Mixing
Difference between ⊂ and ⊆?
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
I was solving the exercises in Discrete Mathematics and its applications book.
Determine whether each of these statements is true or
false.
- 0 ⊂ 0
- ∅ ⊆ ∅
I thought both 1 and 2 are true, but when I checked the answers I found that 1 is false and 2 is true.
I got confused and distracted because I don't know the difference between them.
discrete-mathematics
asked 17 mins ago
Mohamed Magdy
565
 |Â
show 2 more comments
up vote
1
down vote
favorite
I was solving the exercises in Discrete Mathematics and its applications book.
Determine whether each of these statements is true or
false.
- 0 ⊂ 0
- ∅ ⊆ ∅
I thought both 1 and 2 are true, but when I checked the answers I found that 1 is false and 2 is true.
I got confused and distracted because I don't know the difference between them.
discrete-mathematics
asked 17 mins ago
Mohamed Magdy
565
1
$subset$ usually means proper subset. So if $A subset B$, then ($a in A implies a in B$) but there must be an $x in B$ such that $x notin A$. Whereas $A subseteq B$means that either $A$ is a subset of $B$ but $A$ can be equal to $B$ as well. Think of the difference between $x leq 5$ and $x<5$.
– Anurag A
16 mins ago
In this context, $Asubset B$ means that $A$ is a proper subset of $B$, i.e., $Aneq B$
– AnotherJohnDoe
15 mins ago
1
It's matter of context. In many contexts, being a proper subset is denoted with $subsetneq$, for instance.
– Bernard
11 mins ago
2
It depends on the text and the notation convention the text uses. All texts will agree that $subsetneq$ means a proper subset that is not equal. And that is false for both of these because the are equal. And all texts well agree $subseteq$ means a subset that might or might not be equal. Ad that is true for both of these (every subset is a subset of itself). And all texts agree that $subset$ means is a subset but they do not agree whether it means it can't be equal or whether it might be equal. Would say 1 and 2 are both true because of how i interpret $subset$....
– fleablood
9 mins ago
1
... but it not that I am right or wrong. It's a matter of how I interpret the statements. What is right is: Both those are improper subsets that are equal. That is right. What is wrong is that 1) is a proper subset. It isn't. What matters is what is 1 saying is it saying $0$ is a proper subset of $0$ (WRONG!) or $0$ is a subset of $0$ (RIGHT). And that depends on the book. Not on any math.
– fleablood
6 mins ago
 |Â
show 2 more comments
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I was solving the exercises in Discrete Mathematics and its applications book.
Determine whether each of these statements is true or
false.
- 0 ⊂ 0
- ∅ ⊆ ∅
I thought both 1 and 2 are true, but when I checked the answers I found that 1 is false and 2 is true.
I got confused and distracted because I don't know the difference between them.
discrete-mathematics
asked 17 mins ago
Mohamed Magdy
565
I was solving the exercises in Discrete Mathematics and its applications book.
Determine whether each of these statements is true or
false.
- 0 ⊂ 0
- ∅ ⊆ ∅
I thought both 1 and 2 are true, but when I checked the answers I found that 1 is false and 2 is true.
I got confused and distracted because I don't know the difference between them.
discrete-mathematics
discrete-mathematics
asked 17 mins ago
Mohamed Magdy
565
asked 17 mins ago
Mohamed Magdy
565
asked 17 mins ago
Mohamed Magdy
565
asked 17 mins ago
Mohamed Magdy
565
asked 17 mins ago
Mohamed Magdy
565
565
1
$subset$ usually means proper subset. So if $A subset B$, then ($a in A implies a in B$) but there must be an $x in B$ such that $x notin A$. Whereas $A subseteq B$means that either $A$ is a subset of $B$ but $A$ can be equal to $B$ as well. Think of the difference between $x leq 5$ and $x<5$.
– Anurag A
16 mins ago
In this context, $Asubset B$ means that $A$ is a proper subset of $B$, i.e., $Aneq B$
– AnotherJohnDoe
15 mins ago
1
It's matter of context. In many contexts, being a proper subset is denoted with $subsetneq$, for instance.
