Mixing
What does it mean for a set to be contained in another?
$begingroup$
If $A$ & $B$ are two sets, what does '$A$ is contained in $B$' mean ? Does it mean that $A$ is a subset of $B$ or $A$ is a proper subset of $B$?
elementary-set-theory definition
edited Jan 12 at 15:47
Andrés E. Caicedo
65.4k8158249
asked Jan 12 at 15:02
SiddharthSiddharth
936
$endgroup$
add a comment |
$begingroup$
If $A$ & $B$ are two sets, what does '$A$ is contained in $B$' mean ? Does it mean that $A$ is a subset of $B$ or $A$ is a proper subset of $B$?
elementary-set-theory definition
edited Jan 12 at 15:47
Andrés E. Caicedo
65.4k8158249
asked Jan 12 at 15:02
SiddharthSiddharth
936
$endgroup$
$begingroup$
There is no universal convention, check the one used by the particular author.
$endgroup$
– Yves Daoust
Jan 12 at 15:28
add a comment |
$begingroup$
If $A$ & $B$ are two sets, what does '$A$ is contained in $B$' mean ? Does it mean that $A$ is a subset of $B$ or $A$ is a proper subset of $B$?
elementary-set-theory definition
edited Jan 12 at 15:47
Andrés E. Caicedo
65.4k8158249
asked Jan 12 at 15:02
SiddharthSiddharth
936
$endgroup$
If $A$ & $B$ are two sets, what does '$A$ is contained in $B$' mean ? Does it mean that $A$ is a subset of $B$ or $A$ is a proper subset of $B$?
elementary-set-theory definition
elementary-set-theory definition
edited Jan 12 at 15:47
Andrés E. Caicedo
65.4k8158249
asked Jan 12 at 15:02
SiddharthSiddharth
936
edited Jan 12 at 15:47
Andrés E. Caicedo
65.4k8158249
asked Jan 12 at 15:02
SiddharthSiddharth
936
edited Jan 12 at 15:47
Andrés E. Caicedo
65.4k8158249
edited Jan 12 at 15:47
Andrés E. Caicedo
65.4k8158249
edited Jan 12 at 15:47
Andrés E. Caicedo
65.4k8158249
65.4k8158249
asked Jan 12 at 15:02
SiddharthSiddharth
936
asked Jan 12 at 15:02
SiddharthSiddharth
936
asked Jan 12 at 15:02
SiddharthSiddharth
936
936
$begingroup$
There is no universal convention, check the one used by the particular author.
$endgroup$
– Yves Daoust
Jan 12 at 15:28
add a comment |
$begingroup$
There is no universal convention, check the one used by the particular author.
$endgroup$
– Yves Daoust
Jan 12 at 15:28
$begingroup$
There is no universal convention, check the one used by the particular author.
$endgroup$
– Yves Daoust
Jan 12 at 15:28
$begingroup$
There is no universal convention, check the one used by the particular author.
$endgroup$
– Yves Daoust
Jan 12 at 15:28
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
It's a bit ambiguous, but hopefully it's only used when there is little chance of confusion. If there is a chance, it is better to use notation. Either $Ain B$ for $A$ being an element of $B$, or $Asubseteq B$ for $A$ being a subset of $B$.
If it's the latter, generally it is not necessarily a proper subset. Authors however can do whatever they want, as long as they define it. You have to watch out for nonstandard language and notation.
answered Jan 12 at 15:03
Matt SamuelMatt Samuel
38.4k63768
$endgroup$
1
$begingroup$
@Thomas Started writing it on my phone, had to do something else, must've accidentally posted it without finishing. Thanks.
$endgroup$
– Matt Samuel
Jan 12 at 15:27
$begingroup$
I would go as far as saying that writing "$A$ is contained in $B$" and meaning $Ain B$ amounts to gaslighting...
$endgroup$
– JuliusL33t
Jan 12 at 15:56
$begingroup$
@Julius Sets can contain elements. And in set theory everything is a set. So in situations like that it can legitimately be ambiguous.
$endgroup$
– Matt Samuel
Jan 12 at 16:19
$begingroup$
I'm not commenting on the mathematical content, but on pedagogy and custom. Some people define "cat" to mean "dog" and "up" to mean "down". Saying "$A$ is contained in $B$" and meaning "$A$ is an element of $B$" seems so cruel to me, especially when this is very standard terminology. But hey, whatever floats your boat.
$endgroup$
– JuliusL33t
Jan 12 at 16:36
$begingroup$
I DO NOT disagree with you answer, so don't take offense.
