cube Number multiplied by Cube Number is Cube]], if $a$ and $b$ are [[Definition:Cube Number|cube numbers]] then $a b$ is a [[Definition:Cube Number|cube number]].
The result follows by setting $b = a$.
{{qed}} | Square of Cube Number is Cube/Proof 2 | https://proofwiki.org/wiki/Square_of_Cube_Number_is_Cube | https://proofwiki.org/wiki/Square_of_Cube_Number_is_Cube/Proof_2 | [
"Euclidean Number Theory",
"Square of Cube Number is Cube"
] | [
"Definition:Natural Numbers",
"Definition:Cube Number",
"Definition:Cube Number"
] | [
"Cube Number multiplied by Cube Number is Cube",
"Definition:Cube Number",
"Definition:Cube Number"
] |
proofwiki-10039 | Cube Number multiplied by Cube Number is Cube | Let $a, b \in \N$ be natural numbers.
Let $a$ and $b$ be cube numbers.
Then $a b$ is also a cube number.
{{:Euclid:Proposition/IX/4}} | By the definition of cube number:
:$\exists r \in \N: r^3 = a$
:$\exists s \in \N: s^3 = b$
Thus:
{{begin-eqn}}
{{eqn | l = a b
| r = \paren {r^3} \paren {s^3}
| c = Power of Product
}}
{{eqn | r = \paren {r s}^3
| c =
}}
{{end-eqn}}
Thus:
:$\exists k = r s \in \N: a = k^3$
Hence the result by defini... | Let $a, b \in \N$ be [[Definition:Natural Number|natural numbers]].
Let $a$ and $b$ be [[Definition:Cube Number|cube numbers]].
Then $a b$ is also a [[Definition:Cube Number|cube number]].
{{:Euclid:Proposition/IX/4}} | By the definition of [[Definition:Cube Number|cube number]]:
:$\exists r \in \N: r^3 = a$
:$\exists s \in \N: s^3 = b$
Thus:
{{begin-eqn}}
{{eqn | l = a b
| r = \paren {r^3} \paren {s^3}
| c = [[Power of Product]]
}}
{{eqn | r = \paren {r s}^3
| c =
}}
{{end-eqn}}
Thus:
:$\exists k = r s \in \N: a ... | Cube Number multiplied by Cube Number is Cube | https://proofwiki.org/wiki/Cube_Number_multiplied_by_Cube_Number_is_Cube | https://proofwiki.org/wiki/Cube_Number_multiplied_by_Cube_Number_is_Cube | [
"Euclidean Number Theory",
"Cube Numbers"
] | [
"Definition:Natural Numbers",
"Definition:Cube Number",
"Definition:Cube Number"
] | [
"Definition:Cube Number",
"Exponent Combination Laws/Power of Product",
"Definition:Cube Number"
] |
proofwiki-10040 | Number Squared making Cube is itself Cube | Let $a \in \Z$ be an integer.
Let $a^2$ be a cube number.
Then $a$ is a cube number.
{{:Euclid:Proposition/IX/6}} | $a^2$ and $a^3$ are both cube numbers.
From Between two Cubes exist two Mean Proportionals the sequence:
:$a^3, m_1, m_2, a^2$
is a geometric sequence of integers for some $m_1, m_2 \in \Z$.
By Geometric Sequences in Proportion have Same Number of Elements:
:$a^2, m_3, m_4, a$
is a geometric sequence of integers for so... | Let $a \in \Z$ be an [[Definition:Integer|integer]].
Let $a^2$ be a [[Definition:Cube Number|cube number]].
Then $a$ is a [[Definition:Cube Number|cube number]].
{{:Euclid:Proposition/IX/6}} | $a^2$ and $a^3$ are both [[Definition:Cube Number|cube numbers]].
From [[Between two Cubes exist two Mean Proportionals]] the [[Definition:Finite Sequence|sequence]]:
:$a^3, m_1, m_2, a^2$
is a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] for some $m_1, m_2 \in \Z$.
By [[Geometric Sequ... | Number Squared making Cube is itself Cube | https://proofwiki.org/wiki/Number_Squared_making_Cube_is_itself_Cube | https://proofwiki.org/wiki/Number_Squared_making_Cube_is_itself_Cube | [
"Euclidean Number Theory"
] | [
"Definition:Integer",
"Definition:Cube Number",
"Definition:Cube Number"
] | [
"Definition:Cube Number",
"Between two Cubes exist two Mean Proportionals",
"Definition:Finite Sequence",
"Definition:Geometric Sequence/Integers",
"Geometric Sequences in Proportion have Same Number of Elements",
"Definition:Geometric Sequence/Integers",
"If First of Four Numbers in Geometric Sequence ... |
proofwiki-10041 | Product of Composite Number with Number is Solid Number | Let $a, b \in \Z$ be positive integers.
Let $a$ be a composite number.
Then $a b$ is a solid number.
{{:Euclid:Proposition/IX/7}} | By definition of composite number:
:$\exists p, q \in \Z_{>1}: a = p q$
Then:
:$a b = p q b$
Hence the result by definition of solid number.
{{qed}}
{{Euclid Note|7|IX}} | Let $a, b \in \Z$ be [[Definition:Positive Integer|positive integers]].
Let $a$ be a [[Definition:Composite Number|composite number]].
Then $a b$ is a [[Definition:Solid Number|solid number]].
{{:Euclid:Proposition/IX/7}} | By definition of [[Definition:Composite Number|composite number]]:
:$\exists p, q \in \Z_{>1}: a = p q$
Then:
:$a b = p q b$
Hence the result by definition of [[Definition:Solid Number|solid number]].
{{qed}}
{{Euclid Note|7|IX}} | Product of Composite Number with Number is Solid Number | https://proofwiki.org/wiki/Product_of_Composite_Number_with_Number_is_Solid_Number | https://proofwiki.org/wiki/Product_of_Composite_Number_with_Number_is_Solid_Number | [
"Euclidean Number Theory"
] | [
"Definition:Positive/Integer",
"Definition:Composite Number",
"Definition:Solid Number"
] | [
"Definition:Composite Number",
"Definition:Solid Number"
] |
proofwiki-10042 | Square Numbers are Similar Plane Numbers | Let $a$ and $b$ be square numbers.
Then $a$ and $b$ are similar plane numbers. | Let $a = c^2$ and $b = d^2$.
We have (trivially) that:
:$c : d = c : d$
The result follows by definition of similar plane numbers.
{{qed}}
Category:Square Numbers
Category:Euclidean Number Theory
04hgpyj8d33y1qzsa7zr8c6p917xs8x | Let $a$ and $b$ be [[Definition:Square Number|square numbers]].
Then $a$ and $b$ are [[Definition:Similar Plane Numbers|similar plane numbers]]. | Let $a = c^2$ and $b = d^2$.
We have (trivially) that:
:$c : d = c : d$
The result follows by definition of [[Definition:Similar Plane Numbers|similar plane numbers]].
{{qed}}
[[Category:Square Numbers]]
[[Category:Euclidean Number Theory]]
04hgpyj8d33y1qzsa7zr8c6p917xs8x | Square Numbers are Similar Plane Numbers | https://proofwiki.org/wiki/Square_Numbers_are_Similar_Plane_Numbers | https://proofwiki.org/wiki/Square_Numbers_are_Similar_Plane_Numbers | [
"Square Numbers",
"Euclidean Number Theory"
] | [
"Definition:Square Number",
"Definition:Plane Number/Similar Numbers"
] | [
"Definition:Plane Number/Similar Numbers",
"Category:Square Numbers",
"Category:Euclidean Number Theory"
] |
proofwiki-10043 | Elements of Geometric Sequence from One which are Powers of Number | Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a geometric sequence of integers.
Let $a_0 = 1$.
Then:
:$\forall m, k \in \set {1, \ldots, n}: k \divides m \implies a_m$ is a power of $k$
where $\divides$ denotes divisibility.
{{:Euclid:Proposition/IX/8}} | By Form of Geometric Sequence of Integers from One, the general term of $G_n$ can be expressed as:
:$a_j = q^j$
for some $q \in \Z$.
Let $k, m \in \set {1, 2, \ldots, n}$ such that $k \divides m$.
By definition of divisibility:
:$\exists r \in \Z: m = r k$
Then:
{{begin-eqn}}
{{eqn | l = a_m
| r = q^m
| c =... | Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]].
Let $a_0 = 1$.
Then:
:$\forall m, k \in \set {1, \ldots, n}: k \divides m \implies a_m$ is a [[Definition:Integer Power|power]] of $k$
where $\divides$ denotes [[Definition:Di... | By [[Form of Geometric Sequence of Integers from One]], the general [[Definition:Term of Geometric Sequence|term]] of $G_n$ can be expressed as:
:$a_j = q^j$
for some $q \in \Z$.
Let $k, m \in \set {1, 2, \ldots, n}$ such that $k \divides m$.
By definition of [[Definition:Divisor of Integer|divisibility]]:
:$\exists... | Elements of Geometric Sequence from One which are Powers of Number | https://proofwiki.org/wiki/Elements_of_Geometric_Sequence_from_One_which_are_Powers_of_Number | https://proofwiki.org/wiki/Elements_of_Geometric_Sequence_from_One_which_are_Powers_of_Number | [
"Geometric Sequences of Integers"
] | [
"Definition:Geometric Sequence/Integers",
"Definition:Power (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Form of Geometric Sequence of Integers from One",
"Definition:Geometric Sequence/Term",
"Definition:Divisor (Algebra)/Integer",
"Definition:Power (Algebra)/Integer"
] |
proofwiki-10044 | Elements of Geometric Sequence from One where First Element is Power of Number | Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a geometric sequence of integers.
Let $a_0 = 1$.
Let $m \in \Z_{> 0}$.
Let $a_1$ be the $m$th power of an integer.
Then all the terms of $G_n$ are $m$th powers of integers.
{{:Euclid:Proposition/IX/9}} | By Form of Geometric Sequence of Integers from One, the general term of $G_n$ can be expressed as:
:$a_j = q^j$
for some $q \in \Z$.
Let $a_2 = k^m$.
By definition of geometric sequence:
:$\forall j \in \set {1, 2, \ldots, n}: a_j = r a_{j - 1}$
where $r$ is the common ratio.
This holds specifically for $j = 1$:
:$k^m ... | Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]].
Let $a_0 = 1$.
Let $m \in \Z_{> 0}$.
Let $a_1$ be the $m$th [[Definition:Integer Power|power]] of an [[Definition:Integer|integer]].
Then all the [[Definition:Term of Geome... | By [[Form of Geometric Sequence of Integers from One]], the general [[Definition:Term of Geometric Sequence|term]] of $G_n$ can be expressed as:
:$a_j = q^j$
for some $q \in \Z$.
Let $a_2 = k^m$.
By definition of [[Definition:Geometric Sequence|geometric sequence]]:
:$\forall j \in \set {1, 2, \ldots, n}: a_j = r a_... | Elements of Geometric Sequence from One where First Element is Power of Number | https://proofwiki.org/wiki/Elements_of_Geometric_Sequence_from_One_where_First_Element_is_Power_of_Number | https://proofwiki.org/wiki/Elements_of_Geometric_Sequence_from_One_where_First_Element_is_Power_of_Number | [
"Geometric Sequences"
] | [
"Definition:Geometric Sequence/Integers",
"Definition:Power (Algebra)/Integer",
"Definition:Integer",
"Definition:Geometric Sequence/Term",
"Definition:Power (Algebra)/Integer",
"Definition:Integer"
] | [
"Form of Geometric Sequence of Integers from One",
"Definition:Geometric Sequence/Term",
"Definition:Geometric Sequence",
"Definition:Geometric Sequence/Common Ratio"
] |
proofwiki-10045 | Elements of Geometric Sequence from One where First Element is not Power of Number | Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a geometric sequence of integers.
Let $a_0 = 1$.
Let $k \in \Z_{> 1}$.
Let $a_1$ not be a power of $k$.
Then $a_m$ is not a power of $k$ except for:
:$\forall m, k \in \set {1, 2, \ldots, n}: k \divides m$
where $\divides$ denotes divisibility.
{{:Euclid:Pr... | By Form of Geometric Sequence of Integers from One, the general term of $G_n$ can be expressed as:
:$a_j = q^j$
for some $q \in \Z$.
Let $k \nmid m$.
Then by the Division Theorem there exists a unique $q \in \Z$ such that:
:$m = k q + b$
for some $b$ such that $0 < b < k$.
Thus:
:$a_m = a^{k q} a^b$
which is not a powe... | Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]].
Let $a_0 = 1$.
Let $k \in \Z_{> 1}$.
Let $a_1$ not be a [[Definition:Integer Power|power]] of $k$.
Then $a_m$ is not a [[Definition:Integer Power|power]] of $k$ except for:... | By [[Form of Geometric Sequence of Integers from One]], the general [[Definition:Term of Geometric Sequence|term]] of $G_n$ can be expressed as:
:$a_j = q^j$
for some $q \in \Z$.
Let $k \nmid m$.
Then by the [[Division Theorem]] there exists a unique $q \in \Z$ such that:
:$m = k q + b$
for some $b$ such that $0 < b... | Elements of Geometric Sequence from One where First Element is not Power of Number | https://proofwiki.org/wiki/Elements_of_Geometric_Sequence_from_One_where_First_Element_is_not_Power_of_Number | https://proofwiki.org/wiki/Elements_of_Geometric_Sequence_from_One_where_First_Element_is_not_Power_of_Number | [
"Geometric Sequences"
] | [
"Definition:Geometric Sequence/Integers",
"Definition:Power (Algebra)/Integer",
"Definition:Power (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Form of Geometric Sequence of Integers from One",
"Definition:Geometric Sequence/Term",
"Division Theorem",
"Definition:Power (Algebra)/Integer"
] |
proofwiki-10046 | Elements of Geometric Sequence from One which Divide Later Elements | Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a geometric sequence of integers.
Let $a_0 = 1$.
Let $m \in \Z_{> 0}$.
Then:
:$\forall r \in \set {0, 1, \ldots, m}: a_k \divides a_m$
where $\divides$ denotes divisibility.
{{:Euclid:Proposition/IX/11}} | By Form of Geometric Sequence of Integers from One, the general term of $G_n$ can be expressed as:
:$a_j = q^j$
for some $q \in \Z$.
Hence the result from Divisors of Power of Prime.
{{qed}}
{{Euclid Note|11|IX}} | Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]].
Let $a_0 = 1$.
Let $m \in \Z_{> 0}$.
Then:
:$\forall r \in \set {0, 1, \ldots, m}: a_k \divides a_m$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].... | By [[Form of Geometric Sequence of Integers from One]], the general [[Definition:Term of Geometric Sequence|term]] of $G_n$ can be expressed as:
:$a_j = q^j$
for some $q \in \Z$.
Hence the result from [[Divisors of Power of Prime]].
{{qed}}
{{Euclid Note|11|IX}} | Elements of Geometric Sequence from One which Divide Later Elements | https://proofwiki.org/wiki/Elements_of_Geometric_Sequence_from_One_which_Divide_Later_Elements | https://proofwiki.org/wiki/Elements_of_Geometric_Sequence_from_One_which_Divide_Later_Elements | [
"Geometric Sequences"
] | [
"Definition:Geometric Sequence/Integers",
"Definition:Divisor (Algebra)/Integer"
] | [
"Form of Geometric Sequence of Integers from One",
"Definition:Geometric Sequence/Term",
"Divisors of Power of Prime"
] |
proofwiki-10047 | Elements of Geometric Sequence from One Divisible by Prime | Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a geometric sequence of integers.