– Bernard
11 mins ago
2
It depends on the text and the notation convention the text uses. All texts will agree that $subsetneq$ means a proper subset that is not equal. And that is false for both of these because the are equal. And all texts well agree $subseteq$ means a subset that might or might not be equal. Ad that is true for both of these (every subset is a subset of itself). And all texts agree that $subset$ means is a subset but they do not agree whether it means it can't be equal or whether it might be equal. Would say 1 and 2 are both true because of how i interpret $subset$....
– fleablood
9 mins ago
1
... but it not that I am right or wrong. It's a matter of how I interpret the statements. What is right is: Both those are improper subsets that are equal. That is right. What is wrong is that 1) is a proper subset. It isn't. What matters is what is 1 saying is it saying $0$ is a proper subset of $0$ (WRONG!) or $0$ is a subset of $0$ (RIGHT). And that depends on the book. Not on any math.
– fleablood
6 mins ago
 |Â
show 2 more comments
1
$subset$ usually means proper subset. So if $A subset B$, then ($a in A implies a in B$) but there must be an $x in B$ such that $x notin A$. Whereas $A subseteq B$means that either $A$ is a subset of $B$ but $A$ can be equal to $B$ as well. Think of the difference between $x leq 5$ and $x<5$.
– Anurag A
16 mins ago
In this context, $Asubset B$ means that $A$ is a proper subset of $B$, i.e., $Aneq B$
– AnotherJohnDoe
15 mins ago
1
It's matter of context. In many contexts, being a proper subset is denoted with $subsetneq$, for instance.
– Bernard
11 mins ago
2
It depends on the text and the notation convention the text uses. All texts will agree that $subsetneq$ means a proper subset that is not equal. And that is false for both of these because the are equal. And all texts well agree $subseteq$ means a subset that might or might not be equal. Ad that is true for both of these (every subset is a subset of itself). And all texts agree that $subset$ means is a subset but they do not agree whether it means it can't be equal or whether it might be equal. Would say 1 and 2 are both true because of how i interpret $subset$....
– fleablood
9 mins ago
1
... but it not that I am right or wrong. It's a matter of how I interpret the statements. What is right is: Both those are improper subsets that are equal. That is right. What is wrong is that 1) is a proper subset. It isn't. What matters is what is 1 saying is it saying $0$ is a proper subset of $0$ (WRONG!) or $0$ is a subset of $0$ (RIGHT). And that depends on the book. Not on any math.
– fleablood
6 mins ago
1
1
$subset$ usually means proper subset. So if $A subset B$, then ($a in A implies a in B$) but there must be an $x in B$ such that $x notin A$. Whereas $A subseteq B$means that either $A$ is a subset of $B$ but $A$ can be equal to $B$ as well. Think of the difference between $x leq 5$ and $x<5$.
– Anurag A
16 mins ago
$subset$ usually means proper subset. So if $A subset B$, then ($a in A implies a in B$) but there must be an $x in B$ such that $x notin A$. Whereas $A subseteq B$means that either $A$ is a subset of $B$ but $A$ can be equal to $B$ as well. Think of the difference between $x leq 5$ and $x<5$.
– Anurag A
16 mins ago
In this context, $Asubset B$ means that $A$ is a proper subset of $B$, i.e., $Aneq B$
– AnotherJohnDoe
15 mins ago
In this context, $Asubset B$ means that $A$ is a proper subset of $B$, i.e., $Aneq B$
– AnotherJohnDoe
15 mins ago
1
1
It's matter of context. In many contexts, being a proper subset is denoted with $subsetneq$, for instance.
– Bernard
11 mins ago
It's matter of context. In many contexts, being a proper subset is denoted with $subsetneq$, for instance.
– Bernard
11 mins ago
2
2
It depends on the text and the notation convention the text uses. All texts will agree that $subsetneq$ means a proper subset that is not equal. And that is false for both of these because the are equal. And all texts well agree $subseteq$ means a subset that might or might not be equal. Ad that is true for both of these (every subset is a subset of itself). And all texts agree that $subset$ means is a subset but they do not agree whether it means it can't be equal or whether it might be equal. Would say 1 and 2 are both true because of how i interpret $subset$....