$endgroup$
– JuliusL33t
Jan 12 at 16:39
|
show 1 more comment
$begingroup$
Because we do know nothing about the two sets let me give you two examples.
1) We know that $N subset R$. This means that $N$ is contained in $R$.
2) Let us define $A = {x,y}$ and $B = {x,y}$ then A is contained in B and B is contained in A.
To answer your question, both of them can be correct.
answered Jan 12 at 15:22
Dimitris DimitriadisDimitris Dimitriadis
478
$endgroup$
add a comment |
$begingroup$
The only thing that "$A$ is contained in $B$" means is that the implication
$$x in A implies x in B $$
is satisfied. It could be a proper or non-proper subset. Often, when the subset is proper, most authors explicitely state this.
answered Jan 12 at 15:27
JuliusL33tJuliusL33t
1,343917
$endgroup$
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
It's a bit ambiguous, but hopefully it's only used when there is little chance of confusion. If there is a chance, it is better to use notation. Either $Ain B$ for $A$ being an element of $B$, or $Asubseteq B$ for $A$ being a subset of $B$.
If it's the latter, generally it is not necessarily a proper subset. Authors however can do whatever they want, as long as they define it. You have to watch out for nonstandard language and notation.
answered Jan 12 at 15:03
Matt SamuelMatt Samuel
38.4k63768
$endgroup$
1
$begingroup$
@Thomas Started writing it on my phone, had to do something else, must've accidentally posted it without finishing. Thanks.
$endgroup$
– Matt Samuel
Jan 12 at 15:27
$begingroup$
I would go as far as saying that writing "$A$ is contained in $B$" and meaning $Ain B$ amounts to gaslighting...
$endgroup$
– JuliusL33t
Jan 12 at 15:56
$begingroup$
@Julius Sets can contain elements. And in set theory everything is a set. So in situations like that it can legitimately be ambiguous.
$endgroup$
– Matt Samuel
Jan 12 at 16:19
$begingroup$
I'm not commenting on the mathematical content, but on pedagogy and custom. Some people define "cat" to mean "dog" and "up" to mean "down". Saying "$A$ is contained in $B$" and meaning "$A$ is an element of $B$" seems so cruel to me, especially when this is very standard terminology. But hey, whatever floats your boat.
$endgroup$
– JuliusL33t
Jan 12 at 16:36
$begingroup$
I DO NOT disagree with you answer, so don't take offense.
$endgroup$
– JuliusL33t
Jan 12 at 16:39
|
show 1 more comment
$begingroup$
It's a bit ambiguous, but hopefully it's only used when there is little chance of confusion. If there is a chance, it is better to use notation. Either $Ain B$ for $A$ being an element of $B$, or $Asubseteq B$ for $A$ being a subset of $B$.
If it's the latter, generally it is not necessarily a proper subset. Authors however can do whatever they want, as long as they define it. You have to watch out for nonstandard language and notation.
answered Jan 12 at 15:03
Matt SamuelMatt Samuel
38.4k63768
$endgroup$
1
$begingroup$
@Thomas Started writing it on my phone, had to do something else, must've accidentally posted it without finishing. Thanks.
$endgroup$
– Matt Samuel
Jan 12 at 15:27
$begingroup$
I would go as far as saying that writing "$A$ is contained in $B$" and meaning $Ain B$ amounts to gaslighting...
$endgroup$
– JuliusL33t
Jan 12 at 15:56
$begingroup$
@Julius Sets can contain elements. And in set theory everything is a set. So in situations like that it can legitimately be ambiguous.
$endgroup$
– Matt Samuel
Jan 12 at 16:19
$begingroup$
I'm not commenting on the mathematical content, but on pedagogy and custom. Some people define "cat" to mean "dog" and "up" to mean "down". Saying "$A$ is contained in $B$" and meaning "$A$ is an element of $B$" seems so cruel to me, especially when this is very standard terminology. But hey, whatever floats your boat.
$endgroup$
– JuliusL33t
Jan 12 at 16:36
$begingroup$
I DO NOT disagree with you answer, so don't take offense.
$endgroup$
– JuliusL33t
Jan 12 at 16:39
|
show 1 more comment
$begingroup$
It's a bit ambiguous, but hopefully it's only used when there is little chance of confusion. If there is a chance, it is better to use notation. Either $Ain B$ for $A$ being an element of $B$, or $Asubseteq B$ for $A$ being a subset of $B$.
