Mixing
What is the correct notation for the set of rational numbers $frac{n}{m}$ with the constraint that $1le n,m...
I have set that I want to neatly write down/present with set notation. The set contains:
All the rational numbers $frac{n}{m}$ with the constraint that $1le n,m le5, n,min mathbb{Z}$.
I have come up with a few ways to write it down but I am not sure which one (if any) is correct.
$$begin{align}
left{ frac{m}{n} vert n,min mathbb{Z} land1le nle5 land1le mle5right} tag 1\
left{ frac{m}{n} in mathbb{Q}vert n,min mathbb{Z} land1le nle5 land1le mle5right} tag 2\
left{frac{n}{m}in mathbb{Q}vert n,m in mathbb{Z}, 1le n,mle 5right} tag 3
end{align}$$
elementary-set-theory
edited Nov 20 '18 at 9:37
user334732
4,24311240
asked Nov 17 '18 at 12:03
Nullspace
307110
add a comment |
I have set that I want to neatly write down/present with set notation. The set contains:
All the rational numbers $frac{n}{m}$ with the constraint that $1le n,m le5, n,min mathbb{Z}$.
I have come up with a few ways to write it down but I am not sure which one (if any) is correct.
$$begin{align}
left{ frac{m}{n} vert n,min mathbb{Z} land1le nle5 land1le mle5right} tag 1\
left{ frac{m}{n} in mathbb{Q}vert n,min mathbb{Z} land1le nle5 land1le mle5right} tag 2\
left{frac{n}{m}in mathbb{Q}vert n,m in mathbb{Z}, 1le n,mle 5right} tag 3
end{align}$$
elementary-set-theory
edited Nov 20 '18 at 9:37
user334732
4,24311240
asked Nov 17 '18 at 12:03
Nullspace
307110
3
it might be a matter of taste, and depends on the context and conventions, that is it might be possible to automatically assume that $m,n$ denote integers (which you do in the title), then ${frac m n: 1le n,m le5}$. The abbreviation $n,min mathbb{Z}$ is not used by some authors (perhaps it doesn't say that $nin mathbb{Z}$), who prefer $nin mathbb{Z},min mathbb{Z}$, in this case ${ frac{m}{n} | nin mathbb{Z},min mathbb{Z}, 1le nle5, 1le mle5}$. It is a matter of taste, unless you may want to write a formal theorem prover and syntax becomes important (don't know much).
– Mirko
Nov 17 '18 at 12:14
@Mirko Thank you. I will problably use the first or second and combine it with your suggestion of writing $n in mathbb{Z}, m in mathbb{z}$. Thanks again.
– Nullspace
Nov 17 '18 at 12:24
I just added left and right to your brackets so they matched the height of their contents.
– user334732
Nov 20 '18 at 9:38
add a comment |
I have set that I want to neatly write down/present with set notation. The set contains:
All the rational numbers $frac{n}{m}$ with the constraint that $1le n,m le5, n,min mathbb{Z}$.
I have come up with a few ways to write it down but I am not sure which one (if any) is correct.
$$begin{align}
left{ frac{m}{n} vert n,min mathbb{Z} land1le nle5 land1le mle5right} tag 1\
left{ frac{m}{n} in mathbb{Q}vert n,min mathbb{Z} land1le nle5 land1le mle5right} tag 2\
left{frac{n}{m}in mathbb{Q}vert n,m in mathbb{Z}, 1le n,mle 5right} tag 3
end{align}$$
elementary-set-theory
edited Nov 20 '18 at 9:37
user334732
4,24311240
asked Nov 17 '18 at 12:03
Nullspace
307110
I have set that I want to neatly write down/present with set notation. The set contains:
All the rational numbers $frac{n}{m}$ with the constraint that $1le n,m le5, n,min mathbb{Z}$.