– fleablood
9 mins ago
It depends on the text and the notation convention the text uses. All texts will agree that $subsetneq$ means a proper subset that is not equal. And that is false for both of these because the are equal. And all texts well agree $subseteq$ means a subset that might or might not be equal. Ad that is true for both of these (every subset is a subset of itself). And all texts agree that $subset$ means is a subset but they do not agree whether it means it can't be equal or whether it might be equal. Would say 1 and 2 are both true because of how i interpret $subset$....
– fleablood
9 mins ago
1
1
... but it not that I am right or wrong. It's a matter of how I interpret the statements. What is right is: Both those are improper subsets that are equal. That is right. What is wrong is that 1) is a proper subset. It isn't. What matters is what is 1 saying is it saying $0$ is a proper subset of $0$ (WRONG!) or $0$ is a subset of $0$ (RIGHT). And that depends on the book. Not on any math.
– fleablood
6 mins ago
... but it not that I am right or wrong. It's a matter of how I interpret the statements. What is right is: Both those are improper subsets that are equal. That is right. What is wrong is that 1) is a proper subset. It isn't. What matters is what is 1 saying is it saying $0$ is a proper subset of $0$ (WRONG!) or $0$ is a subset of $0$ (RIGHT). And that depends on the book. Not on any math.
– fleablood
6 mins ago
 |Â
show 2 more comments
3 Answers
3
active
oldest
votes
up vote
3
down vote
accepted
If the book distinguishes between $subset$ and $subseteq$, then most likely the former symbol denotes proper inclusion, so $0subset0$ is false. The latter symbol instead will denote inclusion (with possible equality).
However it's very common to find $subset$ denoting inclusion (with possible equality), so one always has to check or try and infer from the context. Don't take Wikipedia pages as revealed truth.
It's so common that $subset$ denotes nonstrict inclusion that somebody uses $subsetneq$ to denote proper inclusion, for safety.
answered 11 mins ago
egreg
168k1281190
add a comment |Â
up vote
1
down vote
https://en.wikipedia.org/wiki/List_of_mathematical_symbols
$A subseteq B$ means that $A$ is a subset of $B$ (in other words, every element of $A$ is an element of $B$).
$A subset B$ means that $A$ is a proper subset of $B$ (every element of $A$ is an element of $B$, but $A ne B$).
answered 12 mins ago
Kurt Schwanda
1358
1
The second is not a universal convention (Bourbaki, for instance).
– Bernard
10 mins ago
Fair enough, but that seems to be the meaning his book was using.
– Kurt Schwanda
8 mins ago
Wikipedia is not gospel. "but that seems to be the meaning his book was using" True. But we'd be doing the OP a disservice to imply this is universally agreed upon.
– fleablood
3 mins ago
add a comment |Â
up vote
1
down vote
The strict inclusion $ Asubset B$ is used when $ Asubseteq B$ and $B$ has an element which is not in $A$
Thus if $A=B$ they count $ Asubset B$ as false.
On the other hand if $A=B$, both $ Asubseteq B$ and $ Bsubseteq A$ are true.
answered 11 mins ago
Mohammad Riazi-Kermani
33.2k41854
add a comment |Â
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
If the book distinguishes between $subset$ and $subseteq$, then most likely the former symbol denotes proper inclusion, so $0subset0$ is false. The latter symbol instead will denote inclusion (with possible equality).
However it's very common to find $subset$ denoting inclusion (with possible equality), so one always has to check or try and infer from the context. Don't take Wikipedia pages as revealed truth.
It's so common that $subset$ denotes nonstrict inclusion that somebody uses $subsetneq$ to denote proper inclusion, for safety.
answered 11 mins ago
egreg
168k1281190
add a comment |Â
up vote
3
down vote
accepted
If the book distinguishes between $subset$ and $subseteq$, then most likely the former symbol denotes proper inclusion, so $0subset0$ is false. The latter symbol instead will denote inclusion (with possible equality).
However it's very common to find $subset$ denoting inclusion (with possible equality), so one always has to check or try and infer from the context. Don't take Wikipedia pages as revealed truth.
It's so common that $subset$ denotes nonstrict inclusion that somebody uses $subsetneq$ to denote proper inclusion, for safety.
answered 11 mins ago
egreg
168k1281190
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
If the book distinguishes between $subset$ and $subseteq$, then most likely the former symbol denotes proper inclusion, so $0subset0$ is false. The latter symbol instead will denote inclusion (with possible equality).
