Confusion about the notation of empty interval
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Currently, I'm studying about intervals. I got the basic understanding and knowledge about them but while reading the page about intervals in Wikipedia, under the classification of intervals tab, I didn't understand some of the notations used for the empty interval. They are
$(b,a)$
$(b,a]$
$[b,a)$
where $b>a$
What does the upper-bound (b) written before the lower-bound (a) in these three notations mean? And how do these notations work?
Thank you for your valuable suggestions.
elementary-set-theory notation
asked Jan 16 at 9:36
user8718165user8718165
125
$endgroup$
|
show 4 more comments
$begingroup$
Currently, I'm studying about intervals. I got the basic understanding and knowledge about them but while reading the page about intervals in Wikipedia, under the classification of intervals tab, I didn't understand some of the notations used for the empty interval. They are
$(b,a)$
$(b,a]$
$[b,a)$
where $b>a$
What does the upper-bound (b) written before the lower-bound (a) in these three notations mean? And how do these notations work?
Thank you for your valuable suggestions.
elementary-set-theory notation
asked Jan 16 at 9:36
user8718165user8718165
125
$endgroup$
1
$begingroup$
Because $(b,a) = { r in mathbb R mid x < a < b <x }$.
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 9:45
$begingroup$
See also the post Why is the empty set considered an interval?
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 9:50
$begingroup$
Yes, I've seen it but why's there nothing after the $≤$ in (A,≤) in the first line ?
$endgroup$
– user8718165
Jan 16 at 10:06
$begingroup$
$(A, le)$ is not an "interval"... It is the "environment"; in calculus it is $mathbb R$ with the usual ordering $le$ between real numbers. An interval $(a,b)$ or $[a,b]$ is a subset of $mathbb R$.
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 10:18
$begingroup$
Thank you so much!! the domain is a set of real numbers that contains a non-empty open interval could you please tell me what the highlighted line means? This is the last thing I'd like to know.
$endgroup$
– user8718165
Jan 16 at 10:27
|
show 4 more comments
$begingroup$
Currently, I'm studying about intervals. I got the basic understanding and knowledge about them but while reading the page about intervals in Wikipedia, under the classification of intervals tab, I didn't understand some of the notations used for the empty interval. They are
$(b,a)$
$(b,a]$
$[b,a)$
where $b>a$
What does the upper-bound (b) written before the lower-bound (a) in these three notations mean? And how do these notations work?
Thank you for your valuable suggestions.
elementary-set-theory notation
asked Jan 16 at 9:36
user8718165user8718165
125
$endgroup$
Currently, I'm studying about intervals. I got the basic understanding and knowledge about them but while reading the page about intervals in Wikipedia, under the classification of intervals tab, I didn't understand some of the notations used for the empty interval. They are
$(b,a)$
$(b,a]$
$[b,a)$
where $b>a$
What does the upper-bound (b) written before the lower-bound (a) in these three notations mean? And how do these notations work?
Thank you for your valuable suggestions.
elementary-set-theory notation
elementary-set-theory notation
asked Jan 16 at 9:36
user8718165user8718165
125
asked Jan 16 at 9:36
user8718165user8718165
125
asked Jan 16 at 9:36
user8718165user8718165
125
asked Jan 16 at 9:36
user8718165user8718165
125
asked Jan 16 at 9:36
user8718165user8718165
125
125
1
$begingroup$
Because $(b,a) = { r in mathbb R mid x < a < b <x }$.
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 9:45
$begingroup$
See also the post Why is the empty set considered an interval?
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 9:50
$begingroup$
Yes, I've seen it but why's there nothing after the $≤$ in (A,≤) in the first line ?
$endgroup$
– user8718165
Jan 16 at 10:06
$begingroup$
$(A, le)$ is not an "interval"... It is the "environment"; in calculus it is $mathbb R$ with the usual ordering $le$ between real numbers. An interval $(a,b)$ or $[a,b]$ is a subset of $mathbb R$.
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 10:18
$begingroup$
Thank you so much!! the domain is a set of real numbers that contains a non-empty open interval could you please tell me what the highlighted line means? This is the last thing I'd like to know.
$endgroup$
– user8718165
Jan 16 at 10:27
|
show 4 more comments
1
$begingroup$
Because $(b,a) = { r in mathbb R mid x < a < b <x }$.
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 9:45
$begingroup$
See also the post Why is the empty set considered an interval?
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 9:50
$begingroup$
Yes, I've seen it but why's there nothing after the $≤$ in (A,≤) in the first line ?
$endgroup$
– user8718165
Jan 16 at 10:06
$begingroup$
$(A, le)$ is not an "interval"... It is the "environment"; in calculus it is $mathbb R$ with the usual ordering $le$ between real numbers. An interval $(a,b)$ or $[a,b]$ is a subset of $mathbb R$.
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 10:18
$begingroup$
Thank you so much!! the domain is a set of real numbers that contains a non-empty open interval could you please tell me what the highlighted line means? This is the last thing I'd like to know.
$endgroup$
– user8718165
Jan 16 at 10:27
1
1
$begingroup$
Because $(b,a) = { r in mathbb R mid x < a < b <x }$.
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 9:45
$begingroup$
Because $(b,a) = { r in mathbb R mid x < a < b <x }$.
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 9:45
$begingroup$
See also the post Why is the empty set considered an interval?
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 9:50
$begingroup$
See also the post Why is the empty set considered an interval?