Let $a_0 = 1$.
Let $p$ be a prime number such that:
:$p \divides a_n$
where $\divides$ denotes divisibility.
Then $p \divides a_1$.
{{:Euclid:Proposition/IX/12}} | By Form of Geometric Sequence of Integers from One, the general term of $G_n$ can be expressed as:
:$a_j = q^j$
for some $q \in \Z$.
Thus by hypothesis:
:$p \divides q^n$
From Euclid's Lemma for Prime Divisors: General Result:
:$p \divides q$
Hence the result.
{{qed}}
{{Euclid Note|12|IX}} | Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]].
Let $a_0 = 1$.
Let $p$ be a [[Definition:Prime Number|prime number]] such that:
:$p \divides a_n$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
Th... | By [[Form of Geometric Sequence of Integers from One]], the general [[Definition:Term of Geometric Sequence|term]] of $G_n$ can be expressed as:
:$a_j = q^j$
for some $q \in \Z$.
Thus [[Definition:By Hypothesis|by hypothesis]]:
:$p \divides q^n$
From [[Euclid's Lemma for Prime Divisors/General Result|Euclid's Lemma f... | Elements of Geometric Sequence from One Divisible by Prime | https://proofwiki.org/wiki/Elements_of_Geometric_Sequence_from_One_Divisible_by_Prime | https://proofwiki.org/wiki/Elements_of_Geometric_Sequence_from_One_Divisible_by_Prime | [
"Geometric Sequences"
] | [
"Definition:Geometric Sequence/Integers",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer"
] | [
"Form of Geometric Sequence of Integers from One",
"Definition:Geometric Sequence/Term",
"Definition:By Hypothesis",
"Euclid's Lemma for Prime Divisors/General Result"
] |
proofwiki-10048 | Divisibility of Elements of Geometric Sequence from One where First Element is Prime | Let $Q_n = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of length $n$ consisting of integers only.
Let $a_0 = 1$.
Let $a_1$ be a prime number.
Then the only divisors of $a_n$ are $a_j$ for $j \in \set {1, 2, \ldots, n}$.
{{:Euclid:Proposition/IX/13}} | From Form of Geometric Sequence of Integers from One, the elements of $Q_n$ are given by:
:$Q_n = \tuple {1, a, a^2, \ldots, a^n}$
From Divisors of Power of Prime, each of $a_j$ for $j \in \set {1, 2, \ldots, n}$ are the only divisors of $a_n$.
{{qed}}
{{Euclid Note|13|IX}} | Let $Q_n = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a [[Definition:Geometric Sequence|geometric sequence]] of [[Definition:Length of Sequence|length]] $n$ consisting of [[Definition:Integer|integers]] only.
Let $a_0 = 1$.
Let $a_1$ be a [[Definition:Prime Number|prime number]].
Then the only [[Definition... | From [[Form of Geometric Sequence of Integers from One]], the elements of $Q_n$ are given by:
:$Q_n = \tuple {1, a, a^2, \ldots, a^n}$
From [[Divisors of Power of Prime]], each of $a_j$ for $j \in \set {1, 2, \ldots, n}$ are the only [[Definition:Divisor of Integer|divisors]] of $a_n$.
{{qed}}
{{Euclid Note|13|IX}} | Divisibility of Elements of Geometric Sequence from One where First Element is Prime | https://proofwiki.org/wiki/Divisibility_of_Elements_of_Geometric_Sequence_from_One_where_First_Element_is_Prime | https://proofwiki.org/wiki/Divisibility_of_Elements_of_Geometric_Sequence_from_One_where_First_Element_is_Prime | [
"Geometric Sequences of Integers"
] | [
"Definition:Geometric Sequence",
"Definition:Length of Sequence",
"Definition:Integer",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer"
] | [
"Form of Geometric Sequence of Integers from One",
"Divisors of Power of Prime",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-10049 | Sum of Pair of Elements of Geometric Sequence with Three Elements in Lowest Terms is Coprime to other Element | Let $P = \tuple {a, b, c}$ be a geometric sequence of integers in its lowest terms.
Then $\paren {a + b}$, $\paren {b + c}$ and $\paren {a + c}$ are all coprime to each of $a$, $b$ and $c$.
{{:Euclid:Proposition/IX/15}} | Let the common ratio of $P$ in canonical form be $\dfrac q p$.
By Form of Geometric Sequence of Integers in Lowest Terms:
:$P = \tuple {p^2, p q, q^2}$
Then:
{{begin-eqn}}
{{eqn | l = p
| o = \perp
| r = q
| c = {{Defof|Canonical Form of Rational Number}}
}}
{{eqn | ll= \leadsto
| l = q
| ... | Let $P = \tuple {a, b, c}$ be a [[Definition:Geometric Sequence of Integers in Lowest Terms|geometric sequence of integers in its lowest terms]].
Then $\paren {a + b}$, $\paren {b + c}$ and $\paren {a + c}$ are all [[Definition:Coprime Integers|coprime]] to each of $a$, $b$ and $c$.
{{:Euclid:Proposition/IX/15}} | Let the [[Definition:Common Ratio|common ratio]] of $P$ in [[Definition:Canonical Form of Rational Number|canonical form]] be $\dfrac q p$.
By [[Form of Geometric Sequence of Integers in Lowest Terms]]:
:$P = \tuple {p^2, p q, q^2}$
Then:
{{begin-eqn}}
{{eqn | l = p
| o = \perp
| r = q
| c = {{Defof... | Sum of Pair of Elements of Geometric Sequence with Three Elements in Lowest Terms is Coprime to other Element | https://proofwiki.org/wiki/Sum_of_Pair_of_Elements_of_Geometric_Sequence_with_Three_Elements_in_Lowest_Terms_is_Coprime_to_other_Element | https://proofwiki.org/wiki/Sum_of_Pair_of_Elements_of_Geometric_Sequence_with_Three_Elements_in_Lowest_Terms_is_Coprime_to_other_Element | [
"Geometric Sequences"
] | [
"Definition:Geometric Sequence of Integers in Lowest Terms",
"Definition:Coprime/Integers"
] | [
"Definition:Geometric Sequence/Common Ratio",
"Definition:Rational Number/Canonical Form",
"Form of Geometric Sequence of Integers in Lowest Terms",
"Numbers are Coprime iff Sum is Coprime to Both",
"Integer Coprime to Factors is Coprime to Whole",
"Square of Coprime Number is Coprime",
"Numbers are Cop... |
proofwiki-10050 | Two Coprime Integers have no Third Integer Proportional | Let $a, b \in \Z_{>0}$ be integers such that $a$ and $b$ are coprime.
Then there is no integer $c \in \Z$ such that:
:$\dfrac a b = \dfrac b c$
{{:Euclid:Proposition/IX/16}} | Suppose such a $c$ exists.
From Coprime Numbers form Fraction in Lowest Terms, $\dfrac a b$ is in canonical form.
From Ratios of Fractions in Lowest Terms:
:$a \divides b$
where $\divides$ denotes divisibility.
This contradicts the fact that $a$ and $b$ are coprime.
Hence such a $c$ cannot exist.
{{qed}}
{{Euclid Note|... | Let $a, b \in \Z_{>0}$ be [[Definition:Integer|integers]] such that $a$ and $b$ are [[Definition:Coprime Integers|coprime]].
Then there is no [[Definition:Integer|integer]] $c \in \Z$ such that:
:$\dfrac a b = \dfrac b c$
{{:Euclid:Proposition/IX/16}} | Suppose such a $c$ exists.
From [[Coprime Numbers form Fraction in Lowest Terms]], $\dfrac a b$ is in [[Definition:Canonical Form of Rational Number|canonical form]].
From [[Ratios of Fractions in Lowest Terms]]:
:$a \divides b$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
This contradict... | Two Coprime Integers have no Third Integer Proportional | https://proofwiki.org/wiki/Two_Coprime_Integers_have_no_Third_Integer_Proportional | https://proofwiki.org/wiki/Two_Coprime_Integers_have_no_Third_Integer_Proportional | [
"Coprime Integers",
"Ratios"
] | [
"Definition:Integer",
"Definition:Coprime/Integers",
"Definition:Integer"
] | [
"Coprime Numbers form Fraction in Lowest Terms",
"Definition:Rational Number/Canonical Form",
"Ratios of Fractions in Lowest Terms",
"Definition:Divisor (Algebra)/Integer",
"Definition:Coprime/Integers"
] |
proofwiki-10051 | Last Element of Geometric Sequence with Coprime Extremes has no Integer Proportional as First to Second | Let $G_n = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers such that $a_0 \ne 1$.
Let $a_0 \perp a_n$, where $\perp$ denotes coprimality.
Then there does not exist an integer $b$ such that:
:$\dfrac {a_0} {a_1} = \dfrac {a_n} b$
{{:Euclid:Proposition/IX/17}} | {{AimForCont}} there exists $b$ such that $\dfrac {a_0} {a_1} = \dfrac {a_n} b$.
Then:
:$\dfrac {a_0} {a_n} = \dfrac {a_1} b$
By Ratios of Fractions in Lowest Terms:
:$a_0 \divides a_1$
where $\divides$ denotes divisibility.
From Divisibility of Elements in Geometric Sequence of Integers:
:$a_0 \divides a_n$
But $a_0 \... | Let $G_n = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] such that $a_0 \ne 1$.
Let $a_0 \perp a_n$, where $\perp$ denotes [[Definition:Coprime Integers|coprimality]].
Then there does not exist an [[Definition:Integer|integer]] $b$ s... | {{AimForCont}} there exists $b$ such that $\dfrac {a_0} {a_1} = \dfrac {a_n} b$.
Then:
:$\dfrac {a_0} {a_n} = \dfrac {a_1} b$
By [[Ratios of Fractions in Lowest Terms]]:
:$a_0 \divides a_1$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
From [[Divisibility of Elements in Geometric Sequence ... | Last Element of Geometric Sequence with Coprime Extremes has no Integer Proportional as First to Second | https://proofwiki.org/wiki/Last_Element_of_Geometric_Sequence_with_Coprime_Extremes_has_no_Integer_Proportional_as_First_to_Second | https://proofwiki.org/wiki/Last_Element_of_Geometric_Sequence_with_Coprime_Extremes_has_no_Integer_Proportional_as_First_to_Second | [
"Geometric Sequences"
] | [
"Definition:Geometric Sequence/Integers",
"Definition:Coprime/Integers",
"Definition:Integer"
] | [
"Ratios of Fractions in Lowest Terms",
"Definition:Divisor (Algebra)/Integer",
"Divisibility of Elements in Geometric Sequence of Integers",
"Proof by Contradiction"
] |
proofwiki-10052 | Condition for Existence of Third Number Proportional to Two Numbers | Let $a, b, c \in \Z$ be integers.
Let $\tuple {a, b, c}$ be a geometric sequence.
In order for this to be possible, both of these conditions must be true:
:$(1): \quad a$ and $b$ cannot be coprime
:$(2): \quad a \divides b^2$
where $\divides$ denotes divisibility.
{{:Euclid:Proposition/IX/18}} | Let $P = \tuple {a, b, c}$ be a geometric sequence.
Then by definition their common ratio is:
:$\dfrac b a = \dfrac c b$
From Two Coprime Integers have no Third Integer Proportional it cannot be the case that $a$ and $b$ are coprime.
Thus condition $(1)$ is satisfied.
From Form of Geometric Sequence of Integers, $P$ is... | Let $a, b, c \in \Z$ be [[Definition:Integer|integers]].
Let $\tuple {a, b, c}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence]].
In order for this to be possible, both of these conditions must be true:
:$(1): \quad a$ and $b$ cannot be [[Definition:Coprime Integers|coprime]]
:$(2): \quad a \divi... | Let $P = \tuple {a, b, c}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence]].
Then by definition their [[Definition:Common Ratio|common ratio]] is:
:$\dfrac b a = \dfrac c b$
From [[Two Coprime Integers have no Third Integer Proportional]] it cannot be the case that $a$ and $b$ are [[Definition:Co... | Condition for Existence of Third Number Proportional to Two Numbers | https://proofwiki.org/wiki/Condition_for_Existence_of_Third_Number_Proportional_to_Two_Numbers | https://proofwiki.org/wiki/Condition_for_Existence_of_Third_Number_Proportional_to_Two_Numbers | [
"Proportion"
] | [
"Definition:Integer",
"Definition:Geometric Sequence/Integers",
"Definition:Coprime/Integers",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Geometric Sequence/Integers",
"Definition:Geometric Sequence/Common Ratio",
"Two Coprime Integers have no Third Integer Proportional",
"Definition:Coprime/Integers",
"Form of Geometric Sequence of Integers"
] |
proofwiki-10053 | Odd Divisor of Even Number also divides its Half | Let $a, b \in \Z$ be integers.
Let $a$ be odd and $b$ be even.
Let:
:$a \divides b$
where $\divides$ denotes divisibility.
Then:
:$a \divides \dfrac b 2$
{{:Euclid:Proposition/IX/30}} | By definition of an even number:
:$\exists r \in \Z: b = 2 r$
By definition of an odd number:
:$\exists s \in \Z: a = 2 s + 1$
Thus:
:$a \divides 2 \iff a = \pm 1$
in which case from One Divides all Integers:
:$a \divides \dfrac b 2$
We have that:
:$a \divides 2 r$
and so by Euclid's Lemma:
:$a \divides r = \dfrac b 2$... | Let $a, b \in \Z$ be [[Definition:Integer|integers]].
Let $a$ be [[Definition:Odd Integer|odd]] and $b$ be [[Definition:Even Integer|even]].
Let:
:$a \divides b$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
Then:
:$a \divides \dfrac b 2$
{{:Euclid:Proposition/IX/30}} | By definition of an [[Definition:Even Number|even number]]:
:$\exists r \in \Z: b = 2 r$
By definition of an [[Definition:Odd Number|odd number]]:
:$\exists s \in \Z: a = 2 s + 1$
Thus:
:$a \divides 2 \iff a = \pm 1$
in which case from [[One Divides all Integers]]:
:$a \divides \dfrac b 2$
We have that:
:$a \divides... | Odd Divisor of Even Number also divides its Half | https://proofwiki.org/wiki/Odd_Divisor_of_Even_Number_also_divides_its_Half | https://proofwiki.org/wiki/Odd_Divisor_of_Even_Number_also_divides_its_Half | [
"Euclidean Number Theory"
] | [
"Definition:Integer",
"Definition:Odd Integer",
"Definition:Even Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Even Integer",
"Definition:Odd Integer",
"Integer Divisor Results/One Divides all Integers",
"Euclid's Lemma"
] |
proofwiki-10054 | Odd Number Coprime to Number is also Coprime to its Double | Let $a, b \in \Z$ be integers.
Let $a$ be odd.
Let:
:$a \perp b$
where $\perp$ denotes coprimality.
Then:
:$a \perp 2 b$
{{:Euclid:Proposition/IX/31}} | By definition of odd number:
:$a \perp 2$
The result follows from Integer Coprime to Factors is Coprime to Whole.
{{qed}}
{{Euclid Note|31|IX}} | Let $a, b \in \Z$ be [[Definition:Integer|integers]].
Let $a$ be [[Definition:Odd Integer|odd]].
