Difference between $[a,b]in mathbb R$ and $[a,b]subset mathbb R$?
$begingroup$
What is the difference between the following?
Are they both mathematically correct?
begin{align}
[a,b]in mathbb R tag 1 \
[a,b]subset mathbb R tag 2
end{align}
And also, which one should I use If I want to say "the interval between $a$ and $b$ is real"? Feel free to correct me if this phrasing is inaccurate.
real-analysis calculus elementary-set-theory notation
$endgroup$
add a comment |
$begingroup$
What is the difference between the following?
Are they both mathematically correct?
begin{align}
[a,b]in mathbb R tag 1 \
[a,b]subset mathbb R tag 2
end{align}
And also, which one should I use If I want to say "the interval between $a$ and $b$ is real"? Feel free to correct me if this phrasing is inaccurate.
real-analysis calculus elementary-set-theory notation
$endgroup$
1
$begingroup$
$in$ is for elements and $subset$ is for sets. The first would be valid only if $[a,b]$ were a real number.
$endgroup$
– Yanko
Jan 12 at 16:05
1
$begingroup$
As has already been explained, (1) is incorrect while (2) is correct. This misconception may have come from the fact both elements and subsets of $Bbb R$ are sometimes described as "in" $Bbb R$. We sometimes say for clarity that $Bbb R$ contains or includes its subsets, and owns its elements. In other words, (1) is wrong because it claims ownership rather than inclusion.
$endgroup$
– J.G.
Jan 12 at 16:10
$begingroup$
@J.G. I don't think I've ever heard "owns" used in that context in English. (On the other hand the Danish pronunciation of $in$ literally means "is owned by", so I agree that metaphor is not crazy -- just that it doesn't seem to be common in English).
$endgroup$
– Henning Makholm
Jan 12 at 16:29
$begingroup$
@HenningMakholm Also, $xin y$ can be rewritten as $yowns x$; $owns$ comes from the LaTeXowns
.
$endgroup$
– J.G.
Jan 12 at 16:34
add a comment |
$begingroup$
What is the difference between the following?
Are they both mathematically correct?
begin{align}
[a,b]in mathbb R tag 1 \
[a,b]subset mathbb R tag 2
end{align}
And also, which one should I use If I want to say "the interval between $a$ and $b$ is real"? Feel free to correct me if this phrasing is inaccurate.
real-analysis calculus elementary-set-theory notation
$endgroup$
What is the difference between the following?
Are they both mathematically correct?
begin{align}
[a,b]in mathbb R tag 1 \
[a,b]subset mathbb R tag 2
end{align}
And also, which one should I use If I want to say "the interval between $a$ and $b$ is real"? Feel free to correct me if this phrasing is inaccurate.
real-analysis calculus elementary-set-theory notation
real-analysis calculus elementary-set-theory notation
asked Jan 12 at 16:04
JDoeDoeJDoeDoe
7701614
7701614
1
$begingroup$
$in$ is for elements and $subset$ is for sets. The first would be valid only if $[a,b]$ were a real number.
$endgroup$
– Yanko
Jan 12 at 16:05
1
$begingroup$
As has already been explained, (1) is incorrect while (2) is correct. This misconception may have come from the fact both elements and subsets of $Bbb R$ are sometimes described as "in" $Bbb R$. We sometimes say for clarity that $Bbb R$ contains or includes its subsets, and owns its elements. In other words, (1) is wrong because it claims ownership rather than inclusion.
$endgroup$
– J.G.
Jan 12 at 16:10
$begingroup$
@J.G. I don't think I've ever heard "owns" used in that context in English. (On the other hand the Danish pronunciation of $in$ literally means "is owned by", so I agree that metaphor is not crazy -- just that it doesn't seem to be common in English).
$endgroup$
– Henning Makholm
Jan 12 at 16:29
$begingroup$
@HenningMakholm Also, $xin y$ can be rewritten as $yowns x$; $owns$ comes from the LaTeXowns
.
$endgroup$
– J.G.
Jan 12 at 16:34
add a comment |
1
$begingroup$
$in$ is for elements and $subset$ is for sets. The first would be valid only if $[a,b]$ were a real number.