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 9:50
$begingroup$
Yes, I've seen it but why's there nothing after the $≤$ in (A,≤) in the first line ?
$endgroup$
– user8718165
Jan 16 at 10:06
$begingroup$
Yes, I've seen it but why's there nothing after the $≤$ in (A,≤) in the first line ?
$endgroup$
– user8718165
Jan 16 at 10:06
$begingroup$
$(A, le)$ is not an "interval"... It is the "environment"; in calculus it is $mathbb R$ with the usual ordering $le$ between real numbers. An interval $(a,b)$ or $[a,b]$ is a subset of $mathbb R$.
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 10:18
$begingroup$
$(A, le)$ is not an "interval"... It is the "environment"; in calculus it is $mathbb R$ with the usual ordering $le$ between real numbers. An interval $(a,b)$ or $[a,b]$ is a subset of $mathbb R$.
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 10:18
$begingroup$
Thank you so much!! the domain is a set of real numbers that contains a non-empty open interval could you please tell me what the highlighted line means? This is the last thing I'd like to know.
$endgroup$
– user8718165
Jan 16 at 10:27
$begingroup$
Thank you so much!! the domain is a set of real numbers that contains a non-empty open interval could you please tell me what the highlighted line means? This is the last thing I'd like to know.
$endgroup$
– user8718165
Jan 16 at 10:27
|
show 4 more comments
1 Answer
1
active
oldest
votes
$begingroup$
The empty interval is obviously ... empty; it is an empty set of numbers.
Why e.g. $(b,a) = emptyset$ when $b > a$ ?
Because, in general :
$(a,b) = { x in mathbb R mid a < x text { and } x < b }$.
When $b > a$ we have that :
$(b,a) = { x in mathbb R mid b < x text { and } x < a }$
and there are no $x$ that satisfies the condition.
answered Jan 16 at 9:47
Mauro ALLEGRANZAMauro ALLEGRANZA
65.8k449114
$endgroup$
add a comment |
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$begingroup$
The empty interval is obviously ... empty; it is an empty set of numbers.
Why e.g. $(b,a) = emptyset$ when $b > a$ ?
Because, in general :
$(a,b) = { x in mathbb R mid a < x text { and } x < b }$.
When $b > a$ we have that :
$(b,a) = { x in mathbb R mid b < x text { and } x < a }$
and there are no $x$ that satisfies the condition.
answered Jan 16 at 9:47
Mauro ALLEGRANZAMauro ALLEGRANZA
65.8k449114
$endgroup$
add a comment |
$begingroup$
The empty interval is obviously ... empty; it is an empty set of numbers.
Why e.g. $(b,a) = emptyset$ when $b > a$ ?
Because, in general :
$(a,b) = { x in mathbb R mid a < x text { and } x < b }$.
When $b > a$ we have that :
$(b,a) = { x in mathbb R mid b < x text { and } x < a }$
and there are no $x$ that satisfies the condition.
answered Jan 16 at 9:47
Mauro ALLEGRANZAMauro ALLEGRANZA
65.8k449114
$endgroup$
add a comment |
$begingroup$
The empty interval is obviously ... empty; it is an empty set of numbers.
Why e.g. $(b,a) = emptyset$ when $b > a$ ?
Because, in general :
$(a,b) = { x in mathbb R mid a < x text { and } x < b }$.
When $b > a$ we have that :
$(b,a) = { x in mathbb R mid b < x text { and } x < a }$
and there are no $x$ that satisfies the condition.
answered Jan 16 at 9:47
Mauro ALLEGRANZAMauro ALLEGRANZA
65.8k449114
$endgroup$
The empty interval is obviously ... empty; it is an empty set of numbers.
Why e.g. $(b,a) = emptyset$ when $b > a$ ?
Because, in general :
$(a,b) = { x in mathbb R mid a < x text { and } x < b }$.
When $b > a$ we have that :
$(b,a) = { x in mathbb R mid b < x text { and } x < a }$
and there are no $x$ that satisfies the condition.
answered Jan 16 at 9:47
Mauro ALLEGRANZAMauro ALLEGRANZA
65.8k449114
answered Jan 16 at 9:47
Mauro ALLEGRANZAMauro ALLEGRANZA
65.8k449114
answered Jan 16 at 9:47
Mauro ALLEGRANZAMauro ALLEGRANZA
65.8k449114
answered Jan 16 at 9:47
Mauro ALLEGRANZAMauro ALLEGRANZA
65.8k449114
65.8k449114
add a comment |
add a comment |
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1
$begingroup$
Because $(b,a) = { r in mathbb R mid x < a < b <x }$.
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 9:45
$begingroup$
See also the post Why is the empty set considered an interval?
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 9:50
$begingroup$
Yes, I've seen it but why's there nothing after the $≤$ in (A,≤) in the first line ?
$endgroup$
– user8718165
Jan 16 at 10:06
$begingroup$
$(A, le)$ is not an "interval"... It is the "environment"; in calculus it is $mathbb R$ with the usual ordering $le$ between real numbers. An interval $(a,b)$ or $[a,b]$ is a subset of $mathbb R$.
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 10:18
$begingroup$
Thank you so much!! the domain is a set of real numbers that contains a non-empty open interval could you please tell me what the highlighted line means? This is the last thing I'd like to know.
$endgroup$
– user8718165
Jan 16 at 10:27