Let:
:$a \perp b$
where $\perp$ denotes [[Definition:Coprime Integers|coprimality]].
Then:
:$a \perp 2 b$
{{:Euclid:Proposition/IX/31}} | By definition of [[Definition:Odd Integer|odd number]]:
:$a \perp 2$
The result follows from [[Integer Coprime to Factors is Coprime to Whole]].
{{qed}}
{{Euclid Note|31|IX}} | Odd Number Coprime to Number is also Coprime to its Double | https://proofwiki.org/wiki/Odd_Number_Coprime_to_Number_is_also_Coprime_to_its_Double | https://proofwiki.org/wiki/Odd_Number_Coprime_to_Number_is_also_Coprime_to_its_Double | [
"Euclidean Number Theory"
] | [
"Definition:Integer",
"Definition:Odd Integer",
"Definition:Coprime/Integers"
] | [
"Definition:Odd Integer",
"Integer Coprime to Factors is Coprime to Whole"
] |
proofwiki-10055 | Power of Two is Even-Times Even Only | Let $a > 2$ be a power of $2$.
Then $a$ is even-times even only.
{{:Euclid:Proposition/IX/32}} | As $a$ is a power of $2$ greater than $2$:
:$\exists k \in \Z_{>1}: a = 2^k$
Thus:
:$a = 2^2 2^{k - 2}$
and so has $2^2 = 4$ as a divisor.
Let $b$ be an odd number.
By definition:
:$b \nmid 2$
The result follows by Integer Coprime to Factors is Coprime to Whole.
{{qed}}
{{Euclid Note|32|IX}} | Let $a > 2$ be a [[Definition:Integer Power|power]] of $2$.
Then $a$ is [[Definition:Even-Times Even Integer|even-times even]] only.
{{:Euclid:Proposition/IX/32}} | As $a$ is a [[Definition:Integer Power|power]] of $2$ greater than $2$:
:$\exists k \in \Z_{>1}: a = 2^k$
Thus:
:$a = 2^2 2^{k - 2}$
and so has $2^2 = 4$ as a [[Definition:Divisor of Integer|divisor]].
Let $b$ be an [[Definition:Odd Number|odd number]].
By definition:
:$b \nmid 2$
The result follows by [[Integer ... | Power of Two is Even-Times Even Only | https://proofwiki.org/wiki/Power_of_Two_is_Even-Times_Even_Only | https://proofwiki.org/wiki/Power_of_Two_is_Even-Times_Even_Only | [
"Euclidean Number Theory"
] | [
"Definition:Power (Algebra)/Integer",
"Definition:Even Integer/Even-Times Even"
] | [
"Definition:Power (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Odd Integer",
"Integer Coprime to Factors is Coprime to Whole"
] |
proofwiki-10056 | Number whose Half is Odd is Even-Times Odd | Let $a \in \Z$ be an integer such that $\dfrac a 2$ is an odd integer.
Then $a$ is even-times odd.
{{:Euclid:Proposition/IX/33}} | By definition:
:$a = 2 r$
where $r$ is an odd integer.
Thus:
:$a$ has an even divisor
and:
:$a$ has an odd divisor.
Hence the result by definition of even-times odd integer.
As $r$ is an odd integer it follows that $2 \nmid r$.
Thus $a$ is not divisible by $4$.
Hence $a$ is not even-times even.
{{qed}}
{{Euclid Note|33... | Let $a \in \Z$ be an [[Definition:Integer|integer]] such that $\dfrac a 2$ is an [[Definition:Odd Integer|odd integer]].
Then $a$ is [[Definition:Even-Times Odd Integer|even-times odd]].
{{:Euclid:Proposition/IX/33}} | By definition:
:$a = 2 r$
where $r$ is an [[Definition:Odd Integer|odd integer]].
Thus:
:$a$ has an [[Definition:Even Integer|even]] [[Definition:Divisor of Integer|divisor]]
and:
:$a$ has an [[Definition:Odd Integer|odd]] [[Definition:Divisor of Integer|divisor]].
Hence the result by definition of [[Definition:Even-... | Number whose Half is Odd is Even-Times Odd | https://proofwiki.org/wiki/Number_whose_Half_is_Odd_is_Even-Times_Odd | https://proofwiki.org/wiki/Number_whose_Half_is_Odd_is_Even-Times_Odd | [
"Euclidean Number Theory"
] | [
"Definition:Integer",
"Definition:Odd Integer",
"Definition:Even Integer/Even-Times Odd"
] | [
"Definition:Odd Integer",
"Definition:Even Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Odd Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Even Integer/Even-Times Odd",
"Definition:Odd Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Even Integer/Even-Tim... |
proofwiki-10057 | Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd | Let $a \in \Z$ be an integer such that:
:$(1): \quad a$ is not a power of $2$
:$(2): \quad \dfrac a 2$ is an even integer.
Then $a$ is both even-times even and even-times odd.
{{:Euclid:Proposition/IX/34}} | As $\dfrac a 2$ is an even integer it follows that:
:$\dfrac a 2 = 2 r$
for some $r \in \Z$.
That is:
:$a = 2^2 r$
and so $a$ is even-times even by definition.
{{qed|lemma}}
{{AimForCont}} $a$ is not even-times odd.
Then $a$ does not have an odd divisor.
Thus in its prime decomposition there are no odd primes.
Thus $a$... | Let $a \in \Z$ be an [[Definition:Integer|integer]] such that:
:$(1): \quad a$ is not a [[Definition:Integer Power|power]] of $2$
:$(2): \quad \dfrac a 2$ is an [[Definition:Even Integer|even integer]].
Then $a$ is both [[Definition:Even-Times Even Integer|even-times even]] and [[Definition:Even-Times Odd Integer|eve... | As $\dfrac a 2$ is an [[Definition:Even Integer|even integer]] it follows that:
:$\dfrac a 2 = 2 r$
for some $r \in \Z$.
That is:
:$a = 2^2 r$
and so $a$ is [[Definition:Even-Times Even Integer|even-times even]] by definition.
{{qed|lemma}}
{{AimForCont}} $a$ is not [[Definition:Even-Times Odd Integer|even-times odd... | Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd | https://proofwiki.org/wiki/Number_neither_whose_Half_is_Odd_nor_Power_of_Two_is_both_Even-Times_Even_and_Even-Times_Odd | https://proofwiki.org/wiki/Number_neither_whose_Half_is_Odd_nor_Power_of_Two_is_both_Even-Times_Even_and_Even-Times_Odd | [
"Euclidean Number Theory"
] | [
"Definition:Integer",
"Definition:Power (Algebra)/Integer",
"Definition:Even Integer",
"Definition:Even Integer/Even-Times Even",
"Definition:Even Integer/Even-Times Odd"
] | [
"Definition:Even Integer",
"Definition:Even Integer/Even-Times Even",
"Definition:Even Integer/Even-Times Odd",
"Definition:Odd Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Prime Decomposition",
"Definition:Odd Prime",
"Definition:Power (Algebra)/Integer",
"Proof by Contradiction",
... |
proofwiki-10058 | Scaled Real Function that Increases Without Bound | Let $f: \R \to \R$ be a real function.
Let $\lambda \in \R_{\ne 0}$ be a nonzero constant.
Then:
For $\lambda > 0$:
:$\ds \lim_{x \mathop \to +\infty} \map f x = +\infty \implies \lim_{x \mathop \to +\infty} \lambda \map f x = +\infty$
:$\ds \lim_{x \mathop \to -\infty} \map f x = +\infty \implies \lim_{x \mathop \to -... | Let $\ds \lim_{x \mathop \to +\infty} \map f x = +\infty$.
From the definition of infinite limit at infinity, this means that:
:$\forall M > 0: \exists N > 0: x > N \implies \map f x > M$.
Suppose $\lambda > 0$.
Then $M > 0 \iff \lambda^{-1} M > 0$.
Also, $\map f x > \lambda^{-1} M \iff \lambda f\map f x > M$.
So:
:$\f... | Let $f: \R \to \R$ be a [[Definition:Real Function|real function]].
Let $\lambda \in \R_{\ne 0}$ be a nonzero [[Definition:Constant|constant]].
Then:
For $\lambda > 0$:
:$\ds \lim_{x \mathop \to +\infty} \map f x = +\infty \implies \lim_{x \mathop \to +\infty} \lambda \map f x = +\infty$
:$\ds \lim_{x \mathop \to... | Let $\ds \lim_{x \mathop \to +\infty} \map f x = +\infty$.
From the definition of [[Definition:Infinite Limit at Infinity|infinite limit at infinity]], this means that:
:$\forall M > 0: \exists N > 0: x > N \implies \map f x > M$.
Suppose $\lambda > 0$.
Then $M > 0 \iff \lambda^{-1} M > 0$.
Also, $\map f x > \lam... | Scaled Real Function that Increases Without Bound | https://proofwiki.org/wiki/Scaled_Real_Function_that_Increases_Without_Bound | https://proofwiki.org/wiki/Scaled_Real_Function_that_Increases_Without_Bound | [
"Unbounded Mappings"
] | [
"Definition:Real Function",
"Definition:Constant"
] | [
"Definition:Limit of Real Function/Limit at Infinity/Positive/Increasing Without Bound",
"Definition:Limit of Real Function/Limit at Infinity/Positive/Increasing Without Bound",
"Negative of Real Function that Increases Without Bound"
] |
proofwiki-10059 | Scaled Real Function that Decreases Without Bound | Let $f: \R \to \R$ be a real function.
Let $\lambda \in \R_{\ne 0}$ be a nonzero constant.
Then:
For $\lambda > 0$:
:$\ds \lim_{x \mathop \to +\infty} \map f x = -\infty \implies \lim_{x \mathop \to +\infty} \lambda \map f x = -\infty$
:$\ds \lim_{x \mathop \to -\infty} \map f x = -\infty \implies \lim_{x \mathop \to -... | Let $\ds \lim_{x \mathop \to +\infty} \map f x = -\infty$.
From the definition of infinite limits at infinity, this means that:
:$\forall M < 0: \exists N > 0: x > N \implies \map f x < M$
Suppose $\lambda > 0$.
Then $M < 0 \iff \lambda^{-1}M < 0$.
Also, $\map f x < \lambda^{-1}M \iff \lambda \map f x < M$
So:
:$\foral... | Let $f: \R \to \R$ be a [[Definition:Real Function|real function]].
Let $\lambda \in \R_{\ne 0}$ be a nonzero [[Definition:Constant|constant]].
Then:
For $\lambda > 0$:
:$\ds \lim_{x \mathop \to +\infty} \map f x = -\infty \implies \lim_{x \mathop \to +\infty} \lambda \map f x = -\infty$
:$\ds \lim_{x \mathop \to... | Let $\ds \lim_{x \mathop \to +\infty} \map f x = -\infty$.
From the definition of [[Definition:Infinite Limit at Infinity|infinite limits at infinity]], this means that:
:$\forall M < 0: \exists N > 0: x > N \implies \map f x < M$
Suppose $\lambda > 0$.
Then $M < 0 \iff \lambda^{-1}M < 0$.
Also, $\map f x < \lamb... | Scaled Real Function that Decreases Without Bound | https://proofwiki.org/wiki/Scaled_Real_Function_that_Decreases_Without_Bound | https://proofwiki.org/wiki/Scaled_Real_Function_that_Decreases_Without_Bound | [
"Unbounded Mappings"
] | [
"Definition:Real Function",
"Definition:Constant"
] | [
"Definition:Limit of Real Function/Limit at Infinity/Positive/Increasing Without Bound",
"Definition:Limit of Real Function/Limit at Infinity/Positive/Increasing Without Bound",
"Negative of Real Function that Decreases Without Bound"
] |
proofwiki-10060 | Existence of Fraction of Number Smaller than Given | Let $a, b \in \R_{>0}$ be two (strictly) positive real numbers, such that $a > b$.
Let the sequence $\sequence {a_n}$ be defined recursively as:
:$a_i = \begin{cases} a & : i = 1 \\
a_{i - 1} - c: \dfrac {a_{i - 1} } 2 < c < a_{i - 1} & : i > 1
\end{cases}$
Then:
: $\exists n \in \N_{>0}: a_n < b$
{{:Euclid:Proposition... | :400px
Let $AB$ and $C$ be two unequal magnitudes such that $AB > C$.
From {{EuclidDefLink|V|4|Existence of Ratio}} there exists a multiple of $C$ which is greater than $AB$.
Thus let $DE$ be a multiple of $C$ which is greater than $AB$.
Let $DE$ be divided into parts $DF$, $FG$ and $GE$ (for example) which are equal t... | Let $a, b \in \R_{>0}$ be two [[Definition:Strictly Positive Real Number|(strictly) positive real numbers]], such that $a > b$.
Let the [[Definition:Sequence|sequence]] $\sequence {a_n}$ be defined [[Definition:Recursive Sequence|recursively]] as:
:$a_i = \begin{cases} a & : i = 1 \\
a_{i - 1} - c: \dfrac {a_{i - 1} }... | :[[File:Euclid-X-1.png|400px]]
Let $AB$ and $C$ be two unequal [[Definition:Strictly Positive Real Number|magnitudes]] such that $AB > C$.
From {{EuclidDefLink|V|4|Existence of Ratio}} there exists a [[Definition:Multiple|multiple]] of $C$ which is greater than $AB$.
Thus let $DE$ be a [[Definition:Multiple|multiple... | Existence of Fraction of Number Smaller than Given | https://proofwiki.org/wiki/Existence_of_Fraction_of_Number_Smaller_than_Given | https://proofwiki.org/wiki/Existence_of_Fraction_of_Number_Smaller_than_Given | [
"Euclidean Number Theory"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Sequence",
"Definition:Recursive Sequence"
] | [
"File:Euclid-X-1.png",
"Definition:Strictly Positive/Real Number",
"Definition:Multiple",
"Definition:Multiple",
"Definition:Strictly Positive/Real Number",
"Definition:Strictly Positive/Real Number",
"Definition:Strictly Positive/Real Number"
] |
proofwiki-10061 | Lines Through Endpoints of One Side of Triangle to Point Inside Triangle is Less than Sum of Other Sides/Corollary | The angle between the two line segments from the endpoints of one side to a point inside the triangle is greater than the angle between the other two sides of the triangle. | :250px
From External Angle of Triangle is Greater than Internal Opposite:
:$\angle BDC > \angle CED$
Similarly:
: $\angle CEB > \angle BEC$
Since $\angle CED$ is the same angle as $\angle CEB$:
: $\angle BDC > \angle CEB > \angle BEC$
{{qed}}
{{Euclid Note|21|I}} | The [[Definition:Angle|angle]] between the two [[Definition:Line Segment|line segments]] from the [[Definition:Endpoint of Line|endpoints]] of one [[Definition:Side of Polygon|side]] to a [[Definition:Point|point]] inside the [[Definition:Triangle (Geometry)|triangle]] is greater than the [[Definition:Angle|angle]] bet... | :[[File:Point Inside Triangle.png|250px]]
From [[External Angle of Triangle is Greater than Internal Opposite]]:
:$\angle BDC > \angle CED$
Similarly:
: $\angle CEB > \angle BEC$
Since $\angle CED$ is the same angle as $\angle CEB$:
: $\angle BDC > \angle CEB > \angle BEC$
{{qed}}
{{Euclid Note|21|I}} | Lines Through Endpoints of One Side of Triangle to Point Inside Triangle is Less than Sum of Other Sides/Corollary | https://proofwiki.org/wiki/Lines_Through_Endpoints_of_One_Side_of_Triangle_to_Point_Inside_Triangle_is_Less_than_Sum_of_Other_Sides/Corollary | https://proofwiki.org/wiki/Lines_Through_Endpoints_of_One_Side_of_Triangle_to_Point_Inside_Triangle_is_Less_than_Sum_of_Other_Sides/Corollary | [
"Triangles"
] | [
"Definition:Angle",
"Definition:Line/Segment",
"Definition:Line/Endpoint",
"Definition:Polygon/Side",
"Definition:Point",
"Definition:Triangle (Geometry)",
"Definition:Angle",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)"
] | [
"File:Point Inside Triangle.png",
"External Angle of Triangle is Greater than Internal Opposite"
] |
proofwiki-10062 | Restriction of Continuous Mapping is Continuous/Topological Spaces | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $M_1 \subseteq S_1$ be a subset of $S_1$.