$endgroup$
– Yanko
Jan 12 at 16:05
1
$begingroup$
As has already been explained, (1) is incorrect while (2) is correct. This misconception may have come from the fact both elements and subsets of $Bbb R$ are sometimes described as "in" $Bbb R$. We sometimes say for clarity that $Bbb R$ contains or includes its subsets, and owns its elements. In other words, (1) is wrong because it claims ownership rather than inclusion.
$endgroup$
– J.G.
Jan 12 at 16:10
$begingroup$
@J.G. I don't think I've ever heard "owns" used in that context in English. (On the other hand the Danish pronunciation of $in$ literally means "is owned by", so I agree that metaphor is not crazy -- just that it doesn't seem to be common in English).
$endgroup$
– Henning Makholm
Jan 12 at 16:29
$begingroup$
@HenningMakholm Also, $xin y$ can be rewritten as $yowns x$; $owns$ comes from the LaTeXowns
.
$endgroup$
– J.G.
Jan 12 at 16:34
1
1
$begingroup$
$in$ is for elements and $subset$ is for sets. The first would be valid only if $[a,b]$ were a real number.
$endgroup$
– Yanko
Jan 12 at 16:05
$begingroup$
$in$ is for elements and $subset$ is for sets. The first would be valid only if $[a,b]$ were a real number.
$endgroup$
– Yanko
Jan 12 at 16:05
1
1
$begingroup$
As has already been explained, (1) is incorrect while (2) is correct. This misconception may have come from the fact both elements and subsets of $Bbb R$ are sometimes described as "in" $Bbb R$. We sometimes say for clarity that $Bbb R$ contains or includes its subsets, and owns its elements. In other words, (1) is wrong because it claims ownership rather than inclusion.
$endgroup$
– J.G.
Jan 12 at 16:10
$begingroup$
As has already been explained, (1) is incorrect while (2) is correct. This misconception may have come from the fact both elements and subsets of $Bbb R$ are sometimes described as "in" $Bbb R$. We sometimes say for clarity that $Bbb R$ contains or includes its subsets, and owns its elements. In other words, (1) is wrong because it claims ownership rather than inclusion.
$endgroup$
– J.G.
Jan 12 at 16:10
$begingroup$
@J.G. I don't think I've ever heard "owns" used in that context in English. (On the other hand the Danish pronunciation of $in$ literally means "is owned by", so I agree that metaphor is not crazy -- just that it doesn't seem to be common in English).
$endgroup$
– Henning Makholm
Jan 12 at 16:29
$begingroup$
@J.G. I don't think I've ever heard "owns" used in that context in English. (On the other hand the Danish pronunciation of $in$ literally means "is owned by", so I agree that metaphor is not crazy -- just that it doesn't seem to be common in English).
$endgroup$
– Henning Makholm
Jan 12 at 16:29
$begingroup$
@HenningMakholm Also, $xin y$ can be rewritten as $yowns x$; $owns$ comes from the LaTeX
owns
.$endgroup$
– J.G.
Jan 12 at 16:34
$begingroup$
@HenningMakholm Also, $xin y$ can be rewritten as $yowns x$; $owns$ comes from the LaTeX
owns
.$endgroup$
– J.G.
Jan 12 at 16:34
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Since $[a,b]$ is a set then only second is validate:
$$ [a,b] := {xin mathbb{R}; aleq xleq b} implies [a,b]subset mathbb{R}$$
If the question was, is $[a,b]in mathcal{P}(mathbb{R})$ (that is the power set of $mathbb{R}$) then the answer would be yes.
$endgroup$
add a comment |
$begingroup$
Assuming that $a$ and be are real numbers and that $aleqslant b$, then the assertion $[a,b]inmathbb R$ is false, since it means that the interval $[a,b]$ is an element of $mathbb R$. If you want to assert that $[a,b]$ is a subset of $mathbb R$, you write $[a,b]subsetmathbb R$.
$endgroup$
add a comment |
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2 Answers
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active
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2 Answers
2
active
oldest
votes
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oldest
votes
$begingroup$
Since $[a,b]$ is a set then only second is validate:
$$ [a,b] := {xin mathbb{R}; aleq xleq b} implies [a,b]subset mathbb{R}$$
If the question was, is $[a,b]in mathcal{P}(mathbb{R})$ (that is the power set of $mathbb{R}$) then the answer would be yes.