I have come up with a few ways to write it down but I am not sure which one (if any) is correct.
$$begin{align}
left{ frac{m}{n} vert n,min mathbb{Z} land1le nle5 land1le mle5right} tag 1\
left{ frac{m}{n} in mathbb{Q}vert n,min mathbb{Z} land1le nle5 land1le mle5right} tag 2\
left{frac{n}{m}in mathbb{Q}vert n,m in mathbb{Z}, 1le n,mle 5right} tag 3
end{align}$$
elementary-set-theory
elementary-set-theory
edited Nov 20 '18 at 9:37
user334732
4,24311240
asked Nov 17 '18 at 12:03
Nullspace
307110
edited Nov 20 '18 at 9:37
user334732
4,24311240
asked Nov 17 '18 at 12:03
Nullspace
307110
edited Nov 20 '18 at 9:37
user334732
4,24311240
edited Nov 20 '18 at 9:37
user334732
4,24311240
edited Nov 20 '18 at 9:37
user334732
4,24311240
4,24311240
asked Nov 17 '18 at 12:03
Nullspace
307110
asked Nov 17 '18 at 12:03
Nullspace
307110
asked Nov 17 '18 at 12:03
Nullspace
307110
307110
3
it might be a matter of taste, and depends on the context and conventions, that is it might be possible to automatically assume that $m,n$ denote integers (which you do in the title), then ${frac m n: 1le n,m le5}$. The abbreviation $n,min mathbb{Z}$ is not used by some authors (perhaps it doesn't say that $nin mathbb{Z}$), who prefer $nin mathbb{Z},min mathbb{Z}$, in this case ${ frac{m}{n} | nin mathbb{Z},min mathbb{Z}, 1le nle5, 1le mle5}$. It is a matter of taste, unless you may want to write a formal theorem prover and syntax becomes important (don't know much).
– Mirko
Nov 17 '18 at 12:14
@Mirko Thank you. I will problably use the first or second and combine it with your suggestion of writing $n in mathbb{Z}, m in mathbb{z}$. Thanks again.
– Nullspace
Nov 17 '18 at 12:24
I just added left and right to your brackets so they matched the height of their contents.
– user334732
Nov 20 '18 at 9:38
add a comment |
3
it might be a matter of taste, and depends on the context and conventions, that is it might be possible to automatically assume that $m,n$ denote integers (which you do in the title), then ${frac m n: 1le n,m le5}$. The abbreviation $n,min mathbb{Z}$ is not used by some authors (perhaps it doesn't say that $nin mathbb{Z}$), who prefer $nin mathbb{Z},min mathbb{Z}$, in this case ${ frac{m}{n} | nin mathbb{Z},min mathbb{Z}, 1le nle5, 1le mle5}$. It is a matter of taste, unless you may want to write a formal theorem prover and syntax becomes important (don't know much).
– Mirko
Nov 17 '18 at 12:14
@Mirko Thank you. I will problably use the first or second and combine it with your suggestion of writing $n in mathbb{Z}, m in mathbb{z}$. Thanks again.
– Nullspace
Nov 17 '18 at 12:24
I just added left and right to your brackets so they matched the height of their contents.
– user334732
Nov 20 '18 at 9:38
3
3
it might be a matter of taste, and depends on the context and conventions, that is it might be possible to automatically assume that $m,n$ denote integers (which you do in the title), then ${frac m n: 1le n,m le5}$. The abbreviation $n,min mathbb{Z}$ is not used by some authors (perhaps it doesn't say that $nin mathbb{Z}$), who prefer $nin mathbb{Z},min mathbb{Z}$, in this case ${ frac{m}{n} | nin mathbb{Z},min mathbb{Z}, 1le nle5, 1le mle5}$. It is a matter of taste, unless you may want to write a formal theorem prover and syntax becomes important (don't know much).
– Mirko
Nov 17 '18 at 12:14
it might be a matter of taste, and depends on the context and conventions, that is it might be possible to automatically assume that $m,n$ denote integers (which you do in the title), then ${frac m n: 1le n,m le5}$. The abbreviation $n,min mathbb{Z}$ is not used by some authors (perhaps it doesn't say that $nin mathbb{Z}$), who prefer $nin mathbb{Z},min mathbb{Z}$, in this case ${ frac{m}{n} | nin mathbb{Z},min mathbb{Z}, 1le nle5, 1le mle5}$. It is a matter of taste, unless you may want to write a formal theorem prover and syntax becomes important (don't know much).
– Mirko
Nov 17 '18 at 12:14
@Mirko Thank you. I will problably use the first or second and combine it with your suggestion of writing $n in mathbb{Z}, m in mathbb{z}$. Thanks again.
– Nullspace
Nov 17 '18 at 12:24
@Mirko Thank you. I will problably use the first or second and combine it with your suggestion of writing $n in mathbb{Z}, m in mathbb{z}$. Thanks again.
– Nullspace
Nov 17 '18 at 12:24
I just added left and right to your brackets so they matched the height of their contents.