Let $f: S_1 \to S_2$ be a mapping which is continuous.
Let $M_2 \subseteq S_2$ be a subset of $S_2$ such that $f \sqbrk {M_1} \subseteq M_2$.
Let $f \restriction_{M_1 \times M_2}: M_1... | Consider first the restriction $f \restriction_{M_1}$.
From Composition of Mapping with Inclusion is Restriction:
:$f \restriction_{M_1} = f \circ i_{M_1}$
where $i_{M_1}$ is the inclusion of $M_1$ into $S_1$.
From Continuity of Composite with Inclusion: Mapping on Inclusion, it follows that $f \circ i_{M_1} = f \restr... | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $M_1 \subseteq S_1$ be a [[Definition:Subset|subset]] of $S_1$.
Let $f: S_1 \to S_2$ be a [[Definition:Mapping|mapping]] which is [[Definition:Continuous Mapping (Topological Spaces)|continu... | Consider first the [[Definition:Restriction of Mapping|restriction]] $f \restriction_{M_1}$.
From [[Composition of Mapping with Inclusion is Restriction]]:
:$f \restriction_{M_1} = f \circ i_{M_1}$
where $i_{M_1}$ is the [[Definition:Inclusion Mapping|inclusion]] of $M_1$ into $S_1$.
From [[Continuity of Composite wi... | Restriction of Continuous Mapping is Continuous/Topological Spaces/Proof 2 | https://proofwiki.org/wiki/Restriction_of_Continuous_Mapping_is_Continuous/Topological_Spaces | https://proofwiki.org/wiki/Restriction_of_Continuous_Mapping_is_Continuous/Topological_Spaces/Proof_2 | [
"Topology",
"Continuous Mappings"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Mapping",
"Definition:Continuous Mapping (Topology)",
"Definition:Subset",
"Definition:Restriction of Mapping ",
"Definition:Continuous Mapping (Topology)",
"Definition:Topological Subspace"
] | [
"Definition:Restriction/Mapping",
"Composition of Mapping with Inclusion is Restriction",
"Definition:Inclusion Mapping",
"Continuity of Composite with Inclusion/Mapping on Inclusion",
"Definition:Continuous_Mapping_(Topology)",
"Definition:Mapping",
"Definition:Inclusion Mapping",
"Continuity of Comp... |
proofwiki-10063 | Restriction of Continuous Mapping is Continuous/Metric Spaces | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $S \subseteq M_1$ be a subset of $M_1$.
Let $f: M_1 \to M_2$ be a mapping which is continuous at a point $\alpha \in S$.
Let $f \restriction_S = g: S \to M_2$ be the restriction of $f$ to $S$.
Then $g$ is continuous at $\alpha$. | Let $\sequence {z_n}$ be a sequence in $S$ such that:
:$\ds \lim_{n \mathop \to \infty} z_n = \alpha$
Since $\sequence {z_n}$ and $\alpha$ both lie in $S$:
:$\ds \lim_{n \mathop \to \infty} \map f {z_n} \to \alpha$
But:
:$\forall n \in \N: \map g {z_n} = \map f {z_n}$
and also:
:$\map g \alpha = \map f \alpha$
The resu... | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $S \subseteq M_1$ be a [[Definition:Subset|subset]] of $M_1$.
Let $f: M_1 \to M_2$ be a [[Definition:Mapping|mapping]] which is [[Definition:Continuous Mapping (Metric Spaces)|continuous]] at a point $\alph... | Let $\sequence {z_n}$ be a [[Definition:Sequence|sequence in $S$]] such that:
:$\ds \lim_{n \mathop \to \infty} z_n = \alpha$
Since $\sequence {z_n}$ and $\alpha$ both lie in $S$:
:$\ds \lim_{n \mathop \to \infty} \map f {z_n} \to \alpha$
But:
:$\forall n \in \N: \map g {z_n} = \map f {z_n}$
and also:
:$\map g \alpha... | Restriction of Continuous Mapping is Continuous/Metric Spaces | https://proofwiki.org/wiki/Restriction_of_Continuous_Mapping_is_Continuous/Metric_Spaces | https://proofwiki.org/wiki/Restriction_of_Continuous_Mapping_is_Continuous/Metric_Spaces | [
"Metric Spaces",
"Continuous Mappings on Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Subset",
"Definition:Mapping",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Restriction of Mapping ",
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:Sequence",
"Limit of Function by Convergent Sequences",
"Category:Metric Spaces",
"Category:Continuous Mappings on Metric Spaces"
] |
proofwiki-10064 | Restriction of Inverse is Inverse of Restriction | Let $S_1$ and $S_2$ be sets.
Let $f: S_1 \to S_2$ be a bijection.
Let $S \subseteq S_1$ be a subset of $S_1$.
Let $f \sqbrk S$ be the image of $S$ under $f$.
Let $f^{-1}$ be the inverse of $f$.
Let $f {\restriction_{S \mathop \times f \sqbrk S}}$ be the restriction of $f$ to $S \times f \sqbrk S$.
Let $f^{-1} {\restric... | Let $y \in f \sqbrk S$.
By the definition of image:
:$\exists z \in S : \map f z = y$
Since $f$ is a bijection, $f^{-1}$ is a mapping.
Let $x = \map {f^{-1} } y$.
{{AimForCont}} $x \notin S$.
Suppose that $x = z$.
Then $x \in S$.
Thus $x \ne z$.
By the definition of inverse mapping:
:$\map f x = y$
By Equality is Trans... | Let $S_1$ and $S_2$ be [[Definition:Set|sets]].
Let $f: S_1 \to S_2$ be a [[Definition:Bijection|bijection]].
Let $S \subseteq S_1$ be a [[Definition:Subset|subset]] of $S_1$.
Let $f \sqbrk S$ be the [[Definition:Image of Subset under Mapping|image]] of $S$ under $f$.
Let $f^{-1}$ be the [[Definition:Inverse Mappin... | Let $y \in f \sqbrk S$.
By the definition of [[Definition:Image of Subset under Mapping|image]]:
:$\exists z \in S : \map f z = y$
Since $f$ is a [[Definition:Bijection|bijection]], $f^{-1}$ is a [[Definition:Mapping|mapping]].
Let $x = \map {f^{-1} } y$.
{{AimForCont}} $x \notin S$.
Suppose that $x = z$.
Then $... | Restriction of Inverse is Inverse of Restriction | https://proofwiki.org/wiki/Restriction_of_Inverse_is_Inverse_of_Restriction | https://proofwiki.org/wiki/Restriction_of_Inverse_is_Inverse_of_Restriction | [
"Restrictions",
"Inverse Mappings"
] | [
"Definition:Set",
"Definition:Bijection",
"Definition:Subset",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Inverse Mapping",
"Definition:Restriction/Mapping",
"Definition:Restriction/Mapping",
"Definition:Bijection"
] | [
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Bijection",
"Definition:Mapping",
"Definition:Inverse Mapping",
"Equality is Transitive",
"Definition:Injection",
"Definition:Bijection",
"Definition:Restriction/Mapping",
"Equality is Transitive",
"Definition:Restriction/Mapping",
"Equ... |
proofwiki-10065 | Restriction of Homeomorphism is Homeomorphism | Let $T_1 = \struct {S_1, \tau_1}$, $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $f: S_1 \to S_2$ be a homeomorphism between $T_1$ and $T_2$.
Let $S$ be a subset of $S_1$.
Let $f {\restriction_{S \times f \sqbrk S} } : S \to f \sqbrk S$ be the restriction of $f$ to $S \times f \sqbrk S$.
Let $S$ and $f \sqbr... | By Restriction of Continuous Mapping is Continuous, $f {\restriction_{S \times f \sqbrk S} }$ is continuous.
By Restriction of Inverse is Inverse of Restriction, $\left({f {\restriction_{S \times f \sqbrk S} } }\right)^{-1}$ is well-defined and equal to $f^{-1} {\restriction_{f \sqbrk S \times S} }$.
By Restriction of ... | Let $T_1 = \struct {S_1, \tau_1}$, $T_2 = \struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $f: S_1 \to S_2$ be a [[Definition:Homeomorphism (Topological Spaces)|homeomorphism]] between $T_1$ and $T_2$.
Let $S$ be a [[Definition:Subset|subset]] of $S_1$.
Let $f {\restriction_{S \time... | By [[Restriction of Continuous Mapping is Continuous/Topological Spaces|Restriction of Continuous Mapping is Continuous]], $f {\restriction_{S \times f \sqbrk S} }$ is [[Definition:Continuous Mapping on Set|continuous]].
By [[Restriction of Inverse is Inverse of Restriction]], $\left({f {\restriction_{S \times f \sqbr... | Restriction of Homeomorphism is Homeomorphism | https://proofwiki.org/wiki/Restriction_of_Homeomorphism_is_Homeomorphism | https://proofwiki.org/wiki/Restriction_of_Homeomorphism_is_Homeomorphism | [
"Homeomorphisms (Topological Spaces)"
] | [
"Definition:Topological Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Subset",
"Definition:Restriction/Mapping",
"Definition:Topological Subspace",
"Definition:Homeomorphism/Topological Spaces"
] | [
"Restriction of Continuous Mapping is Continuous/Topological Spaces",
"Definition:Continuous Mapping (Topology)/Set",
"Restriction of Inverse is Inverse of Restriction",
"Definition:Well-Defined/Mapping",
"Restriction of Continuous Mapping is Continuous/Topological Spaces",
"Definition:Continuous Mapping ... |
proofwiki-10066 | Hölder's Inequality for Integrals/General | Let $\struct {X, \Sigma, \mu}$ be a measure space.
For $i = 1, \ldots, n$ let $p_i \in \R_{>0}$ such that:
:$\ds \sum_{i \mathop = 1}^n \frac 1 {p_i} = 1$
Let $f_i \in \map {\LL^{p_i} } \mu, f_i: X \to \R$, where $\LL$ denotes Lebesgue space.
Then their pointwise product $\ds \prod_{i \mathop = 1}^n f_i$ is integrable,... | {{explain|Has it been demonstrated for $i {{=}} 1$? Best if this were put into the house style of induction proofs.}}
We use the Principle of Mathematical Induction.
Let it be assumed that the result holds for $i = n - 1$.
We show that the result holds for $i = n$.
Define:
:$q_n := \dfrac {p_n} {p_n - 1}$
and for $i =... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
For $i = 1, \ldots, n$ let $p_i \in \R_{>0}$ such that:
:$\ds \sum_{i \mathop = 1}^n \frac 1 {p_i} = 1$
Let $f_i \in \map {\LL^{p_i} } \mu, f_i: X \to \R$, where $\LL$ denotes [[Definition:Lebesgue Space|Lebesgue space]].
Then their [[D... | {{explain|Has it been demonstrated for $i {{=}} 1$? Best if this were put into the house style of induction proofs.}}
We use the [[Principle of Mathematical Induction]].
Let it be assumed that the result holds for $i = n - 1$.
We show that the result holds for $i = n$.
Define:
:$q_n := \dfrac {p_n} {p_n - 1}$
and... | Hölder's Inequality for Integrals/General | https://proofwiki.org/wiki/Hölder's_Inequality_for_Integrals/General | https://proofwiki.org/wiki/Hölder's_Inequality_for_Integrals/General | [
"Hölder's Inequality for Integrals"
] | [
"Definition:Measure Space",
"Definition:Lebesgue Space",
"Definition:Pointwise Multiplication",
"Definition:Integrable Function/Measure Space",
"Definition:P-Seminorm"
] | [
"Principle of Mathematical Induction",
"Hölder's Inequality for Integrals",
"Category:Hölder's Inequality for Integrals"
] |
proofwiki-10067 | Theorem of Even Perfect Numbers/Sufficient Condition | Let $n \in \N$ be such that $2^n - 1$ is prime.
Then $2^{n - 1} \paren {2^n - 1}$ is perfect.
{{:Euclid:Proposition/IX/36}} | Suppose $2^n - 1$ is prime.
Let $a = 2^{n - 1} \paren {2^n - 1}$.
Then $n \ge 2$ which means $2^{n - 1}$ is even and hence so is $a = 2^{n - 1} \paren {2^n - 1}$.
Note that $2^n - 1$ is odd.
Since all divisors (except $1$) of $2^{n - 1}$ are even it follows that $2^{n - 1}$ and $2^n - 1$ are coprime.
Let $\map {\sigma_... | Let $n \in \N$ be such that $2^n - 1$ is [[Definition:Prime Number|prime]].
Then $2^{n - 1} \paren {2^n - 1}$ is [[Definition:Perfect Number|perfect]].
{{:Euclid:Proposition/IX/36}} | Suppose $2^n - 1$ is [[Definition:Prime Number|prime]].
Let $a = 2^{n - 1} \paren {2^n - 1}$.
Then $n \ge 2$ which means $2^{n - 1}$ is [[Definition:Even Integer|even]] and hence so is $a = 2^{n - 1} \paren {2^n - 1}$.
Note that $2^n - 1$ is [[Definition:Odd Integer|odd]].
Since all [[Definition:Divisor of Integer|... | Theorem of Even Perfect Numbers/Sufficient Condition | https://proofwiki.org/wiki/Theorem_of_Even_Perfect_Numbers/Sufficient_Condition | https://proofwiki.org/wiki/Theorem_of_Even_Perfect_Numbers/Sufficient_Condition | [
"Theorem of Even Perfect Numbers"
] | [
"Definition:Prime Number",
"Definition:Perfect Number"
] | [
"Definition:Prime Number",
"Definition:Even Integer",
"Definition:Odd Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Even Integer",
"Definition:Coprime/Integers",
"Definition:Divisor Sum Function",
"Definition:Divisor (Algebra)/Integer",
"Divisor Sum Function is Multiplicative",
"Di... |
proofwiki-10068 | Theorem of Even Perfect Numbers/Necessary Condition | Let $a \in \N$ be an even perfect number.
Then $a$ is in the form:
:$2^{n - 1} \paren {2^n - 1}$
where $2^n - 1$ is prime. | Let $a \in \N$ be an even perfect number.
We can extract the highest power of $2$ out of $a$ that we can, and write $a$ in the form:
:$a = m 2^{n - 1}$
where $n \ge 2$ and $m$ is odd.
Since $a$ is perfect and therefore $\map {\sigma_1} a = 2 a$:
{{begin-eqn}}
{{eqn| l = m 2^n
| r = 2 a
| c =
}}
{{eqn| r ... | Let $a \in \N$ be an [[Definition:Even Integer|even]] [[Definition:Perfect Number|perfect number]].