$endgroup$
add a comment |
$begingroup$
Since $[a,b]$ is a set then only second is validate:
$$ [a,b] := {xin mathbb{R}; aleq xleq b} implies [a,b]subset mathbb{R}$$
If the question was, is $[a,b]in mathcal{P}(mathbb{R})$ (that is the power set of $mathbb{R}$) then the answer would be yes.
$endgroup$
add a comment |
$begingroup$
Since $[a,b]$ is a set then only second is validate:
$$ [a,b] := {xin mathbb{R}; aleq xleq b} implies [a,b]subset mathbb{R}$$
If the question was, is $[a,b]in mathcal{P}(mathbb{R})$ (that is the power set of $mathbb{R}$) then the answer would be yes.
$endgroup$
Since $[a,b]$ is a set then only second is validate:
$$ [a,b] := {xin mathbb{R}; aleq xleq b} implies [a,b]subset mathbb{R}$$
If the question was, is $[a,b]in mathcal{P}(mathbb{R})$ (that is the power set of $mathbb{R}$) then the answer would be yes.
edited Jan 12 at 16:25
David C. Ullrich
61.3k43994
61.3k43994
answered Jan 12 at 16:05
Maria MazurMaria Mazur
46.9k1260120
46.9k1260120
add a comment |
add a comment |
$begingroup$
Assuming that $a$ and be are real numbers and that $aleqslant b$, then the assertion $[a,b]inmathbb R$ is false, since it means that the interval $[a,b]$ is an element of $mathbb R$. If you want to assert that $[a,b]$ is a subset of $mathbb R$, you write $[a,b]subsetmathbb R$.
$endgroup$
add a comment |
$begingroup$
Assuming that $a$ and be are real numbers and that $aleqslant b$, then the assertion $[a,b]inmathbb R$ is false, since it means that the interval $[a,b]$ is an element of $mathbb R$. If you want to assert that $[a,b]$ is a subset of $mathbb R$, you write $[a,b]subsetmathbb R$.
$endgroup$
add a comment |
$begingroup$
Assuming that $a$ and be are real numbers and that $aleqslant b$, then the assertion $[a,b]inmathbb R$ is false, since it means that the interval $[a,b]$ is an element of $mathbb R$. If you want to assert that $[a,b]$ is a subset of $mathbb R$, you write $[a,b]subsetmathbb R$.
$endgroup$
Assuming that $a$ and be are real numbers and that $aleqslant b$, then the assertion $[a,b]inmathbb R$ is false, since it means that the interval $[a,b]$ is an element of $mathbb R$. If you want to assert that $[a,b]$ is a subset of $mathbb R$, you write $[a,b]subsetmathbb R$.
edited Jan 12 at 16:21
answered Jan 12 at 16:08
José Carlos SantosJosé Carlos Santos
167k22132235
167k22132235
add a comment |
add a comment |
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1
$begingroup$
$in$ is for elements and $subset$ is for sets. The first would be valid only if $[a,b]$ were a real number.
$endgroup$
– Yanko
Jan 12 at 16:05
1
$begingroup$
As has already been explained, (1) is incorrect while (2) is correct. This misconception may have come from the fact both elements and subsets of $Bbb R$ are sometimes described as "in" $Bbb R$. We sometimes say for clarity that $Bbb R$ contains or includes its subsets, and owns its elements. In other words, (1) is wrong because it claims ownership rather than inclusion.
$endgroup$
– J.G.
Jan 12 at 16:10
$begingroup$
@J.G. I don't think I've ever heard "owns" used in that context in English. (On the other hand the Danish pronunciation of $in$ literally means "is owned by", so I agree that metaphor is not crazy -- just that it doesn't seem to be common in English).
$endgroup$
– Henning Makholm
Jan 12 at 16:29
$begingroup$
@HenningMakholm Also, $xin y$ can be rewritten as $yowns x$; $owns$ comes from the LaTeX
owns
.$endgroup$
– J.G.
Jan 12 at 16:34