What is the difference of the following two mathematical sentences? (Walter Rudin's Principles of...
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I am very poor at English.
But I am reading Walter Rudin's Principles of Mathematical Analysis now.
In the book, I found the following sentence:
A point $p$ is a limit point of the set $E$ if every neighborhood of $p$ contains a point $q neq p$ such that $q in E$.
I cannot understand why "the set $E$" instead of "a set $E$".
What is the difference in meaning between the following two mathematical sentences?
(1)
A point $p$ is a limit point of the set $E$ if every neighborhood of $p$ contains a point $q neq p$ such that $q in E$.
(2)
A point $p$ is a limit point of a set $E$ if every neighborhood of $p$ contains a point $q neq p$ such that $q in E$.
soft-question
asked Dec 29 '18 at 4:36
tchappy hatchappy ha
773412
$endgroup$
add a comment |
$begingroup$
I am very poor at English.
But I am reading Walter Rudin's Principles of Mathematical Analysis now.
In the book, I found the following sentence:
A point $p$ is a limit point of the set $E$ if every neighborhood of $p$ contains a point $q neq p$ such that $q in E$.
I cannot understand why "the set $E$" instead of "a set $E$".
What is the difference in meaning between the following two mathematical sentences?
(1)
A point $p$ is a limit point of the set $E$ if every neighborhood of $p$ contains a point $q neq p$ such that $q in E$.
(2)
A point $p$ is a limit point of a set $E$ if every neighborhood of $p$ contains a point $q neq p$ such that $q in E$.
soft-question
asked Dec 29 '18 at 4:36
tchappy hatchappy ha
773412
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6
$begingroup$
There's no difference.
$endgroup$
– Chris Custer
Dec 29 '18 at 4:40
3
$begingroup$
No difference really. For the first may be the set $E$ was already introduced so "the" would be used to refer to that specific one.
$endgroup$
– Pratyush Sarkar
Dec 29 '18 at 4:43
$begingroup$
Thank you very much, Chris Custer and Pratyush Sarkar.
$endgroup$
– tchappy ha
Dec 29 '18 at 5:07
add a comment |
$begingroup$
I am very poor at English.
But I am reading Walter Rudin's Principles of Mathematical Analysis now.
In the book, I found the following sentence:
A point $p$ is a limit point of the set $E$ if every neighborhood of $p$ contains a point $q neq p$ such that $q in E$.
I cannot understand why "the set $E$" instead of "a set $E$".
What is the difference in meaning between the following two mathematical sentences?
(1)
A point $p$ is a limit point of the set $E$ if every neighborhood of $p$ contains a point $q neq p$ such that $q in E$.
(2)
A point $p$ is a limit point of a set $E$ if every neighborhood of $p$ contains a point $q neq p$ such that $q in E$.
soft-question
asked Dec 29 '18 at 4:36
tchappy hatchappy ha
773412
$endgroup$
I am very poor at English.
But I am reading Walter Rudin's Principles of Mathematical Analysis now.
In the book, I found the following sentence:
A point $p$ is a limit point of the set $E$ if every neighborhood of $p$ contains a point $q neq p$ such that $q in E$.
I cannot understand why "the set $E$" instead of "a set $E$".
What is the difference in meaning between the following two mathematical sentences?
(1)
A point $p$ is a limit point of the set $E$ if every neighborhood of $p$ contains a point $q neq p$ such that $q in E$.
(2)
A point $p$ is a limit point of a set $E$ if every neighborhood of $p$ contains a point $q neq p$ such that $q in E$.
soft-question
soft-question
asked Dec 29 '18 at 4:36
tchappy hatchappy ha
773412
asked Dec 29 '18 at 4:36
tchappy hatchappy ha
773412
asked Dec 29 '18 at 4:36
tchappy hatchappy ha
773412
asked Dec 29 '18 at 4:36
tchappy hatchappy ha
773412
asked Dec 29 '18 at 4:36
tchappy hatchappy ha
773412
773412
6
$begingroup$
There's no difference.
$endgroup$
– Chris Custer
Dec 29 '18 at 4:40
3
$begingroup$
No difference really. For the first may be the set $E$ was already introduced so "the" would be used to refer to that specific one.
$endgroup$
– Pratyush Sarkar
Dec 29 '18 at 4:43
$begingroup$
Thank you very much, Chris Custer and Pratyush Sarkar.
$endgroup$
– tchappy ha
Dec 29 '18 at 5:07
add a comment |
6
$begingroup$
There's no difference.
$endgroup$
– Chris Custer
Dec 29 '18 at 4:40
3
$begingroup$
No difference really. For the first may be the set $E$ was already introduced so "the" would be used to refer to that specific one.
$endgroup$
– Pratyush Sarkar
Dec 29 '18 at 4:43
$begingroup$
Thank you very much, Chris Custer and Pratyush Sarkar.
$endgroup$
– tchappy ha
Dec 29 '18 at 5:07
6
6
$begingroup$
There's no difference.