Then $a$ is in the form:
:$2^{n - 1} \paren {2^n - 1}$
where $2^n - 1$ is [[Definition:Prime Number|prime]]. | Let $a \in \N$ be an [[Definition:Even Integer|even]] [[Definition:Perfect Number|perfect number]].
We can extract the highest power of $2$ out of $a$ that we can, and write $a$ in the form:
:$a = m 2^{n - 1}$
where $n \ge 2$ and $m$ is [[Definition:Odd Integer|odd]].
Since $a$ is [[Definition:Perfect Number|perfect... | Theorem of Even Perfect Numbers/Necessary Condition | https://proofwiki.org/wiki/Theorem_of_Even_Perfect_Numbers/Necessary_Condition | https://proofwiki.org/wiki/Theorem_of_Even_Perfect_Numbers/Necessary_Condition | [
"Theorem of Even Perfect Numbers"
] | [
"Definition:Even Integer",
"Definition:Perfect Number",
"Definition:Prime Number"
] | [
"Definition:Even Integer",
"Definition:Perfect Number",
"Definition:Odd Integer",
"Definition:Perfect Number",
"Divisor Sum Function is Multiplicative",
"Divisor Sum of Power of Prime",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Consecutive Integers are Coprime",
"Definition:Co... |
proofwiki-10069 | Condition for Commensurability of Roots of Quadratic Equation | Consider the quadratic equation:
:$(1): \quad a x - x^2 = \dfrac {b^2} 4$
Then $x$ and $a - x$ are commensurable {{iff}} $\sqrt{a^2 - b^2}$ and $a$ are commensurable.
{{:Euclid:Proposition/X/17}} | We have that:
{{begin-eqn}}
{{eqn | l = x \paren {a - x} + \paren {\frac a 2 - x}^2
| r = a x - x^2 + \frac {a^2} 4 - 2 \frac a 2 x + x^2
| c =
}}
{{eqn | r = \frac {a^2} 4
| c = simplifying
}}
{{eqn | ll= \leadsto
| l = 4 x \paren {a - x} + 4 \paren {\frac a 2 - x}^2
| r = a^2
| c ... | Consider the [[Definition:Quadratic Equation|quadratic equation]]:
:$(1): \quad a x - x^2 = \dfrac {b^2} 4$
Then $x$ and $a - x$ are [[Definition:Commensurable|commensurable]] {{iff}} $\sqrt{a^2 - b^2}$ and $a$ are [[Definition:Commensurable|commensurable]].
{{:Euclid:Proposition/X/17}} | We have that:
{{begin-eqn}}
{{eqn | l = x \paren {a - x} + \paren {\frac a 2 - x}^2
| r = a x - x^2 + \frac {a^2} 4 - 2 \frac a 2 x + x^2
| c =
}}
{{eqn | r = \frac {a^2} 4
| c = simplifying
}}
{{eqn | ll= \leadsto
| l = 4 x \paren {a - x} + 4 \paren {\frac a 2 - x}^2
| r = a^2
| c ... | Condition for Commensurability of Roots of Quadratic Equation | https://proofwiki.org/wiki/Condition_for_Commensurability_of_Roots_of_Quadratic_Equation | https://proofwiki.org/wiki/Condition_for_Commensurability_of_Roots_of_Quadratic_Equation | [
"Euclidean Number Theory"
] | [
"Definition:Quadratic Equation",
"Definition:Commensurable",
"Definition:Commensurable"
] | [
"Definition:Commensurable"
] |
proofwiki-10070 | Condition for Incommensurability of Roots of Quadratic Equation | Consider the quadratic equation:
:$(1): \quad a x - x^2 = \dfrac {b^2} 4$
Then $x$ and $a - x$ are incommensurable {{iff}} $\sqrt {a^2 - b^2}$ and $a$ are incommensurable.
{{:Euclid:Proposition/X/18}} | We have that:
{{begin-eqn}}
{{eqn | l = x \paren {a - x} + \paren {\frac a 2 - x}^2
| r = a x - x^2 + \frac {a^2} 4 - 2 \frac a 2 x + x^2
| c =
}}
{{eqn | r = \frac {a^2} 4
| c = simplifying
}}
{{eqn | ll= \leadsto
| l = 4 x \paren {a - x} + 4 \paren {\frac a 2 - x}^2
| r = a^2
| c ... | Consider the [[Definition:Quadratic Equation|quadratic equation]]:
:$(1): \quad a x - x^2 = \dfrac {b^2} 4$
Then $x$ and $a - x$ are [[Definition:Incommensurable|incommensurable]] {{iff}} $\sqrt {a^2 - b^2}$ and $a$ are [[Definition:Incommensurable|incommensurable]].
{{:Euclid:Proposition/X/18}} | We have that:
{{begin-eqn}}
{{eqn | l = x \paren {a - x} + \paren {\frac a 2 - x}^2
| r = a x - x^2 + \frac {a^2} 4 - 2 \frac a 2 x + x^2
| c =
}}
{{eqn | r = \frac {a^2} 4
| c = simplifying
}}
{{eqn | ll= \leadsto
| l = 4 x \paren {a - x} + 4 \paren {\frac a 2 - x}^2
| r = a^2
| c ... | Condition for Incommensurability of Roots of Quadratic Equation | https://proofwiki.org/wiki/Condition_for_Incommensurability_of_Roots_of_Quadratic_Equation | https://proofwiki.org/wiki/Condition_for_Incommensurability_of_Roots_of_Quadratic_Equation | [
"Euclidean Number Theory"
] | [
"Definition:Quadratic Equation",
"Definition:Incommensurable",
"Definition:Incommensurable"
] | [
"Definition:Incommensurable",
"Definition:Commensurable"
] |
proofwiki-10071 | Apotome is Irrational | Every apotome is irrational, i.e.:
:$\ds \forall a, b \in \set {x \in \R_{>0} : x^2 \in \Q}: \paren {\frac a b \notin \Q \land \paren {\frac a b}^2 \in \Q} \implies \paren {\paren {a - b} \notin \Q \land \paren {a - b}^2 \notin \Q}$
{{:Euclid:Proposition/X/73}} | :300px
Let $AB$ be a rational straight line.
Let a rational straight line $BC$ which is commensurable in square only with $AB$ be cut off from $AB$.
We have that $AB$ is incommensurable in length with $BC$.
We also have:
:$AB : BC = AB^2 : AB \cdot AC$
Therefore from {{EuclidPropLink|book = X|prop = 11|title = Commensu... | Every [[Definition:Apotome|apotome]] is [[Definition:Irrational Number|irrational]], i.e.:
:$\ds \forall a, b \in \set {x \in \R_{>0} : x^2 \in \Q}: \paren {\frac a b \notin \Q \land \paren {\frac a b}^2 \in \Q} \implies \paren {\paren {a - b} \notin \Q \land \paren {a - b}^2 \notin \Q}$
{{:Euclid:Proposition/X/73}} | :[[File:Euclid-X-73.png|300px]]
Let $AB$ be a [[Definition:Rational Line Segment|rational straight line]].
Let a [[Definition:Rational Line Segment|rational straight line]] $BC$ which is [[Definition:Commensurable in Square Only|commensurable in square only]] with $AB$ be cut off from $AB$.
We have that $AB$ is [[D... | Apotome is Irrational | https://proofwiki.org/wiki/Apotome_is_Irrational | https://proofwiki.org/wiki/Apotome_is_Irrational | [
"Apotome"
] | [
"Definition:Apotome",
"Definition:Irrational Number"
] | [
"File:Euclid-X-73.png",
"Definition:Rational Line Segment",
"Definition:Rational Line Segment",
"Definition:Commensurable in Square Only",
"Definition:Incommensurable",
"Definition:Incommensurable",
"Definition:Commensurable",
"Definition:Commensurable",
"Definition:Incommensurable",
"Definition:R... |
proofwiki-10072 | Construction of Apotome is Unique | Let $D$ be the domain $\left\{{x \in \R_{>0} : x^2 \in \Q}\right\}$, the rationally expressible numbers.
Let $a, b \in D$ be two rationally expressible numbers such that $a - b$ is an apotome.
Then, there exists only one $x \in D$ such that $a - b + x$ and $a$ are commensurable in square only.
{{:Euclid:Proposition/X/7... | :300px
Let $AB$ be an apotome.
Let $BC$ be added to $AB$ so that $AC$ and $CB$ are rational straight lines which are commensurable in square only.
It is to be proved that no other rational straight line can be added to $AB$ which is commensurable in square only with the whole.
Suppose $BD$ can be added to $AB$ so as to... | Let $D$ be the domain $\left\{{x \in \R_{>0} : x^2 \in \Q}\right\}$, the [[Definition:Rationally Expressible Number|rationally expressible numbers]].
Let $a, b \in D$ be two [[Definition:Rationally Expressible Number|rationally expressible numbers]] such that $a - b$ is an [[Definition:Apotome|apotome]].
Then, there ... | :[[File:Euclid-X-79.png|300px]]
Let $AB$ be an [[Definition:Apotome|apotome]].
Let $BC$ be added to $AB$ so that $AC$ and $CB$ are [[Definition:Rational Line Segment|rational straight lines]] which are [[Definition:Commensurable in Square Only|commensurable in square only]].
It is to be proved that no other [[Defini... | Construction of Apotome is Unique | https://proofwiki.org/wiki/Construction_of_Apotome_is_Unique | https://proofwiki.org/wiki/Construction_of_Apotome_is_Unique | [
"Euclidean Number Theory"
] | [
"Definition:Rationally Expressible Number",
"Definition:Rationally Expressible Number",
"Definition:Apotome",
"Definition:Commensurable in Square Only"
] | [
"File:Euclid-X-79.png",
"Definition:Apotome",
"Definition:Rational Line Segment",
"Definition:Commensurable in Square Only",
"Definition:Rational Line Segment",
"Definition:Commensurable in Square Only",
"Definition:Rational Line Segment",
"Definition:Commensurable in Square Only",
"Definition:Ratio... |
proofwiki-10073 | Lower Bound of Natural Logarithm | :$\forall x \in \R_{>0}: 1 - \dfrac 1 x \le \ln x$
where $\ln x$ denotes the natural logarithm of $x$. | Let $x > 0$.
{{begin-eqn}}
{{eqn | l = x - 1
| o = \ge
| r = \ln x
| c = Upper Bound of Natural Logarithm
}}
{{eqn | ll= \leadsto
| l = \frac 1 x -1
| o = \ge
| r = \ln \frac 1 x
| c = putting $\frac 1 x$ into the above inequality
}}
{{eqn | r = -\ln x
| c = Logarithm of... | :$\forall x \in \R_{>0}: 1 - \dfrac 1 x \le \ln x$
where $\ln x$ denotes the [[Definition:Natural Logarithm|natural logarithm]] of $x$. | Let $x > 0$.
{{begin-eqn}}
{{eqn | l = x - 1
| o = \ge
| r = \ln x
| c = [[Upper Bound of Natural Logarithm]]
}}
{{eqn | ll= \leadsto
| l = \frac 1 x -1
| o = \ge
| r = \ln \frac 1 x
| c = putting $\frac 1 x$ into the above inequality
}}
{{eqn | r = -\ln x
| c = [[Logar... | Lower Bound of Natural Logarithm/Proof 1 | https://proofwiki.org/wiki/Lower_Bound_of_Natural_Logarithm | https://proofwiki.org/wiki/Lower_Bound_of_Natural_Logarithm/Proof_1 | [
"Real Analysis",
"Logarithms",
"Inequalities",
"Lower Bound of Natural Logarithm"
] | [
"Definition:Natural Logarithm"
] | [
"Upper Bound of Natural Logarithm",
"Logarithm of Reciprocal"
] |
proofwiki-10074 | Lower Bound of Natural Logarithm | :$\forall x \in \R_{>0}: 1 - \dfrac 1 x \le \ln x$
where $\ln x$ denotes the natural logarithm of $x$. | Let $x > 0$.
Note that:
:$1 - \dfrac 1 x \le \ln x$
is logically equivalent to:
:$1 - \dfrac 1 x - \ln x \le 0$
Let $\map f x = 1 - \dfrac 1 x - \ln x$.
Then:
{{begin-eqn}}
{{eqn | l = \map f x
| r = 1 - \dfrac 1 x - \ln x
}}
{{eqn | ll= \leadsto
| l = \map {f'} x
| r = \frac 1 {x^2} - \frac 1 x
... | :$\forall x \in \R_{>0}: 1 - \dfrac 1 x \le \ln x$
where $\ln x$ denotes the [[Definition:Natural Logarithm|natural logarithm]] of $x$. | Let $x > 0$.
Note that:
:$1 - \dfrac 1 x \le \ln x$
is [[Definition:Logically Equivalent|logically equivalent]] to:
:$1 - \dfrac 1 x - \ln x \le 0$
Let $\map f x = 1 - \dfrac 1 x - \ln x$.
Then:
{{begin-eqn}}
{{eqn | l = \map f x
| r = 1 - \dfrac 1 x - \ln x
}}
{{eqn | ll= \leadsto
| l = \map {f'} x
... | Lower Bound of Natural Logarithm/Proof 2 | https://proofwiki.org/wiki/Lower_Bound_of_Natural_Logarithm | https://proofwiki.org/wiki/Lower_Bound_of_Natural_Logarithm/Proof_2 | [
"Real Analysis",
"Logarithms",
"Inequalities",
"Lower Bound of Natural Logarithm"
] | [
"Definition:Natural Logarithm"
] | [
"Definition:Logical Equivalence",
"Derivative of Constant",
"Power Rule for Derivatives",
"Derivative of Natural Logarithm Function",
"Power Rule for Derivatives",
"Second Derivative Test",
"Definition:Maximum Value of Real Function/Local",
"Derivative of Monotone Function",
"Definition:Strictly Inc... |
proofwiki-10075 | Lower Bound of Natural Logarithm | :$\forall x \in \R_{>0}: 1 - \dfrac 1 x \le \ln x$
where $\ln x$ denotes the natural logarithm of $x$. | Let $\sequence {f_n}$ be the sequence of mappings $f_n: \R_{>0} \to \R$ defined as:
:$\map {f_n} x = n \paren {\sqrt [n] x - 1 }$
Let $x \in \R_{>0}$ be fixed.
We first show that:
:$\forall n \in \N : 1 - \dfrac 1 x \le n \paren {\sqrt [n] x - 1}$
Let $n \in \N$.
From Sum of Geometric Sequence:
:$\sqrt [n] x - 1 = \dfr... | :$\forall x \in \R_{>0}: 1 - \dfrac 1 x \le \ln x$
where $\ln x$ denotes the [[Definition:Natural Logarithm|natural logarithm]] of $x$. | Let $\sequence {f_n}$ be the [[Definition:Sequence|sequence]] of [[Definition:Mapping|mappings]] $f_n: \R_{>0} \to \R$ defined as:
:$\map {f_n} x = n \paren {\sqrt [n] x - 1 }$
Let $x \in \R_{>0}$ be fixed.
We first show that:
:$\forall n \in \N : 1 - \dfrac 1 x \le n \paren {\sqrt [n] x - 1}$
Let $n \in \N$.