However it's very common to find $subset$ denoting inclusion (with possible equality), so one always has to check or try and infer from the context. Don't take Wikipedia pages as revealed truth.
It's so common that $subset$ denotes nonstrict inclusion that somebody uses $subsetneq$ to denote proper inclusion, for safety.
answered 11 mins ago
egreg
168k1281190
If the book distinguishes between $subset$ and $subseteq$, then most likely the former symbol denotes proper inclusion, so $0subset0$ is false. The latter symbol instead will denote inclusion (with possible equality).
However it's very common to find $subset$ denoting inclusion (with possible equality), so one always has to check or try and infer from the context. Don't take Wikipedia pages as revealed truth.
It's so common that $subset$ denotes nonstrict inclusion that somebody uses $subsetneq$ to denote proper inclusion, for safety.
answered 11 mins ago
egreg
168k1281190
answered 11 mins ago
egreg
168k1281190
answered 11 mins ago
egreg
168k1281190
answered 11 mins ago
egreg
168k1281190
168k1281190
add a comment |Â
add a comment |Â
up vote
1
down vote
https://en.wikipedia.org/wiki/List_of_mathematical_symbols
$A subseteq B$ means that $A$ is a subset of $B$ (in other words, every element of $A$ is an element of $B$).
$A subset B$ means that $A$ is a proper subset of $B$ (every element of $A$ is an element of $B$, but $A ne B$).
answered 12 mins ago
Kurt Schwanda
1358
1
The second is not a universal convention (Bourbaki, for instance).
– Bernard
10 mins ago
Fair enough, but that seems to be the meaning his book was using.
– Kurt Schwanda
8 mins ago
Wikipedia is not gospel. "but that seems to be the meaning his book was using" True. But we'd be doing the OP a disservice to imply this is universally agreed upon.
– fleablood
3 mins ago
add a comment |Â
up vote
1
down vote
https://en.wikipedia.org/wiki/List_of_mathematical_symbols
$A subseteq B$ means that $A$ is a subset of $B$ (in other words, every element of $A$ is an element of $B$).
$A subset B$ means that $A$ is a proper subset of $B$ (every element of $A$ is an element of $B$, but $A ne B$).
answered 12 mins ago
Kurt Schwanda
1358
1
The second is not a universal convention (Bourbaki, for instance).
– Bernard
10 mins ago
Fair enough, but that seems to be the meaning his book was using.
– Kurt Schwanda
8 mins ago
Wikipedia is not gospel. "but that seems to be the meaning his book was using" True. But we'd be doing the OP a disservice to imply this is universally agreed upon.
– fleablood
3 mins ago
add a comment |Â
up vote
1
down vote
up vote
1
down vote
https://en.wikipedia.org/wiki/List_of_mathematical_symbols
$A subseteq B$ means that $A$ is a subset of $B$ (in other words, every element of $A$ is an element of $B$).
$A subset B$ means that $A$ is a proper subset of $B$ (every element of $A$ is an element of $B$, but $A ne B$).
answered 12 mins ago
Kurt Schwanda
1358
https://en.wikipedia.org/wiki/List_of_mathematical_symbols
$A subseteq B$ means that $A$ is a subset of $B$ (in other words, every element of $A$ is an element of $B$).
$A subset B$ means that $A$ is a proper subset of $B$ (every element of $A$ is an element of $B$, but $A ne B$).
answered 12 mins ago
Kurt Schwanda
1358
answered 12 mins ago
Kurt Schwanda
1358
answered 12 mins ago
Kurt Schwanda
1358
answered 12 mins ago
Kurt Schwanda
1358
1358
1
The second is not a universal convention (Bourbaki, for instance).
– Bernard
10 mins ago
Fair enough, but that seems to be the meaning his book was using.
– Kurt Schwanda
8 mins ago
Wikipedia is not gospel. "but that seems to be the meaning his book was using" True. But we'd be doing the OP a disservice to imply this is universally agreed upon.
– fleablood
3 mins ago
add a comment |Â
1
The second is not a universal convention (Bourbaki, for instance).
– Bernard
10 mins ago
Fair enough, but that seems to be the meaning his book was using.
– Kurt Schwanda
8 mins ago
Wikipedia is not gospel. "but that seems to be the meaning his book was using" True. But we'd be doing the OP a disservice to imply this is universally agreed upon.