$endgroup$
– Chris Custer
Dec 29 '18 at 4:40
$begingroup$
There's no difference.
$endgroup$
– Chris Custer
Dec 29 '18 at 4:40
3
3
$begingroup$
No difference really. For the first may be the set $E$ was already introduced so "the" would be used to refer to that specific one.
$endgroup$
– Pratyush Sarkar
Dec 29 '18 at 4:43
$begingroup$
No difference really. For the first may be the set $E$ was already introduced so "the" would be used to refer to that specific one.
$endgroup$
– Pratyush Sarkar
Dec 29 '18 at 4:43
$begingroup$
Thank you very much, Chris Custer and Pratyush Sarkar.
$endgroup$
– tchappy ha
Dec 29 '18 at 5:07
$begingroup$
Thank you very much, Chris Custer and Pratyush Sarkar.
$endgroup$
– tchappy ha
Dec 29 '18 at 5:07
add a comment |
1 Answer
1
active
oldest
votes
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There's no fundamental difference; to say "the set $E$" or "a set $E$" are both equally valid, grammatically, and Rudin isn't referring to a special or particular set either. It's just how some people word things sometimes.
While not particularly mathematical, this did remind me of another Stack Exchange question I saw recently. You might find it worth looking at: here ya go
answered Dec 29 '18 at 4:44
Eevee TrainerEevee Trainer
8,32421439
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$begingroup$
Thank you very much, Eevee Trainer.
$endgroup$
– tchappy ha
Dec 29 '18 at 5:07
add a comment |
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1 Answer
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1 Answer
1
active
oldest
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active
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$begingroup$
There's no fundamental difference; to say "the set $E$" or "a set $E$" are both equally valid, grammatically, and Rudin isn't referring to a special or particular set either. It's just how some people word things sometimes.
While not particularly mathematical, this did remind me of another Stack Exchange question I saw recently. You might find it worth looking at: here ya go
answered Dec 29 '18 at 4:44
Eevee TrainerEevee Trainer
8,32421439
$endgroup$
$begingroup$
Thank you very much, Eevee Trainer.
$endgroup$
– tchappy ha
Dec 29 '18 at 5:07
add a comment |
$begingroup$
There's no fundamental difference; to say "the set $E$" or "a set $E$" are both equally valid, grammatically, and Rudin isn't referring to a special or particular set either. It's just how some people word things sometimes.
While not particularly mathematical, this did remind me of another Stack Exchange question I saw recently. You might find it worth looking at: here ya go
answered Dec 29 '18 at 4:44
Eevee TrainerEevee Trainer
8,32421439
$endgroup$
$begingroup$
Thank you very much, Eevee Trainer.
$endgroup$
– tchappy ha
Dec 29 '18 at 5:07
add a comment |
$begingroup$
There's no fundamental difference; to say "the set $E$" or "a set $E$" are both equally valid, grammatically, and Rudin isn't referring to a special or particular set either. It's just how some people word things sometimes.
While not particularly mathematical, this did remind me of another Stack Exchange question I saw recently. You might find it worth looking at: here ya go
answered Dec 29 '18 at 4:44
Eevee TrainerEevee Trainer
8,32421439
$endgroup$
There's no fundamental difference; to say "the set $E$" or "a set $E$" are both equally valid, grammatically, and Rudin isn't referring to a special or particular set either. It's just how some people word things sometimes.
While not particularly mathematical, this did remind me of another Stack Exchange question I saw recently. You might find it worth looking at: here ya go
answered Dec 29 '18 at 4:44
Eevee TrainerEevee Trainer
8,32421439
answered Dec 29 '18 at 4:44
Eevee TrainerEevee Trainer
8,32421439
answered Dec 29 '18 at 4:44
Eevee TrainerEevee Trainer
8,32421439
answered Dec 29 '18 at 4:44
Eevee TrainerEevee Trainer
8,32421439
8,32421439
$begingroup$
Thank you very much, Eevee Trainer.
$endgroup$
– tchappy ha
Dec 29 '18 at 5:07
add a comment |
$begingroup$
Thank you very much, Eevee Trainer.
$endgroup$
– tchappy ha
Dec 29 '18 at 5:07
$begingroup$
Thank you very much, Eevee Trainer.
$endgroup$
– tchappy ha
Dec 29 '18 at 5:07
$begingroup$
Thank you very much, Eevee Trainer.
$endgroup$
– tchappy ha
Dec 29 '18 at 5:07
add a comment |
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6
$begingroup$
There's no difference.
$endgroup$
– Chris Custer
Dec 29 '18 at 4:40
3
$begingroup$
No difference really. For the first may be the set $E$ was already introduced so "the" would be used to refer to that specific one.
$endgroup$
– Pratyush Sarkar
Dec 29 '18 at 4:43
$begingroup$
Thank you very much, Chris Custer and Pratyush Sarkar.
$endgroup$
– tchappy ha
Dec 29 '18 at 5:07