From ... | Lower Bound of Natural Logarithm/Proof 3 | https://proofwiki.org/wiki/Lower_Bound_of_Natural_Logarithm | https://proofwiki.org/wiki/Lower_Bound_of_Natural_Logarithm/Proof_3 | [
"Real Analysis",
"Logarithms",
"Inequalities",
"Lower Bound of Natural Logarithm"
] | [
"Definition:Natural Logarithm"
] | [
"Definition:Sequence",
"Definition:Mapping",
"Sum of Geometric Sequence",
"Power Function on Base between Zero and One is Strictly Decreasing/Rational Number",
"Real Number Ordering is Compatible with Addition",
"Ordering of Reciprocals",
"Order of Real Numbers is Dual of Order of their Negatives",
"S... |
proofwiki-10076 | Bounds of Natural Logarithm | 300pxthumbright
Let $\ln x$ be the natural logarithm of $x$ where $x \in \R_{>0}$.
Then $\ln$ satisfies the compound inequality:
:$1 - \dfrac 1 x \le \ln x \le x - 1$ | From Upper Bound of Natural Logarithm:
:$\ln x \le x - 1$
From Lower Bound of Natural Logarithm:
:$1 - \dfrac 1 x \le \ln x$
{{qed}} | [[File:BoundsOfNatLog.png|300px|thumb|right]]
Let $\ln x$ be the [[Definition:Natural Logarithm|natural logarithm]] of $x$ where $x \in \R_{>0}$.
Then $\ln$ satisfies the [[Definition:Compound Inequality|compound inequality]]:
:$1 - \dfrac 1 x \le \ln x \le x - 1$ | From [[Upper Bound of Natural Logarithm]]:
:$\ln x \le x - 1$
From [[Lower Bound of Natural Logarithm]]:
:$1 - \dfrac 1 x \le \ln x$
{{qed}} | Bounds of Natural Logarithm | https://proofwiki.org/wiki/Bounds_of_Natural_Logarithm | https://proofwiki.org/wiki/Bounds_of_Natural_Logarithm | [
"Analysis",
"Logarithms",
"Inequalities"
] | [
"File:BoundsOfNatLog.png",
"Definition:Natural Logarithm",
"Definition:Compound Inequality"
] | [
"Upper Bound of Natural Logarithm",
"Lower Bound of Natural Logarithm"
] |
proofwiki-10077 | Mordell's Theorem | Let $C$ be an elliptic curve over $\Q$.
Then the group of rational points on $C$ is a finitely generated abelian group. | {{ProofWanted}}
{{Namedfor|Louis Joel Mordell|cat = Mordell}} | Let $C$ be an [[Definition:Elliptic Curve|elliptic curve]] over $\Q$.
Then the [[Definition:Group of Rational Points|group of rational points]] on $C$ is a [[Definition:Finitely Generated Group|finitely generated]] [[Definition:Abelian Group|abelian group]]. | {{ProofWanted}}
{{Namedfor|Louis Joel Mordell|cat = Mordell}} | Mordell's Theorem | https://proofwiki.org/wiki/Mordell's_Theorem | https://proofwiki.org/wiki/Mordell's_Theorem | [
"Algebraic Geometry"
] | [
"Definition:Elliptic Curve",
"Definition:Group of Rational Points",
"Definition:Finitely Generated Group",
"Definition:Abelian Group"
] | [] |
proofwiki-10078 | Common Ratio in Rational Geometric Sequence is Rational | Let $\sequence {a_k}$ be a geometric sequence whose terms are rational.
Then the common ratio of $\sequence {a_k}$ is rational. | Let $r$ be the common ratio of $\sequence {a_k}$.
Let $p, q$ be consecutive terms of $r$.
We have {{hypothesis}} that:
:$p, q \in \Q$
Then, by definition of geometric sequence:
:$q = r p$
It follows that:
:$r = \dfrac q p$
From Rational Numbers form Field, $\Q$ is closed under division.
Thus $r \in \Q$ and hence the re... | Let $\sequence {a_k}$ be a [[Definition:Geometric Sequence|geometric sequence]] whose [[Definition:Term of Geometric Sequence|terms]] are [[Definition:Rational Number|rational]].
Then the [[Definition:Common Ratio|common ratio]] of $\sequence {a_k}$ is [[Definition:Rational Number|rational]]. | Let $r$ be the [[Definition:Common Ratio|common ratio]] of $\sequence {a_k}$.
Let $p, q$ be consecutive [[Definition:Term of Geometric Sequence|terms]] of $r$.
We have {{hypothesis}} that:
:$p, q \in \Q$
Then, by definition of [[Definition:Geometric Sequence|geometric sequence]]:
:$q = r p$
It follows that:
:$r = \... | Common Ratio in Rational Geometric Sequence is Rational | https://proofwiki.org/wiki/Common_Ratio_in_Rational_Geometric_Sequence_is_Rational | https://proofwiki.org/wiki/Common_Ratio_in_Rational_Geometric_Sequence_is_Rational | [
"Geometric Sequences"
] | [
"Definition:Geometric Sequence",
"Definition:Geometric Sequence/Term",
"Definition:Rational Number",
"Definition:Geometric Sequence/Common Ratio",
"Definition:Rational Number"
] | [
"Definition:Geometric Sequence/Common Ratio",
"Definition:Geometric Sequence/Term",
"Definition:Geometric Sequence",
"Rational Numbers form Field",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Division/Field/Rational Numbers",
"Category:Geometric Sequences"
] |
proofwiki-10079 | Common Ratio in Integer Geometric Sequence is Rational | Let $\sequence {a_k}$ be a geometric sequence whose terms are all integers.
Then the common ratio of $\sequence {a_k}$ is rational. | From Integers form Subdomain of Rationals it follows that $a_k \in \Q$ for all $0 \le k \le n$.
The result follows from Common Ratio in Rational Geometric Sequence is Rational.
{{qed}}
Category:Geometric Sequences of Integers
0qft406itmha5ood9xvu2bzzz0xivkq | Let $\sequence {a_k}$ be a [[Definition:Geometric Sequence|geometric sequence]] whose [[Definition:Term of Geometric Sequence|terms]] are all [[Definition:Integer|integers]].
Then the [[Definition:Common Ratio|common ratio]] of $\sequence {a_k}$ is [[Definition:Rational Number|rational]]. | From [[Integers form Subdomain of Rationals]] it follows that $a_k \in \Q$ for all $0 \le k \le n$.
The result follows from [[Common Ratio in Rational Geometric Sequence is Rational]].
{{qed}}
[[Category:Geometric Sequences of Integers]]
0qft406itmha5ood9xvu2bzzz0xivkq | Common Ratio in Integer Geometric Sequence is Rational | https://proofwiki.org/wiki/Common_Ratio_in_Integer_Geometric_Sequence_is_Rational | https://proofwiki.org/wiki/Common_Ratio_in_Integer_Geometric_Sequence_is_Rational | [
"Geometric Sequences of Integers"
] | [
"Definition:Geometric Sequence",
"Definition:Geometric Sequence/Term",
"Definition:Integer",
"Definition:Geometric Sequence/Common Ratio",
"Definition:Rational Number"
] | [
"Integers form Subdomain of Rationals",
"Common Ratio in Rational Geometric Sequence is Rational",
"Category:Geometric Sequences of Integers"
] |
proofwiki-10080 | Form of Geometric Sequence of Integers | Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of length $n + 1$ consisting of integers only.
Then the $j$th term of $P$ is given by:
:$a_j = k p^{n - j} q^j$
where:
: the common ratio of $P$ expressed in canonical form is $\dfrac q p$
: $k$ is an integer. | Let $r$ be the common ratio of $P$.
From Common Ratio in Integer Geometric Sequence is Rational, $r$ is a rational number.
Let $r = \dfrac q p$ be in canonical form.
Thus, by definition:
:$p \perp q$
Let $a$ be the first term of $P$.
Then the sequence $P$ is:
:$P = \paren {a, a \dfrac q p, a \dfrac {q^2} {p^2}, \ldots,... | Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a [[Definition:Geometric Sequence|geometric sequence]] of [[Definition:Length of Sequence|length]] $n + 1$ consisting of [[Definition:Integer|integers]] only.
Then the $j$th [[Definition:Term of Geometric Sequence|term]] of $P$ is given by:
:$a_j = k p^{n -... | Let $r$ be the [[Definition:Common Ratio of Geometric Sequence|common ratio]] of $P$.
From [[Common Ratio in Integer Geometric Sequence is Rational]], $r$ is a [[Definition:Rational Number|rational number]].
Let $r = \dfrac q p$ be in [[Definition:Canonical Form of Rational Number|canonical form]].
Thus, by definiti... | Form of Geometric Sequence of Integers | https://proofwiki.org/wiki/Form_of_Geometric_Sequence_of_Integers | https://proofwiki.org/wiki/Form_of_Geometric_Sequence_of_Integers | [
"Geometric Sequences of Integers"
] | [
"Definition:Geometric Sequence",
"Definition:Length of Sequence",
"Definition:Integer",
"Definition:Geometric Sequence/Term",
"Definition:Geometric Sequence/Common Ratio",
"Definition:Rational Number/Canonical Form",
"Definition:Integer"
] | [
"Definition:Geometric Sequence/Common Ratio",
"Common Ratio in Integer Geometric Sequence is Rational",
"Definition:Rational Number",
"Definition:Rational Number/Canonical Form",
"Definition:Geometric Sequence/Term",
"Definition:Sequence",
"Definition:Integer",
"Powers of Coprime Numbers are Coprime",... |
proofwiki-10081 | Geometric Sequence of Integers with Integer Common Ratio | Let $P = \sequence {a_j}_{1 \mathop \le j \mathop \le n}$ be a geometric sequence of length $n$ consisting entirely of integers.
Let $r$ be the common ratio of $P$.
Then $r$ is an integer {{iff}}:
:$\forall i, j \in \set {1, 2, \ldots, n}, i \le j: a_i \divides a_j$
That is, terms of $P$ divide later terms of $P$ {{iff... | === Necessary Condition ===
Let $r$ be an integer.
By definition of geometric sequence, the terms of $P$ are in the form:
:$a_k = b r^{k - 1}$
where $b$ and $k$ are integers.
It follows from Integer Multiplication is Closed that $a_k$ is an integer.
{{qed|lemma}} | Let $P = \sequence {a_j}_{1 \mathop \le j \mathop \le n}$ be a [[Definition:Geometric Sequence|geometric sequence]] of [[Definition:Length of Sequence|length]] $n$ consisting entirely of [[Definition:Integer|integers]].
Let $r$ be the [[Definition:Common Ratio of Geometric Sequence|common ratio]] of $P$.
Then $r$ is... | === Necessary Condition ===
Let $r$ be an [[Definition:Integer|integer]].
By definition of [[Definition:Geometric Sequence|geometric sequence]], the [[Definition:Term of Geometric Sequence|terms]] of $P$ are in the form:
:$a_k = b r^{k - 1}$
where $b$ and $k$ are [[Definition:Integer|integers]].
It follows from [[In... | Geometric Sequence of Integers with Integer Common Ratio | https://proofwiki.org/wiki/Geometric_Sequence_of_Integers_with_Integer_Common_Ratio | https://proofwiki.org/wiki/Geometric_Sequence_of_Integers_with_Integer_Common_Ratio | [
"Geometric Sequences of Integers"
] | [
"Definition:Geometric Sequence",
"Definition:Length of Sequence",
"Definition:Integer",
"Definition:Geometric Sequence/Common Ratio",
"Definition:Integer",
"Definition:Geometric Sequence/Term",
"Definition:Divisor (Algebra)/Integer",
"Definition:Geometric Sequence/Term",
"Definition:Integer"
] | [
"Definition:Integer",
"Definition:Geometric Sequence",
"Definition:Geometric Sequence/Term",
"Definition:Integer",
"Integer Multiplication is Closed",
"Definition:Integer",
"Definition:Geometric Sequence/Term",
"Definition:Integer"
] |
proofwiki-10082 | Integers are Coprime iff Powers are Coprime | Let $a, b \in \Z$ be integers.
Then:
:$a \perp b \iff \forall n \in \N: a^n \perp b^n$
That is, two integers are coprime {{iff}} all their positive integer powers are coprime. | The forward implication is shown in Powers of Coprime Numbers are Coprime.
The reverse implication is shown by substituting $n = 1$.
{{qed}}
Category:Coprime Integers
8k2ky39jzctjiltp3egexjushvinn3s | Let $a, b \in \Z$ be [[Definition:Integer|integers]].
Then:
:$a \perp b \iff \forall n \in \N: a^n \perp b^n$
That is, two [[Definition:Integer|integers]] are [[Definition:Coprime Integers|coprime]] {{iff}} all their [[Definition:Integer Power|positive integer powers]] are [[Definition:Coprime Integers|coprime]]. | The forward implication is shown in [[Powers of Coprime Numbers are Coprime]].
The reverse implication is shown by substituting $n = 1$.
{{qed}}
[[Category:Coprime Integers]]
8k2ky39jzctjiltp3egexjushvinn3s | Integers are Coprime iff Powers are Coprime | https://proofwiki.org/wiki/Integers_are_Coprime_iff_Powers_are_Coprime | https://proofwiki.org/wiki/Integers_are_Coprime_iff_Powers_are_Coprime | [
"Coprime Integers"
] | [
"Definition:Integer",
"Definition:Integer",
"Definition:Coprime/Integers",
"Definition:Power (Algebra)/Integer",
"Definition:Coprime/Integers"
] | [
"Powers of Coprime Numbers are Coprime",
"Category:Coprime Integers"
] |
proofwiki-10083 | Successor Mapping of Peano Structure has no Fixed Point | Let $\PP = \struct {P, s, 0}$ be a Peano structure.
Then:
:$\forall n \in P: \map s n \ne n$
That is, the successor mapping has no fixed points. | Let $T$ be the set:
:$T = \set {n \in P: \map s n \ne n}$
We will use {{PeanoAxiom|5}} to prove that $T = P$. | Let $\PP = \struct {P, s, 0}$ be a [[Definition:Peano Structure|Peano structure]].
Then:
:$\forall n \in P: \map s n \ne n$
That is, the [[Definition:Successor Mapping|successor mapping]] has no [[Definition:Fixed Point|fixed points]]. | Let $T$ be the set:
:$T = \set {n \in P: \map s n \ne n}$
We will use {{PeanoAxiom|5}} to prove that $T = P$. | Successor Mapping of Peano Structure has no Fixed Point | https://proofwiki.org/wiki/Successor_Mapping_of_Peano_Structure_has_no_Fixed_Point | https://proofwiki.org/wiki/Successor_Mapping_of_Peano_Structure_has_no_Fixed_Point | [
"Peano's Axioms"
] | [
"Definition:Peano Structure",
"Definition:Successor Mapping",
"Definition:Fixed Point"
] | [] |
proofwiki-10084 | Integer Addition is Cancellable | The operation of addition on the set of integers $\Z$ is cancellable:
:$\forall x, y, z \in \Z: x + z = y + z \implies x = y$ | Let $x = \eqclass {a, b} {}$, $y = \eqclass {c, d} {}$ and $z = \eqclass {e, f} {}$ for some $x, y, z\in \Z$.
Then:
{{begin-eqn}}
{{eqn | l = x + z
| r = y + z
| c =
}}
{{eqn | ll= \leadsto
| l = \eqclass {a, b} {} + \eqclass {e, f} {}
| r = \eqclass {c, d} {} + \eqclass {e, f} {}
| c =
}... | The operation of [[Definition:Integer Addition|addition]] on the [[Definition:Set|set]] of [[Definition:Integer|integers]] $\Z$ is [[Definition:Cancellable|cancellable]]:
:$\forall x, y, z \in \Z: x + z = y + z \implies x = y$ | Let $x = \eqclass {a, b} {}$, $y = \eqclass {c, d} {}$ and $z = \eqclass {e, f} {}$ for some $x, y, z\in \Z$.