– fleablood
3 mins ago
1
1
The second is not a universal convention (Bourbaki, for instance).
– Bernard
10 mins ago
The second is not a universal convention (Bourbaki, for instance).
– Bernard
10 mins ago
Fair enough, but that seems to be the meaning his book was using.
– Kurt Schwanda
8 mins ago
Fair enough, but that seems to be the meaning his book was using.
– Kurt Schwanda
8 mins ago
Wikipedia is not gospel. "but that seems to be the meaning his book was using" True. But we'd be doing the OP a disservice to imply this is universally agreed upon.
– fleablood
3 mins ago
Wikipedia is not gospel. "but that seems to be the meaning his book was using" True. But we'd be doing the OP a disservice to imply this is universally agreed upon.
– fleablood
3 mins ago
add a comment |Â
up vote
1
down vote
The strict inclusion $ Asubset B$ is used when $ Asubseteq B$ and $B$ has an element which is not in $A$
Thus if $A=B$ they count $ Asubset B$ as false.
On the other hand if $A=B$, both $ Asubseteq B$ and $ Bsubseteq A$ are true.
answered 11 mins ago
Mohammad Riazi-Kermani
33.2k41854
add a comment |Â
up vote
1
down vote
The strict inclusion $ Asubset B$ is used when $ Asubseteq B$ and $B$ has an element which is not in $A$
Thus if $A=B$ they count $ Asubset B$ as false.
On the other hand if $A=B$, both $ Asubseteq B$ and $ Bsubseteq A$ are true.
answered 11 mins ago
Mohammad Riazi-Kermani
33.2k41854
add a comment |Â
up vote
1
down vote
up vote
1
down vote
The strict inclusion $ Asubset B$ is used when $ Asubseteq B$ and $B$ has an element which is not in $A$
Thus if $A=B$ they count $ Asubset B$ as false.
On the other hand if $A=B$, both $ Asubseteq B$ and $ Bsubseteq A$ are true.
answered 11 mins ago
Mohammad Riazi-Kermani
33.2k41854
The strict inclusion $ Asubset B$ is used when $ Asubseteq B$ and $B$ has an element which is not in $A$
Thus if $A=B$ they count $ Asubset B$ as false.
On the other hand if $A=B$, both $ Asubseteq B$ and $ Bsubseteq A$ are true.
answered 11 mins ago
Mohammad Riazi-Kermani
33.2k41854
answered 11 mins ago
Mohammad Riazi-Kermani
33.2k41854
answered 11 mins ago
Mohammad Riazi-Kermani
33.2k41854
answered 11 mins ago
Mohammad Riazi-Kermani
33.2k41854
33.2k41854
add a comment |Â
add a comment |Â
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1
$subset$ usually means proper subset. So if $A subset B$, then ($a in A implies a in B$) but there must be an $x in B$ such that $x notin A$. Whereas $A subseteq B$means that either $A$ is a subset of $B$ but $A$ can be equal to $B$ as well. Think of the difference between $x leq 5$ and $x<5$.
– Anurag A
16 mins ago
In this context, $Asubset B$ means that $A$ is a proper subset of $B$, i.e., $Aneq B$
– AnotherJohnDoe
15 mins ago
1
It's matter of context. In many contexts, being a proper subset is denoted with $subsetneq$, for instance.
– Bernard
11 mins ago
2
It depends on the text and the notation convention the text uses. All texts will agree that $subsetneq$ means a proper subset that is not equal. And that is false for both of these because the are equal. And all texts well agree $subseteq$ means a subset that might or might not be equal. Ad that is true for both of these (every subset is a subset of itself). And all texts agree that $subset$ means is a subset but they do not agree whether it means it can't be equal or whether it might be equal. Would say 1 and 2 are both true because of how i interpret $subset$....
– fleablood
9 mins ago
1
... but it not that I am right or wrong. It's a matter of how I interpret the statements. What is right is: Both those are improper subsets that are equal. That is right. What is wrong is that 1) is a proper subset. It isn't. What matters is what is 1 saying is it saying $0$ is a proper subset of $0$ (WRONG!) or $0$ is a subset of $0$ (RIGHT). And that depends on the book. Not on any math.
– fleablood
6 mins ago