– user334732
Nov 20 '18 at 9:38
I just added left and right to your brackets so they matched the height of their contents.
– user334732
Nov 20 '18 at 9:38
add a comment |
4 Answers
4
active
oldest
votes
$(1)$ and $(2)$ are okay, though I would rather do it with $frac{n}{m}$ instead of $frac{m}{n}$.
In $(2)$ the part $inmathbb Q$ is redundant, but that does not harm correctness.
$(3)$ is wrong (e.g. it demands that $mleq1$)
Actually you cannot speak of "the" correct notation of..., but of "a" correct notation of...
answered Nov 17 '18 at 12:11
drhab
97.9k544129
add a comment |
The definition $1$ and $2$ seems correct to me, we could also use for example
$$Big{ frac{m}{n}in mathbb{Q} ,vert , n,min {1,2,3,4,5}subseteq mathbb{Z}Big} $$
answered Nov 17 '18 at 12:07
gimusi
1
Thanks for your answer. The consesus seems to be that 1 and 2 is ok. I will use on of them with slight modification.
– Nullspace
Nov 17 '18 at 12:22
add a comment |
in an axiomatic view of set theory, certain notations that we use are justified by corresponding axioms.
As the very simplest example of this, the notation ${a,b}$ for a set that has $a$ and $b$ as elements and nothing else, is justified by the Pairing Axiom which states that for any two sets $a,b$ there exists a set $c$ such that $$forall xcolon xin cleftrightarrow x=alor x=b.$$ It also follows that this set is unique and we introduce the notiation ${a,b}$ as a shortcut for "the set $c$ guaranteed to exist by the pairing axiom applied to $a$ and $b$".
Likewise, the Axiom Scheme of Comprehension states that for any set $a$ and predicate $phi$, there exists a set $c$ such that
$$forall xcolon xin cleftrightarrow xin aland phi(x).$$
We usually use the notation ${,xin Amid phi(x),}$ as a shortcut for "the unique set $c$ guaranteed to exist by the comprehension scheme applied to $a$ and $phi$".
Also, the Axiom Scheme of Replacement states that for every set $a$ and function $F$, there exists a set $c$ such that
$$forall xcolon xin cleftrightarrow exists tcolon t in aland x=F(t).$$
We usually use the notation ${,F(t)mid tin a,}$ as a shortcut for "the unique set $c$ guaranteed to exist by the comprehension scheme applied to $a$ and $F$".
With this in mind, your
version $(1)$ is a formally correct notation inspired by the Axiom Schema of Replacement.
This nearly makes version $(2)$ okay as well. However, in order to make it match the Axiom Schema of Comprehension, I would - for extra strictness - prefer to see it in the following form:
$$tag{2'}{,qinBbb Qmid exists m,nin Bbb Zcolon (1le mle 5land 1le nle 5land q=tfrac mn),}$$
Note however, that in most contexts, legibility trumps strict formalism and that certain "colloqialsms" usually enter most texts. For example, in the above I myself used $exists m,ninBbb Zcolon ldots$ as a "colloquial" short form for $exists mcolon exists ncolon (min Bbb Zland ninBbb Zlandldots)$.
Finally, I suppose you mistyped something in $(3)$.
EDIT: After the original question was edited, variant $(3)$ might be understandable and might be interpreted by many to describe the intended set. However, this variant is quite ambiguous: One can interpret $1le n,mle 5$ as "both numbers $n,m$ are $ge 1$ and $le 5$". But in the same instance, you use the comma as a logical and, thus suggesting another possible reading "$n,min Bbb Z$ and $1le n$ and $mle 5$". I consider this ambiguity fatal. This could be saved by making it clear that you do not habitually use the comma to denote logical conjunction, i.e., write $n,minBbb Zland 1le n,mle 5$.
answered Nov 17 '18 at 12:32
Hagen von Eitzen
276k21269496
Thank you for that detailed answer. Yeah you are right. I mistyped something in (3). I will fix that quickly.
– Nullspace
Nov 17 '18 at 13:12
add a comment |
You could use $$left{frac nm mid (n, m) in [[1,5]]^2right}$$ (not to sure how to obtain the notation for an integer interval on this site, but $[[1,5]]$ is a short way of writing ${1, ..., 5}$)
answered Nov 17 '18 at 12:16
krirkrirk
1,458518
in what language is $[[1;5]]$ or $[1;5]$ an integer interval? I think I have seen $[1:5]$ or $[1..5]$ not sure myself, perhaps in Python, Haskell. In math one would want to introduce such a notation before formally using it, I don't think it is standard.