Then:
{{begin-eqn}}
{{eqn | l = x + z
| r = y + z
| c =
}}
{{eqn | ll= \leadsto
| l = \eqclass {a, b} {} + \eqclass {e, f} {}
| r = \eqclass {c, d} {} + \eqclass {e, f} {}
| c =... | Integer Addition is Cancellable | https://proofwiki.org/wiki/Integer_Addition_is_Cancellable | https://proofwiki.org/wiki/Integer_Addition_is_Cancellable | [
"Integer Addition"
] | [
"Definition:Addition/Integers",
"Definition:Set",
"Definition:Integer",
"Definition:Cancellable"
] | [
"Natural Number Addition is Cancellable",
"Natural Number Addition is Cancellable"
] |
proofwiki-10085 | Open Real Interval is Subset of Closed Real Interval | Let $a, b \in \R$ be real numbers.
Then:
:$\openint a b \subseteq \closedint a b$
where:
:$\openint a b$ is the open interval between $a$ and $b$
:$\closedint a b$ is the closed interval between $a$ and $b$. | Let $x \in \openint a b$.
Then by definition of open interval:
:$a < x < b$
Thus:
:$a \le x \le b$
and so by definition of closed interval:
:$x \in \closedint a b$
{{qed}} | Let $a, b \in \R$ be [[Definition:Real Number|real numbers]].
Then:
:$\openint a b \subseteq \closedint a b$
where:
:$\openint a b$ is the [[Definition:Open Real Interval|open interval]] between $a$ and $b$
:$\closedint a b$ is the [[Definition:Closed Real Interval|closed interval]] between $a$ and $b$. | Let $x \in \openint a b$.
Then by definition of [[Definition:Open Real Interval|open interval]]:
:$a < x < b$
Thus:
:$a \le x \le b$
and so by definition of [[Definition:Closed Real Interval|closed interval]]:
:$x \in \closedint a b$
{{qed}} | Open Real Interval is Subset of Closed Real Interval | https://proofwiki.org/wiki/Open_Real_Interval_is_Subset_of_Closed_Real_Interval | https://proofwiki.org/wiki/Open_Real_Interval_is_Subset_of_Closed_Real_Interval | [
"Real Intervals"
] | [
"Definition:Real Number",
"Definition:Real Interval/Open",
"Definition:Real Interval/Closed"
] | [
"Definition:Real Interval/Open",
"Definition:Real Interval/Closed"
] |
proofwiki-10086 | Complement Relative to Subset is Subset of Complement Relative to Superset | Let $A, B, C$ be sets such that $A \subseteq B \subseteq C$.
Then:
:$\relcomp B A \subseteq \relcomp C A$ | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \relcomp B A
| c =
}}
{{eqn | ll= \leadsto
| l = x
| o = \in
| r = B
| c = {{Defof|Relative Complement}}
}}
{{eqn | lo= \land
| l = x
| o = \notin
| r = A
| c = {{Defof|Relative Complement}}
}}
{{eqn | ll= \le... | Let $A, B, C$ be [[Definition:Set|sets]] such that $A \subseteq B \subseteq C$.
Then:
:$\relcomp B A \subseteq \relcomp C A$ | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \relcomp B A
| c =
}}
{{eqn | ll= \leadsto
| l = x
| o = \in
| r = B
| c = {{Defof|Relative Complement}}
}}
{{eqn | lo= \land
| l = x
| o = \notin
| r = A
| c = {{Defof|Relative Complement}}
}}
{{eqn | ll= \le... | Complement Relative to Subset is Subset of Complement Relative to Superset | https://proofwiki.org/wiki/Complement_Relative_to_Subset_is_Subset_of_Complement_Relative_to_Superset | https://proofwiki.org/wiki/Complement_Relative_to_Subset_is_Subset_of_Complement_Relative_to_Superset | [
"Subsets",
"Relative Complement"
] | [
"Definition:Set"
] | [] |
proofwiki-10087 | Complement of Relative Complement is Union with Complement | Let $A, B, C$ be sets such that $A \subseteq B \subseteq C$.
Then:
:$\relcomp C {\relcomp B A} = A \cup \relcomp C B$ | {{begin-eqn}}
{{eqn | l = \relcomp C {\relcomp B A}
| r = C \setminus \paren {B \setminus A}
| c = {{Defof|Relative Complement}}
}}
{{eqn | r = \paren {C \setminus B} \cup \paren {C \cap A}
| c = Set Difference with Set Difference is Union of Set Difference with Intersection
}}
{{eqn | r = \paren {C \... | Let $A, B, C$ be [[Definition:Set|sets]] such that $A \subseteq B \subseteq C$.
Then:
:$\relcomp C {\relcomp B A} = A \cup \relcomp C B$ | {{begin-eqn}}
{{eqn | l = \relcomp C {\relcomp B A}
| r = C \setminus \paren {B \setminus A}
| c = {{Defof|Relative Complement}}
}}
{{eqn | r = \paren {C \setminus B} \cup \paren {C \cap A}
| c = [[Set Difference with Set Difference is Union of Set Difference with Intersection]]
}}
{{eqn | r = \paren ... | Complement of Relative Complement is Union with Complement | https://proofwiki.org/wiki/Complement_of_Relative_Complement_is_Union_with_Complement | https://proofwiki.org/wiki/Complement_of_Relative_Complement_is_Union_with_Complement | [
"Subsets",
"Relative Complement"
] | [
"Definition:Set"
] | [
"Set Difference with Set Difference is Union of Set Difference with Intersection",
"Intersection with Subset is Subset",
"Union is Commutative"
] |
proofwiki-10088 | Set Union Preserves Subsets/Families of Sets | Let $I$ be an indexing set.
Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\alpha}_{\alpha \mathop \in I}$ be indexed families of subsets of a set $S$.
Let:
:$\forall \beta \in I: A_\beta \subseteq B_\beta$
Then:
:$\ds \bigcup_{\alpha \mathop \in I} A_\alpha \subseteq \bigcup_{\alpha \mathop \in I} B_\... | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \bigcup_{\alpha \mathop \in I} A_\alpha
| c =
}}
{{eqn | ll= \leadsto
| q = \exists \alpha \in I
| l = x
| o = \in
| r = A_\alpha
| c = {{Defof|Union of Family}}
}}
{{eqn | ll= \leadsto
| q = \exists \alpha \in I
| ... | Let $I$ be an [[Definition:Indexing Set|indexing set]].
Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\alpha}_{\alpha \mathop \in I}$ be [[Definition:Indexed Family of Subsets|indexed families of subsets]] of a [[Definition:Set|set]] $S$.
Let:
:$\forall \beta \in I: A_\beta \subseteq B_\beta$
Then... | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \bigcup_{\alpha \mathop \in I} A_\alpha
| c =
}}
{{eqn | ll= \leadsto
| q = \exists \alpha \in I
| l = x
| o = \in
| r = A_\alpha
| c = {{Defof|Union of Family}}
}}
{{eqn | ll= \leadsto
| q = \exists \alpha \in I
| ... | Set Union Preserves Subsets/Families of Sets | https://proofwiki.org/wiki/Set_Union_Preserves_Subsets/Families_of_Sets | https://proofwiki.org/wiki/Set_Union_Preserves_Subsets/Families_of_Sets | [
"Set Union Preserves Subsets",
"Indexed Families"
] | [
"Definition:Indexing Set",
"Definition:Indexing Set/Family of Subsets",
"Definition:Set"
] | [
"Definition:Subset"
] |
proofwiki-10089 | Set Union Preserves Subsets/General Result | Let $\mathbb S, \mathbb T$ be sets of sets.
Suppose that for each $S \in \mathbb S$ there exists a $T \in \mathbb T$ such that $S \subseteq T$.
Then $\bigcup \mathbb S \subseteq \bigcup \mathbb T$. | Let $x \in \bigcup \mathbb S$.
By the definition of union, there exists an $S \in \mathbb S$ such that $x \in S$.
By the premise, there exists a $T \in \mathbb T$ such that $S \subseteq T$.
By the definition of Subset:
:$x \in T$
Thus by the definition of union:
:$x \in \bigcup \mathbb T$
{{qed}}
Category:Set Union Pre... | Let $\mathbb S, \mathbb T$ be [[Definition:Set of Sets|sets of sets]].
Suppose that for each $S \in \mathbb S$ there exists a $T \in \mathbb T$ such that $S \subseteq T$.
Then $\bigcup \mathbb S \subseteq \bigcup \mathbb T$. | Let $x \in \bigcup \mathbb S$.
By the definition of [[Definition:Set Union|union]], there exists an $S \in \mathbb S$ such that $x \in S$.
By the premise, there exists a $T \in \mathbb T$ such that $S \subseteq T$.
By the definition of [[Definition:Subset|Subset]]:
:$x \in T$
Thus by the definition of [[Definition:... | Set Union Preserves Subsets/General Result | https://proofwiki.org/wiki/Set_Union_Preserves_Subsets/General_Result | https://proofwiki.org/wiki/Set_Union_Preserves_Subsets/General_Result | [
"Set Union Preserves Subsets"
] | [
"Definition:Set of Sets"
] | [
"Definition:Set Union",
"Definition:Subset",
"Definition:Set Union",
"Category:Set Union Preserves Subsets"
] |
proofwiki-10090 | Intersection of Family is Subset of Intersection of Subset of Family | Let $I$ be an indexing set.
Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be an indexed family of subsets of a set $S$.
Let $J \subseteq I$.
Then:
:$\ds \bigcap_{\alpha \mathop \in I} A_\alpha \subseteq \bigcap_{\alpha \mathop \in J} A_\alpha$
where $\ds \bigcap_{\alpha \mathop \in I} A_\alpha$ denotes the intersecti... | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \bigcap_{\alpha \mathop \in I} A_\alpha
| c =
}}
{{eqn | ll= \leadsto
| q = \forall \alpha \in I
| l = x
| o = \in
| r = A_\alpha
| c = Intersection is Subset
}}
{{eqn | ll= \leadsto
| q = \forall \alpha \in J
| l = ... | Let $I$ be an [[Definition:Indexing Set|indexing set]].
Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family of Subsets|indexed family of subsets]] of a [[Definition:Set|set]] $S$.
Let $J \subseteq I$.
Then:
:$\ds \bigcap_{\alpha \mathop \in I} A_\alpha \subseteq \bigcap_{\alpha \mathop... | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \bigcap_{\alpha \mathop \in I} A_\alpha
| c =
}}
{{eqn | ll= \leadsto
| q = \forall \alpha \in I
| l = x
| o = \in
| r = A_\alpha
| c = [[Intersection is Subset/Family of Sets|Intersection is Subset]]
}}
{{eqn | ll= \leadsto
... | Intersection of Family is Subset of Intersection of Subset of Family | https://proofwiki.org/wiki/Intersection_of_Family_is_Subset_of_Intersection_of_Subset_of_Family | https://proofwiki.org/wiki/Intersection_of_Family_is_Subset_of_Intersection_of_Subset_of_Family | [
"Subsets",
"Set Intersection",
"Indexed Families"
] | [
"Definition:Indexing Set",
"Definition:Indexing Set/Family of Subsets",
"Definition:Set",
"Definition:Set Intersection/Family of Sets"
] | [
"Intersection is Subset/Family of Sets"
] |
proofwiki-10091 | Union of Subset of Family is Subset of Union of Family | Let $I$ be an indexing set.
Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be an indexed family of subsets of a set $S$.
Let $J \subseteq I$
Then:
:$\ds \bigcup_{\alpha \mathop \in J} A_\alpha \subseteq \bigcup_{\alpha \mathop \in I} A_\alpha$
where $\ds \bigcup_{\alpha \mathop \in I} A_\alpha$ denotes the union of $\... | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \bigcup_{\alpha \mathop \in J} A_\alpha
| c =
}}
{{eqn | ll= \leadsto
| q = \exists \alpha \in J
| l = x
| o = \in
| r = A_\alpha
| c = {{Defof|Union of Family}}
}}
{{eqn | ll= \leadsto
| q = \exists \alpha \in I
| l... | Let $I$ be an [[Definition:Indexing Set|indexing set]].
Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family of Subsets|indexed family of subsets]] of a [[Definition:Set|set]] $S$.
Let $J \subseteq I$
Then:
:$\ds \bigcup_{\alpha \mathop \in J} A_\alpha \subseteq \bigcup_{\alpha \mathop ... | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \bigcup_{\alpha \mathop \in J} A_\alpha
| c =
}}
{{eqn | ll= \leadsto
| q = \exists \alpha \in J
| l = x
| o = \in
| r = A_\alpha
| c = {{Defof|Union of Family}}
}}
{{eqn | ll= \leadsto
| q = \exists \alpha \in I
| l... | Union of Subset of Family is Subset of Union of Family | https://proofwiki.org/wiki/Union_of_Subset_of_Family_is_Subset_of_Union_of_Family | https://proofwiki.org/wiki/Union_of_Subset_of_Family_is_Subset_of_Union_of_Family | [
"Subsets",
"Set Union",
"Indexed Families"
] | [
"Definition:Indexing Set",
"Definition:Indexing Set/Family of Subsets",
"Definition:Set",
"Definition:Set Union/Family of Sets"
] | [
"Set is Subset of Union/Family of Sets"
] |
proofwiki-10092 | Intersection of Open Intervals of Positive Reals is Empty | Let $\R_{>0}$ be the set of strictly positive real numbers.
For all $x \in \R_{>0}$, let $A_x$ be the open real interval $\openint 0 x$.
Then:
:$\ds \bigcap_{x \mathop \in \R_{>0} } A_x = \O$ | Let $\ds A = \bigcap_{x \mathop \in \R_{>0} } A_x$.
{{AimForCont}} $A \ne \O$.
Then:
:$\exists y \in \R_{>0}: y \in A$
By definition of open interval:
:$y \notin \openint 0 y = A_y$
and so by definition of intersection of family:
:$y \notin A$
From this contradiction it follows that $A$ has no elements.
That is:
:$\ds ... | Let $\R_{>0}$ be the [[Definition:Set|set]] of [[Definition:Strictly Positive Real Number|strictly positive real numbers]].
For all $x \in \R_{>0}$, let $A_x$ be the [[Definition:Open Real Interval|open real interval]] $\openint 0 x$.
Then:
:$\ds \bigcap_{x \mathop \in \R_{>0} } A_x = \O$ | Let $\ds A = \bigcap_{x \mathop \in \R_{>0} } A_x$.
{{AimForCont}} $A \ne \O$.
Then:
:$\exists y \in \R_{>0}: y \in A$
By definition of [[Definition:Open Real Interval|open interval]]:
:$y \notin \openint 0 y = A_y$
and so by definition of [[Definition:Intersection of Family|intersection of family]]:
:$y \notin A$
... | Intersection of Open Intervals of Positive Reals is Empty | https://proofwiki.org/wiki/Intersection_of_Open_Intervals_of_Positive_Reals_is_Empty | https://proofwiki.org/wiki/Intersection_of_Open_Intervals_of_Positive_Reals_is_Empty | [
"Real Intervals"
] | [
"Definition:Set",
"Definition:Strictly Positive/Real Number",
"Definition:Real Interval/Open"
] | [
"Definition:Real Interval/Open",
"Definition:Set Intersection/Family of Sets",
"Proof by Contradiction",
"Definition:Element"
] |
proofwiki-10093 | Union of Open Intervals of Positive Reals is Set of Strictly Positive Reals | Let $\R_{>0}$ be the set of strictly positive real numbers.