If it's the latter, generally it is not necessarily a proper subset. Authors however can do whatever they want, as long as they define it. You have to watch out for nonstandard language and notation.
answered Jan 12 at 15:03
Matt SamuelMatt Samuel
38.4k63768
$endgroup$
It's a bit ambiguous, but hopefully it's only used when there is little chance of confusion. If there is a chance, it is better to use notation. Either $Ain B$ for $A$ being an element of $B$, or $Asubseteq B$ for $A$ being a subset of $B$.
If it's the latter, generally it is not necessarily a proper subset. Authors however can do whatever they want, as long as they define it. You have to watch out for nonstandard language and notation.
answered Jan 12 at 15:03
Matt SamuelMatt Samuel
38.4k63768
edited Jan 12 at 15:26
answered Jan 12 at 15:03
Matt SamuelMatt Samuel
38.4k63768
answered Jan 12 at 15:03
Matt SamuelMatt Samuel
38.4k63768
answered Jan 12 at 15:03
Matt SamuelMatt Samuel
38.4k63768
38.4k63768
1
$begingroup$
@Thomas Started writing it on my phone, had to do something else, must've accidentally posted it without finishing. Thanks.
$endgroup$
– Matt Samuel
Jan 12 at 15:27
$begingroup$
I would go as far as saying that writing "$A$ is contained in $B$" and meaning $Ain B$ amounts to gaslighting...
$endgroup$
– JuliusL33t
Jan 12 at 15:56
$begingroup$
@Julius Sets can contain elements. And in set theory everything is a set. So in situations like that it can legitimately be ambiguous.
$endgroup$
– Matt Samuel
Jan 12 at 16:19
$begingroup$
I'm not commenting on the mathematical content, but on pedagogy and custom. Some people define "cat" to mean "dog" and "up" to mean "down". Saying "$A$ is contained in $B$" and meaning "$A$ is an element of $B$" seems so cruel to me, especially when this is very standard terminology. But hey, whatever floats your boat.
$endgroup$
– JuliusL33t
Jan 12 at 16:36
$begingroup$
I DO NOT disagree with you answer, so don't take offense.
$endgroup$
– JuliusL33t
Jan 12 at 16:39
|
show 1 more comment
1
$begingroup$
@Thomas Started writing it on my phone, had to do something else, must've accidentally posted it without finishing. Thanks.
$endgroup$
– Matt Samuel
Jan 12 at 15:27
$begingroup$
I would go as far as saying that writing "$A$ is contained in $B$" and meaning $Ain B$ amounts to gaslighting...
$endgroup$
– JuliusL33t
Jan 12 at 15:56
$begingroup$
@Julius Sets can contain elements. And in set theory everything is a set. So in situations like that it can legitimately be ambiguous.
$endgroup$
– Matt Samuel
Jan 12 at 16:19
$begingroup$
I'm not commenting on the mathematical content, but on pedagogy and custom. Some people define "cat" to mean "dog" and "up" to mean "down". Saying "$A$ is contained in $B$" and meaning "$A$ is an element of $B$" seems so cruel to me, especially when this is very standard terminology. But hey, whatever floats your boat.
$endgroup$
– JuliusL33t
Jan 12 at 16:36
$begingroup$
I DO NOT disagree with you answer, so don't take offense.
$endgroup$
– JuliusL33t
Jan 12 at 16:39
1
1
$begingroup$
@Thomas Started writing it on my phone, had to do something else, must've accidentally posted it without finishing. Thanks.
$endgroup$
– Matt Samuel
Jan 12 at 15:27
$begingroup$
@Thomas Started writing it on my phone, had to do something else, must've accidentally posted it without finishing. Thanks.
$endgroup$
– Matt Samuel
Jan 12 at 15:27
$begingroup$
I would go as far as saying that writing "$A$ is contained in $B$" and meaning $Ain B$ amounts to gaslighting...
$endgroup$
– JuliusL33t
Jan 12 at 15:56
$begingroup$
I would go as far as saying that writing "$A$ is contained in $B$" and meaning $Ain B$ amounts to gaslighting...
$endgroup$
– JuliusL33t
Jan 12 at 15:56
$begingroup$
@Julius Sets can contain elements. And in set theory everything is a set. So in situations like that it can legitimately be ambiguous.
$endgroup$
– Matt Samuel
Jan 12 at 16:19
$begingroup$
@Julius Sets can contain elements. And in set theory everything is a set. So in situations like that it can legitimately be ambiguous.
$endgroup$
– Matt Samuel
Jan 12 at 16:19
$begingroup$
I'm not commenting on the mathematical content, but on pedagogy and custom. Some people define "cat" to mean "dog" and "up" to mean "down". Saying "$A$ is contained in $B$" and meaning "$A$ is an element of $B$" seems so cruel to me, especially when this is very standard terminology. But hey, whatever floats your boat.