– Mirko
Nov 17 '18 at 12:20
@Mirko In French litterature, it's a standard notation. Thought it was the case in English too!
– krirkrirk
Nov 17 '18 at 12:23
add a comment |
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
$(1)$ and $(2)$ are okay, though I would rather do it with $frac{n}{m}$ instead of $frac{m}{n}$.
In $(2)$ the part $inmathbb Q$ is redundant, but that does not harm correctness.
$(3)$ is wrong (e.g. it demands that $mleq1$)
Actually you cannot speak of "the" correct notation of..., but of "a" correct notation of...
answered Nov 17 '18 at 12:11
drhab
97.9k544129
add a comment |
$(1)$ and $(2)$ are okay, though I would rather do it with $frac{n}{m}$ instead of $frac{m}{n}$.
In $(2)$ the part $inmathbb Q$ is redundant, but that does not harm correctness.
$(3)$ is wrong (e.g. it demands that $mleq1$)
Actually you cannot speak of "the" correct notation of..., but of "a" correct notation of...
answered Nov 17 '18 at 12:11
drhab
97.9k544129
add a comment |
$(1)$ and $(2)$ are okay, though I would rather do it with $frac{n}{m}$ instead of $frac{m}{n}$.
In $(2)$ the part $inmathbb Q$ is redundant, but that does not harm correctness.
$(3)$ is wrong (e.g. it demands that $mleq1$)
Actually you cannot speak of "the" correct notation of..., but of "a" correct notation of...
answered Nov 17 '18 at 12:11
drhab
97.9k544129
$(1)$ and $(2)$ are okay, though I would rather do it with $frac{n}{m}$ instead of $frac{m}{n}$.
In $(2)$ the part $inmathbb Q$ is redundant, but that does not harm correctness.
$(3)$ is wrong (e.g. it demands that $mleq1$)
Actually you cannot speak of "the" correct notation of..., but of "a" correct notation of...
answered Nov 17 '18 at 12:11
drhab
97.9k544129
answered Nov 17 '18 at 12:11
drhab
97.9k544129
answered Nov 17 '18 at 12:11
drhab
97.9k544129
answered Nov 17 '18 at 12:11
drhab
97.9k544129
97.9k544129
add a comment |
add a comment |
The definition $1$ and $2$ seems correct to me, we could also use for example
$$Big{ frac{m}{n}in mathbb{Q} ,vert , n,min {1,2,3,4,5}subseteq mathbb{Z}Big} $$
answered Nov 17 '18 at 12:07
gimusi
1
Thanks for your answer. The consesus seems to be that 1 and 2 is ok. I will use on of them with slight modification.
– Nullspace
Nov 17 '18 at 12:22
add a comment |
The definition $1$ and $2$ seems correct to me, we could also use for example
$$Big{ frac{m}{n}in mathbb{Q} ,vert , n,min {1,2,3,4,5}subseteq mathbb{Z}Big} $$
answered Nov 17 '18 at 12:07
gimusi
1
Thanks for your answer. The consesus seems to be that 1 and 2 is ok. I will use on of them with slight modification.
– Nullspace
Nov 17 '18 at 12:22
add a comment |
The definition $1$ and $2$ seems correct to me, we could also use for example
$$Big{ frac{m}{n}in mathbb{Q} ,vert , n,min {1,2,3,4,5}subseteq mathbb{Z}Big} $$
answered Nov 17 '18 at 12:07
gimusi
1
The definition $1$ and $2$ seems correct to me, we could also use for example
$$Big{ frac{m}{n}in mathbb{Q} ,vert , n,min {1,2,3,4,5}subseteq mathbb{Z}Big} $$
answered Nov 17 '18 at 12:07
gimusi
1
answered Nov 17 '18 at 12:07
gimusi
1
answered Nov 17 '18 at 12:07
gimusi
1
answered Nov 17 '18 at 12:07
gimusi
1
1
Thanks for your answer. The consesus seems to be that 1 and 2 is ok. I will use on of them with slight modification.
– Nullspace
Nov 17 '18 at 12:22
add a comment |
Thanks for your answer. The consesus seems to be that 1 and 2 is ok. I will use on of them with slight modification.
– Nullspace
Nov 17 '18 at 12:22
Thanks for your answer. The consesus seems to be that 1 and 2 is ok. I will use on of them with slight modification.