For all $x \in \R_{>0}$, let $A_x$ be the open real interval $\openint 0 x$.
Then:
:$\ds \bigcup_{x \mathop \in \R_{>0} } A_x = \R_{>0}$ | Let $\ds A = \bigcup_{x \mathop \in \R_{>0} } A_x$.
Let $y \in A$.
Then by definition of union of family:
:$\exists x \in \R_{>0}: y \in A_x$
As $A_x \subseteq \R_{>0}$ it follows by definition of subset that:
:$y \in \R_{>0}$
So:
:$\ds \bigcup_{x \mathop \in \R_{>0} } A_x \subseteq \R_{>0}$
{{qed|lemma}}
Let $y \in \R... | Let $\R_{>0}$ be the [[Definition:Set|set]] of [[Definition:Strictly Positive Real Number|strictly positive real numbers]].
For all $x \in \R_{>0}$, let $A_x$ be the [[Definition:Open Real Interval|open real interval]] $\openint 0 x$.
Then:
:$\ds \bigcup_{x \mathop \in \R_{>0} } A_x = \R_{>0}$ | Let $\ds A = \bigcup_{x \mathop \in \R_{>0} } A_x$.
Let $y \in A$.
Then by definition of [[Definition:Union of Family|union of family]]:
:$\exists x \in \R_{>0}: y \in A_x$
As $A_x \subseteq \R_{>0}$ it follows by definition of [[Definition:Subset|subset]] that:
:$y \in \R_{>0}$
So:
:$\ds \bigcup_{x \mathop \in \R_... | Union of Open Intervals of Positive Reals is Set of Strictly Positive Reals | https://proofwiki.org/wiki/Union_of_Open_Intervals_of_Positive_Reals_is_Set_of_Strictly_Positive_Reals | https://proofwiki.org/wiki/Union_of_Open_Intervals_of_Positive_Reals_is_Set_of_Strictly_Positive_Reals | [
"Real Intervals"
] | [
"Definition:Set",
"Definition:Strictly Positive/Real Number",
"Definition:Real Interval/Open"
] | [
"Definition:Set Union/Family of Sets",
"Definition:Subset",
"Axiom of Archimedes",
"Definition:Set Union/Family of Sets",
"Definition:Subset",
"Definition:Set Equality/Definition 2"
] |
proofwiki-10094 | Intersection of Closed Intervals of Positive Reals is Zero | Let $\R_{> 0}$ be the set of strictly positive real numbers.
For all $x \in \R_{> 0}$, let $B_x$ be the closed real interval $\closedint 0 x$.
Then:
:$\ds \bigcap_{x \mathop \in \R_{> 0} } B_x = \set 0$ | Let $\ds B = \bigcap_{x \mathop \in \R_{> 0} } B_x$.
We have that:
:$\forall x \in \R_{> 0}: 0 \in \closedint 0 x$
So by definition of intersection:
:$0 \in B$
and so by Singleton of Element is Subset:
:$\ds \set 0 \subseteq \bigcap_{x \mathop \in \R_{> 0} } B_x$
{{AimForCont}} $\exists y \in \R_{> 0}: y \in B$.
By def... | Let $\R_{> 0}$ be the [[Definition:Set|set]] of [[Definition:Strictly Positive Real Number|strictly positive real numbers]].
For all $x \in \R_{> 0}$, let $B_x$ be the [[Definition:Closed Real Interval|closed real interval]] $\closedint 0 x$.
Then:
:$\ds \bigcap_{x \mathop \in \R_{> 0} } B_x = \set 0$ | Let $\ds B = \bigcap_{x \mathop \in \R_{> 0} } B_x$.
We have that:
:$\forall x \in \R_{> 0}: 0 \in \closedint 0 x$
So by definition of [[Definition:Intersection of Family|intersection]]:
:$0 \in B$
and so by [[Singleton of Element is Subset]]:
:$\ds \set 0 \subseteq \bigcap_{x \mathop \in \R_{> 0} } B_x$
{{AimForCo... | Intersection of Closed Intervals of Positive Reals is Zero | https://proofwiki.org/wiki/Intersection_of_Closed_Intervals_of_Positive_Reals_is_Zero | https://proofwiki.org/wiki/Intersection_of_Closed_Intervals_of_Positive_Reals_is_Zero | [
"Real Intervals"
] | [
"Definition:Set",
"Definition:Strictly Positive/Real Number",
"Definition:Real Interval/Closed"
] | [
"Definition:Set Intersection/Family of Sets",
"Singleton of Element is Subset",
"Definition:Set Intersection/Family of Sets",
"Proof by Contradiction",
"Definition:Set Equality/Definition 2"
] |
proofwiki-10095 | Union of Closed Intervals of Positive Reals is Set of Positive Reals | Let $\R_{>0}$ be the set of strictly positive real numbers.
For all $x \in \R_{> 0}$, let $B_x$ be the closed real interval $\closedint 0 x$.
Then:
:$\ds \bigcup_{x \mathop \in \R_{>0} } B_x = \R_{\ge 0}$ | Let $\ds B = \bigcap_{x \mathop \in \R_{>0} } B_x$.
Let $y \in B$.
Then by definition of union of family:
:$\exists x \in \R_{>0}: y \in B_x$
As $B_x \subseteq \R_{>0}$ it follows by definition of subset that:
:$y = 0$
or
:$y \in \R_{>0}$
In either case, $y \in \R_{\ge 0}$
So:
:$\ds \bigcap_{x \mathop \in \R_{>0} } B_x... | Let $\R_{>0}$ be the [[Definition:Set|set]] of [[Definition:Strictly Positive Real Number|strictly positive real numbers]].
For all $x \in \R_{> 0}$, let $B_x$ be the [[Definition:Closed Real Interval|closed real interval]] $\closedint 0 x$.
Then:
:$\ds \bigcup_{x \mathop \in \R_{>0} } B_x = \R_{\ge 0}$ | Let $\ds B = \bigcap_{x \mathop \in \R_{>0} } B_x$.
Let $y \in B$.
Then by definition of [[Definition:Union of Family|union of family]]:
:$\exists x \in \R_{>0}: y \in B_x$
As $B_x \subseteq \R_{>0}$ it follows by definition of [[Definition:Subset|subset]] that:
:$y = 0$
or
:$y \in \R_{>0}$
In either case, $y \in \... | Union of Closed Intervals of Positive Reals is Set of Positive Reals | https://proofwiki.org/wiki/Union_of_Closed_Intervals_of_Positive_Reals_is_Set_of_Positive_Reals | https://proofwiki.org/wiki/Union_of_Closed_Intervals_of_Positive_Reals_is_Set_of_Positive_Reals | [
"Real Intervals"
] | [
"Definition:Set",
"Definition:Strictly Positive/Real Number",
"Definition:Real Interval/Closed"
] | [
"Definition:Set Union/Family of Sets",
"Definition:Subset",
"Axiom of Archimedes",
"Definition:Set Union/Family of Sets",
"Definition:Subset",
"Definition:Set Equality/Definition 2"
] |
proofwiki-10096 | Relative Complement of Cartesian Product | Let $A$ and $B$ be sets.
Let $X \subseteq A$ and $Y \subseteq B$.
Then:
:$\relcomp {A \mathop \times B} {X \times Y} = \paren {A \times \relcomp B Y} \cup \paren {\relcomp A X \times B}$ | From Set with Relative Complement forms Partition:
:$A = \set {X \mid \relcomp A X}$
:$B = \set {Y \mid \relcomp B Y}$
and so by definition of partition:
:$A = X \cup \relcomp A X$
:$B = Y \cup \relcomp B Y$
By Cartesian Product of Unions:
:$A \times B = \paren {X \times Y} \cup \paren {\relcomp A X \times \relcomp B Y... | Let $A$ and $B$ be [[Definition:Set|sets]].
Let $X \subseteq A$ and $Y \subseteq B$.
Then:
:$\relcomp {A \mathop \times B} {X \times Y} = \paren {A \times \relcomp B Y} \cup \paren {\relcomp A X \times B}$ | From [[Set with Relative Complement forms Partition]]:
:$A = \set {X \mid \relcomp A X}$
:$B = \set {Y \mid \relcomp B Y}$
and so by definition of [[Definition:Partition (Set Theory)|partition]]:
:$A = X \cup \relcomp A X$
:$B = Y \cup \relcomp B Y$
By [[Cartesian Product of Unions]]:
:$A \times B = \paren {X \time... | Relative Complement of Cartesian Product/Proof 1 | https://proofwiki.org/wiki/Relative_Complement_of_Cartesian_Product | https://proofwiki.org/wiki/Relative_Complement_of_Cartesian_Product/Proof_1 | [
"Relative Complement of Cartesian Product",
"Cartesian Product",
"Relative Complement"
] | [
"Definition:Set"
] | [
"Set Difference and Intersection form Partition/Corollary 2",
"Definition:Set Partition",
"Cartesian Product of Unions"
] |
proofwiki-10097 | Relative Complement of Cartesian Product | Let $A$ and $B$ be sets.
Let $X \subseteq A$ and $Y \subseteq B$.
Then:
:$\relcomp {A \mathop \times B} {X \times Y} = \paren {A \times \relcomp B Y} \cup \paren {\relcomp A X \times B}$ | {{begin-eqn}}
{{eqn | o =
| r = \relcomp {A \mathop \times B} {X \times Y}
}}
{{eqn | ll= \leadstoandfrom
| o =
| r = A \times B \setminus X \times Y
| c = {{Defof|Relative Complement}}
}}
{{eqn | ll= \leadstoandfrom
| o =
| r = \set {\tuple {x, y}: x \in A \land y \in B \land \neg \... | Let $A$ and $B$ be [[Definition:Set|sets]].
Let $X \subseteq A$ and $Y \subseteq B$.
Then:
:$\relcomp {A \mathop \times B} {X \times Y} = \paren {A \times \relcomp B Y} \cup \paren {\relcomp A X \times B}$ | {{begin-eqn}}
{{eqn | o =
| r = \relcomp {A \mathop \times B} {X \times Y}
}}
{{eqn | ll= \leadstoandfrom
| o =
| r = A \times B \setminus X \times Y
| c = {{Defof|Relative Complement}}
}}
{{eqn | ll= \leadstoandfrom
| o =
| r = \set {\tuple {x, y}: x \in A \land y \in B \land \neg \... | Relative Complement of Cartesian Product/Proof 2 | https://proofwiki.org/wiki/Relative_Complement_of_Cartesian_Product | https://proofwiki.org/wiki/Relative_Complement_of_Cartesian_Product/Proof_2 | [
"Relative Complement of Cartesian Product",
"Cartesian Product",
"Relative Complement"
] | [
"Definition:Set"
] | [
"De Morgan's Laws (Logic)/Disjunction of Negations",
"Definition:Set Equality"
] |
proofwiki-10098 | Subset of Cartesian Product not necessarily Cartesian Product of Subsets | Let $A$ and $B$ be sets.
Let $A$ and $B$ both have at least two distinct elements.
Then there exists $W \subseteq A \times B$ such that $W$ is not the cartesian product of a subset of $A$ and a subset of $B$. | Let $a \in A, b \in A, c \in B, d \in B$ be arbitrary elements of $A$ and $B$.
Let:
:$W = \set {\tuple {a, c}, \tuple {a, d}, \tuple {b, d} }$
Then $W \subseteq A \times B$.
Suppose $W = X \times Y$ such that $X \subseteq A, Y \subseteq B$.
Then $a, b \in X$ and $c, d \in Y$.
But $X \times Y$ also contains $\tuple {b, ... | Let $A$ and $B$ be [[Definition:Set|sets]].
Let $A$ and $B$ both have at least two [[Definition:Distinct|distinct]] [[Definition:Element|elements]].
Then there exists $W \subseteq A \times B$ such that $W$ is not the [[Definition:Cartesian Product|cartesian product]] of a [[Definition:Subset|subset]] of $A$ and a [[... | Let $a \in A, b \in A, c \in B, d \in B$ be arbitrary elements of $A$ and $B$.
Let:
:$W = \set {\tuple {a, c}, \tuple {a, d}, \tuple {b, d} }$
Then $W \subseteq A \times B$.
Suppose $W = X \times Y$ such that $X \subseteq A, Y \subseteq B$.
Then $a, b \in X$ and $c, d \in Y$.
But $X \times Y$ also contains $\tuple... | Subset of Cartesian Product not necessarily Cartesian Product of Subsets | https://proofwiki.org/wiki/Subset_of_Cartesian_Product_not_necessarily_Cartesian_Product_of_Subsets | https://proofwiki.org/wiki/Subset_of_Cartesian_Product_not_necessarily_Cartesian_Product_of_Subsets | [
"Cartesian Product"
] | [
"Definition:Set",
"Definition:Distinct",
"Definition:Element",
"Definition:Cartesian Product",
"Definition:Subset",
"Definition:Subset"
] | [] |
proofwiki-10099 | Subset equals Preimage of Image implies Injection | Let $f: S \to T$ be a mapping.
Let $f^\to: \powerset S \to \powerset T$ be the direct image mapping of $f$.
Similarly, let $f^\gets: \powerset T \to \powerset S$ be the inverse image mapping of $f$.
Let:
:$\forall A \in \powerset S: A = \map {\paren {f^\gets \circ f^\to} } A$
Then $f$ is an injection. | Let $f$ be such that:
:$\forall A \in \powerset S: A = \map {\paren {f^\gets \circ f^\to} } A$
In particular, it holds for all subsets of $A$ which are singletons.
Now, consider any $x, y \in A$.
We have:
{{begin-eqn}}
{{eqn | l = \map f x
| r = \map f y
| c =
}}
{{eqn | ll= \leadsto
| l = \set {\map ... | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $f^\to: \powerset S \to \powerset T$ be the [[Definition:Direct Image Mapping of Mapping|direct image mapping]] of $f$.
Similarly, let $f^\gets: \powerset T \to \powerset S$ be the [[Definition:Inverse Image Mapping of Mapping|inverse image mapping]] of $f$.
... | Let $f$ be such that:
:$\forall A \in \powerset S: A = \map {\paren {f^\gets \circ f^\to} } A$
In particular, it holds for all [[Definition:Subset|subsets]] of $A$ which are [[Definition:Singleton|singletons]].
Now, consider any $x, y \in A$.
We have:
{{begin-eqn}}
{{eqn | l = \map f x
| r = \map f y
|... | Subset equals Preimage of Image implies Injection/Proof 1 | https://proofwiki.org/wiki/Subset_equals_Preimage_of_Image_implies_Injection | https://proofwiki.org/wiki/Subset_equals_Preimage_of_Image_implies_Injection/Proof_1 | [
"Injections",
"Subsets",
"Subset equals Preimage of Image implies Injection"
] | [
"Definition:Mapping",
"Definition:Direct Image Mapping/Mapping",
"Definition:Inverse Image Mapping/Mapping",
"Definition:Injection"
] | [
"Definition:Subset",
"Definition:Singleton",
"Definition:Injection"
] |
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