$endgroup$
– JuliusL33t
Jan 12 at 16:36
$begingroup$
I'm not commenting on the mathematical content, but on pedagogy and custom. Some people define "cat" to mean "dog" and "up" to mean "down". Saying "$A$ is contained in $B$" and meaning "$A$ is an element of $B$" seems so cruel to me, especially when this is very standard terminology. But hey, whatever floats your boat.
$endgroup$
– JuliusL33t
Jan 12 at 16:36
$begingroup$
I DO NOT disagree with you answer, so don't take offense.
$endgroup$
– JuliusL33t
Jan 12 at 16:39
$begingroup$
I DO NOT disagree with you answer, so don't take offense.
$endgroup$
– JuliusL33t
Jan 12 at 16:39
|
show 1 more comment
$begingroup$
Because we do know nothing about the two sets let me give you two examples.
1) We know that $N subset R$. This means that $N$ is contained in $R$.
2) Let us define $A = {x,y}$ and $B = {x,y}$ then A is contained in B and B is contained in A.
To answer your question, both of them can be correct.
answered Jan 12 at 15:22
Dimitris DimitriadisDimitris Dimitriadis
478
$endgroup$
add a comment |
$begingroup$
Because we do know nothing about the two sets let me give you two examples.
1) We know that $N subset R$. This means that $N$ is contained in $R$.
2) Let us define $A = {x,y}$ and $B = {x,y}$ then A is contained in B and B is contained in A.
To answer your question, both of them can be correct.
answered Jan 12 at 15:22
Dimitris DimitriadisDimitris Dimitriadis
478
$endgroup$
add a comment |
$begingroup$
Because we do know nothing about the two sets let me give you two examples.
1) We know that $N subset R$. This means that $N$ is contained in $R$.
2) Let us define $A = {x,y}$ and $B = {x,y}$ then A is contained in B and B is contained in A.
To answer your question, both of them can be correct.
answered Jan 12 at 15:22
Dimitris DimitriadisDimitris Dimitriadis
478
$endgroup$
Because we do know nothing about the two sets let me give you two examples.
1) We know that $N subset R$. This means that $N$ is contained in $R$.
2) Let us define $A = {x,y}$ and $B = {x,y}$ then A is contained in B and B is contained in A.
To answer your question, both of them can be correct.
answered Jan 12 at 15:22
Dimitris DimitriadisDimitris Dimitriadis
478
answered Jan 12 at 15:22
Dimitris DimitriadisDimitris Dimitriadis
478
answered Jan 12 at 15:22
Dimitris DimitriadisDimitris Dimitriadis
478
answered Jan 12 at 15:22
Dimitris DimitriadisDimitris Dimitriadis
478
478
add a comment |
add a comment |
$begingroup$
The only thing that "$A$ is contained in $B$" means is that the implication
$$x in A implies x in B $$
is satisfied. It could be a proper or non-proper subset. Often, when the subset is proper, most authors explicitely state this.
answered Jan 12 at 15:27
JuliusL33tJuliusL33t
1,343917
$endgroup$
add a comment |
$begingroup$
The only thing that "$A$ is contained in $B$" means is that the implication
$$x in A implies x in B $$
is satisfied. It could be a proper or non-proper subset. Often, when the subset is proper, most authors explicitely state this.
answered Jan 12 at 15:27
JuliusL33tJuliusL33t
1,343917
$endgroup$
add a comment |
$begingroup$
The only thing that "$A$ is contained in $B$" means is that the implication
$$x in A implies x in B $$
is satisfied. It could be a proper or non-proper subset. Often, when the subset is proper, most authors explicitely state this.
answered Jan 12 at 15:27
JuliusL33tJuliusL33t
1,343917
$endgroup$
The only thing that "$A$ is contained in $B$" means is that the implication
$$x in A implies x in B $$
is satisfied. It could be a proper or non-proper subset. Often, when the subset is proper, most authors explicitely state this.
answered Jan 12 at 15:27
JuliusL33tJuliusL33t
1,343917
answered Jan 12 at 15:27
JuliusL33tJuliusL33t
1,343917
answered Jan 12 at 15:27
JuliusL33tJuliusL33t
1,343917
answered Jan 12 at 15:27
JuliusL33tJuliusL33t
1,343917
1,343917
add a comment |
add a comment |
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$begingroup$
There is no universal convention, check the one used by the particular author.
$endgroup$
– Yves Daoust
Jan 12 at 15:28