– Nullspace
Nov 17 '18 at 12:22
Thanks for your answer. The consesus seems to be that 1 and 2 is ok. I will use on of them with slight modification.
– Nullspace
Nov 17 '18 at 12:22
add a comment |
in an axiomatic view of set theory, certain notations that we use are justified by corresponding axioms.
As the very simplest example of this, the notation ${a,b}$ for a set that has $a$ and $b$ as elements and nothing else, is justified by the Pairing Axiom which states that for any two sets $a,b$ there exists a set $c$ such that $$forall xcolon xin cleftrightarrow x=alor x=b.$$ It also follows that this set is unique and we introduce the notiation ${a,b}$ as a shortcut for "the set $c$ guaranteed to exist by the pairing axiom applied to $a$ and $b$".
Likewise, the Axiom Scheme of Comprehension states that for any set $a$ and predicate $phi$, there exists a set $c$ such that
$$forall xcolon xin cleftrightarrow xin aland phi(x).$$
We usually use the notation ${,xin Amid phi(x),}$ as a shortcut for "the unique set $c$ guaranteed to exist by the comprehension scheme applied to $a$ and $phi$".
Also, the Axiom Scheme of Replacement states that for every set $a$ and function $F$, there exists a set $c$ such that
$$forall xcolon xin cleftrightarrow exists tcolon t in aland x=F(t).$$
We usually use the notation ${,F(t)mid tin a,}$ as a shortcut for "the unique set $c$ guaranteed to exist by the comprehension scheme applied to $a$ and $F$".
With this in mind, your
version $(1)$ is a formally correct notation inspired by the Axiom Schema of Replacement.
This nearly makes version $(2)$ okay as well. However, in order to make it match the Axiom Schema of Comprehension, I would - for extra strictness - prefer to see it in the following form:
$$tag{2'}{,qinBbb Qmid exists m,nin Bbb Zcolon (1le mle 5land 1le nle 5land q=tfrac mn),}$$
Note however, that in most contexts, legibility trumps strict formalism and that certain "colloqialsms" usually enter most texts. For example, in the above I myself used $exists m,ninBbb Zcolon ldots$ as a "colloquial" short form for $exists mcolon exists ncolon (min Bbb Zland ninBbb Zlandldots)$.
Finally, I suppose you mistyped something in $(3)$.
EDIT: After the original question was edited, variant $(3)$ might be understandable and might be interpreted by many to describe the intended set. However, this variant is quite ambiguous: One can interpret $1le n,mle 5$ as "both numbers $n,m$ are $ge 1$ and $le 5$". But in the same instance, you use the comma as a logical and, thus suggesting another possible reading "$n,min Bbb Z$ and $1le n$ and $mle 5$". I consider this ambiguity fatal. This could be saved by making it clear that you do not habitually use the comma to denote logical conjunction, i.e., write $n,minBbb Zland 1le n,mle 5$.
answered Nov 17 '18 at 12:32
Hagen von Eitzen
276k21269496
Thank you for that detailed answer. Yeah you are right. I mistyped something in (3). I will fix that quickly.
– Nullspace
Nov 17 '18 at 13:12
add a comment |
in an axiomatic view of set theory, certain notations that we use are justified by corresponding axioms.
As the very simplest example of this, the notation ${a,b}$ for a set that has $a$ and $b$ as elements and nothing else, is justified by the Pairing Axiom which states that for any two sets $a,b$ there exists a set $c$ such that $$forall xcolon xin cleftrightarrow x=alor x=b.$$ It also follows that this set is unique and we introduce the notiation ${a,b}$ as a shortcut for "the set $c$ guaranteed to exist by the pairing axiom applied to $a$ and $b$".
Likewise, the Axiom Scheme of Comprehension states that for any set $a$ and predicate $phi$, there exists a set $c$ such that
$$forall xcolon xin cleftrightarrow xin aland phi(x).$$
We usually use the notation ${,xin Amid phi(x),}$ as a shortcut for "the unique set $c$ guaranteed to exist by the comprehension scheme applied to $a$ and $phi$".
Also, the Axiom Scheme of Replacement states that for every set $a$ and function $F$, there exists a set $c$ such that
$$forall xcolon xin cleftrightarrow exists tcolon t in aland x=F(t).$$
We usually use the notation ${,F(t)mid tin a,}$ as a shortcut for "the unique set $c$ guaranteed to exist by the comprehension scheme applied to $a$ and $F$".
With this in mind, your
version $(1)$ is a formally correct notation inspired by the Axiom Schema of Replacement.
This nearly makes version $(2)$ okay as well. However, in order to make it match the Axiom Schema of Comprehension, I would - for extra strictness - prefer to see it in the following form:
$$tag{2'}{,qinBbb Qmid exists m,nin Bbb Zcolon (1le mle 5land 1le nle 5land q=tfrac mn),}$$
Note however, that in most contexts, legibility trumps strict formalism and that certain "colloqialsms" usually enter most texts. For example, in the above I myself used $exists m,ninBbb Zcolon ldots$ as a "colloquial" short form for $exists mcolon exists ncolon (min Bbb Zland ninBbb Zlandldots)$.
Finally, I suppose you mistyped something in $(3)$.
EDIT: After the original question was edited, variant $(3)$ might be understandable and might be interpreted by many to describe the intended set. However, this variant is quite ambiguous: One can interpret $1le n,mle 5$ as "both numbers $n,m$ are $ge 1$ and $le 5$". But in the same instance, you use the comma as a logical and, thus suggesting another possible reading "$n,min Bbb Z$ and $1le n$ and $mle 5$". I consider this ambiguity fatal. This could be saved by making it clear that you do not habitually use the comma to denote logical conjunction, i.e., write $n,minBbb Zland 1le n,mle 5$.
answered Nov 17 '18 at 12:32
Hagen von Eitzen
276k21269496
Thank you for that detailed answer. Yeah you are right. I mistyped something in (3). I will fix that quickly.
– Nullspace
Nov 17 '18 at 13:12
add a comment |
in an axiomatic view of set theory, certain notations that we use are justified by corresponding axioms.
As the very simplest example of this, the notation ${a,b}$ for a set that has $a$ and $b$ as elements and nothing else, is justified by the Pairing Axiom which states that for any two sets $a,b$ there exists a set $c$ such that $$forall xcolon xin cleftrightarrow x=alor x=b.$$ It also follows that this set is unique and we introduce the notiation ${a,b}$ as a shortcut for "the set $c$ guaranteed to exist by the pairing axiom applied to $a$ and $b$".
Likewise, the Axiom Scheme of Comprehension states that for any set $a$ and predicate $phi$, there exists a set $c$ such that
$$forall xcolon xin cleftrightarrow xin aland phi(x).$$
We usually use the notation ${,xin Amid phi(x),}$ as a shortcut for "the unique set $c$ guaranteed to exist by the comprehension scheme applied to $a$ and $phi$".
Also, the Axiom Scheme of Replacement states that for every set $a$ and function $F$, there exists a set $c$ such that
$$forall xcolon xin cleftrightarrow exists tcolon t in aland x=F(t).$$
We usually use the notation ${,F(t)mid tin a,}$ as a shortcut for "the unique set $c$ guaranteed to exist by the comprehension scheme applied to $a$ and $F$".
With this in mind, your
version $(1)$ is a formally correct notation inspired by the Axiom Schema of Replacement.
This nearly makes version $(2)$ okay as well. However, in order to make it match the Axiom Schema of Comprehension, I would - for extra strictness - prefer to see it in the following form:
$$tag{2'}{,qinBbb Qmid exists m,nin Bbb Zcolon (1le mle 5land 1le nle 5land q=tfrac mn),}$$
Note however, that in most contexts, legibility trumps strict formalism and that certain "colloqialsms" usually enter most texts. For example, in the above I myself used $exists m,ninBbb Zcolon ldots$ as a "colloquial" short form for $exists mcolon exists ncolon (min Bbb Zland ninBbb Zlandldots)$.
Finally, I suppose you mistyped something in $(3)$.
EDIT: After the original question was edited, variant $(3)$ might be understandable and might be interpreted by many to describe the intended set. However, this variant is quite ambiguous: One can interpret $1le n,mle 5$ as "both numbers $n,m$ are $ge 1$ and $le 5$". But in the same instance, you use the comma as a logical and, thus suggesting another possible reading "$n,min Bbb Z$ and $1le n$ and $mle 5$". I consider this ambiguity fatal. This could be saved by making it clear that you do not habitually use the comma to denote logical conjunction, i.e., write $n,minBbb Zland 1le n,mle 5$.
answered Nov 17 '18 at 12:32
Hagen von Eitzen
276k21269496
in an axiomatic view of set theory, certain notations that we use are justified by corresponding axioms.
As the very simplest example of this, the notation ${a,b}$ for a set that has $a$ and $b$ as elements and nothing else, is justified by the Pairing Axiom which states that for any two sets $a,b$ there exists a set $c$ such that $$forall xcolon xin cleftrightarrow x=alor x=b.$$ It also follows that this set is unique and we introduce the notiation ${a,b}$ as a shortcut for "the set $c$ guaranteed to exist by the pairing axiom applied to $a$ and $b$".
Likewise, the Axiom Scheme of Comprehension states that for any set $a$ and predicate $phi$, there exists a set $c$ such that
$$forall xcolon xin cleftrightarrow xin aland phi(x).$$
We usually use the notation ${,xin Amid phi(x),}$ as a shortcut for "the unique set $c$ guaranteed to exist by the comprehension scheme applied to $a$ and $phi$".
Also, the Axiom Scheme of Replacement states that for every set $a$ and function $F$, there exists a set $c$ such that
$$forall xcolon xin cleftrightarrow exists tcolon t in aland x=F(t).$$
We usually use the notation ${,F(t)mid tin a,}$ as a shortcut for "the unique set $c$ guaranteed to exist by the comprehension scheme applied to $a$ and $F$".
With this in mind, your
version $(1)$ is a formally correct notation inspired by the Axiom Schema of Replacement.
This nearly makes version $(2)$ okay as well. However, in order to make it match the Axiom Schema of Comprehension, I would - for extra strictness - prefer to see it in the following form:
$$tag{2'}{,qinBbb Qmid exists m,nin Bbb Zcolon (1le mle 5land 1le nle 5land q=tfrac mn),}$$
Note however, that in most contexts, legibility trumps strict formalism and that certain "colloqialsms" usually enter most texts. For example, in the above I myself used $exists m,ninBbb Zcolon ldots$ as a "colloquial" short form for $exists mcolon exists ncolon (min Bbb Zland ninBbb Zlandldots)$.
Finally, I suppose you mistyped something in $(3)$.
EDIT: After the original question was edited, variant $(3)$ might be understandable and might be interpreted by many to describe the intended set. However, this variant is quite ambiguous: One can interpret $1le n,mle 5$ as "both numbers $n,m$ are $ge 1$ and $le 5$". But in the same instance, you use the comma as a logical and, thus suggesting another possible reading "$n,min Bbb Z$ and $1le n$ and $mle 5$". I consider this ambiguity fatal. This could be saved by making it clear that you do not habitually use the comma to denote logical conjunction, i.e., write $n,minBbb Zland 1le n,mle 5$.
answered Nov 17 '18 at 12:32
Hagen von Eitzen
276k21269496
edited Nov 17 '18 at 21:29
answered Nov 17 '18 at 12:32
Hagen von Eitzen
276k21269496
answered Nov 17 '18 at 12:32
Hagen von Eitzen
276k21269496
answered Nov 17 '18 at 12:32
Hagen von Eitzen
276k21269496
276k21269496
Thank you for that detailed answer. Yeah you are right. I mistyped something in (3). I will fix that quickly.
– Nullspace
Nov 17 '18 at 13:12
add a comment |
Thank you for that detailed answer. Yeah you are right. I mistyped something in (3). I will fix that quickly.
– Nullspace
Nov 17 '18 at 13:12
Thank you for that detailed answer. Yeah you are right. I mistyped something in (3). I will fix that quickly.
– Nullspace
Nov 17 '18 at 13:12
Thank you for that detailed answer. Yeah you are right. I mistyped something in (3). I will fix that quickly.
– Nullspace
Nov 17 '18 at 13:12
add a comment |
You could use $$left{frac nm mid (n, m) in [[1,5]]^2right}$$ (not to sure how to obtain the notation for an integer interval on this site, but $[[1,5]]$ is a short way of writing ${1, ..., 5}$)
answered Nov 17 '18 at 12:16
krirkrirk
1,458518
in what language is $[[1;5]]$ or $[1;5]$ an integer interval? I think I have seen $[1:5]$ or $[1..5]$ not sure myself, perhaps in Python, Haskell. In math one would want to introduce such a notation before formally using it, I don't think it is standard.
– Mirko
Nov 17 '18 at 12:20
@Mirko In French litterature, it's a standard notation. Thought it was the case in English too!
– krirkrirk
Nov 17 '18 at 12:23
add a comment |
You could use $$left{frac nm mid (n, m) in [[1,5]]^2right}$$ (not to sure how to obtain the notation for an integer interval on this site, but $[[1,5]]$ is a short way of writing ${1, ..., 5}$)
answered Nov 17 '18 at 12:16
krirkrirk
1,458518
in what language is $[[1;5]]$ or $[1;5]$ an integer interval? I think I have seen $[1:5]$ or $[1..5]$ not sure myself, perhaps in Python, Haskell. In math one would want to introduce such a notation before formally using it, I don't think it is standard.
– Mirko
Nov 17 '18 at 12:20
@Mirko In French litterature, it's a standard notation. Thought it was the case in English too!
– krirkrirk
Nov 17 '18 at 12:23
add a comment |
You could use $$left{frac nm mid (n, m) in [[1,5]]^2right}$$ (not to sure how to obtain the notation for an integer interval on this site, but $[[1,5]]$ is a short way of writing ${1, ..., 5}$)
answered Nov 17 '18 at 12:16
krirkrirk
1,458518
You could use $$left{frac nm mid (n, m) in [[1,5]]^2right}$$ (not to sure how to obtain the notation for an integer interval on this site, but $[[1,5]]$ is a short way of writing ${1, ..., 5}$)
answered Nov 17 '18 at 12:16
krirkrirk
1,458518
edited Nov 17 '18 at 12:23
answered Nov 17 '18 at 12:16
krirkrirk
1,458518
answered Nov 17 '18 at 12:16
krirkrirk
1,458518
answered Nov 17 '18 at 12:16
krirkrirk
1,458518
1,458518
in what language is $[[1;5]]$ or $[1;5]$ an integer interval? I think I have seen $[1:5]$ or $[1..5]$ not sure myself, perhaps in Python, Haskell. In math one would want to introduce such a notation before formally using it, I don't think it is standard.
– Mirko
Nov 17 '18 at 12:20
@Mirko In French litterature, it's a standard notation. Thought it was the case in English too!
– krirkrirk
Nov 17 '18 at 12:23
add a comment |
in what language is $[[1;5]]$ or $[1;5]$ an integer interval? I think I have seen $[1:5]$ or $[1..5]$ not sure myself, perhaps in Python, Haskell. In math one would want to introduce such a notation before formally using it, I don't think it is standard.
– Mirko
Nov 17 '18 at 12:20
@Mirko In French litterature, it's a standard notation. Thought it was the case in English too!
– krirkrirk
Nov 17 '18 at 12:23
in what language is $[[1;5]]$ or $[1;5]$ an integer interval? I think I have seen $[1:5]$ or $[1..5]$ not sure myself, perhaps in Python, Haskell. In math one would want to introduce such a notation before formally using it, I don't think it is standard.
– Mirko
Nov 17 '18 at 12:20
in what language is $[[1;5]]$ or $[1;5]$ an integer interval? I think I have seen $[1:5]$ or $[1..5]$ not sure myself, perhaps in Python, Haskell. In math one would want to introduce such a notation before formally using it, I don't think it is standard.
– Mirko
Nov 17 '18 at 12:20
@Mirko In French litterature, it's a standard notation. Thought it was the case in English too!
– krirkrirk
Nov 17 '18 at 12:23
@Mirko In French litterature, it's a standard notation. Thought it was the case in English too!
– krirkrirk
Nov 17 '18 at 12:23
add a comment |
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it might be a matter of taste, and depends on the context and conventions, that is it might be possible to automatically assume that $m,n$ denote integers (which you do in the title), then ${frac m n: 1le n,m le5}$. The abbreviation $n,min mathbb{Z}$ is not used by some authors (perhaps it doesn't say that $nin mathbb{Z}$), who prefer $nin mathbb{Z},min mathbb{Z}$, in this case ${ frac{m}{n} | nin mathbb{Z},min mathbb{Z}, 1le nle5, 1le mle5}$. It is a matter of taste, unless you may want to write a formal theorem prover and syntax becomes important (don't know much).
– Mirko
Nov 17 '18 at 12:14
@Mirko Thank you. I will problably use the first or second and combine it with your suggestion of writing $n in mathbb{Z}, m in mathbb{z}$. Thanks again.
– Nullspace
Nov 17 '18 at 12:24
I just added left and right to your brackets so they matched the height of their contents.
– user334732
Nov 20 '18 at 9:38