Mixing
Questions about basic logic (why position of “for all” makes difference)
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I am reading Appendix B of "Introduction to Analysis by Arthur Mattuck 1st edition" It says that the following two sentences have different meaning.
The book says that if the epsilon is introduced in the first clause, everything coming later is understood to depend to it. It says that N depends on the epsilon in the first sentence but does not depend on it in the second sentence. However, I don't understand what the difference is. Could you explain me difference? Also, this is my first question on stack exchange. If you think my attitude is not acceptible or doesn't follow the rules of stack exchange, please let me know so that I can fix it. Thank you
logic formal-languages
asked Jan 20 at 6:19
toughshjtoughshj
1261
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|
show 1 more comment
$begingroup$
I am reading Appendix B of "Introduction to Analysis by Arthur Mattuck 1st edition" It says that the following two sentences have different meaning.
The book says that if the epsilon is introduced in the first clause, everything coming later is understood to depend to it. It says that N depends on the epsilon in the first sentence but does not depend on it in the second sentence. However, I don't understand what the difference is. Could you explain me difference? Also, this is my first question on stack exchange. If you think my attitude is not acceptible or doesn't follow the rules of stack exchange, please let me know so that I can fix it. Thank you
logic formal-languages
asked Jan 20 at 6:19
toughshjtoughshj
1261
$endgroup$
$begingroup$
In the first case, I may have to adjust $N$ to account for smaller values of $epsilon$. In the second case, I can find an $N$ that works for all values of $epsilon$.
$endgroup$
– John Douma
Jan 20 at 6:34
1
$begingroup$
Consider the difference between "(for each person P)(there exists a person M)(M is the biological mother of P)" AND "(there exists a person M)(for each person P)(M is the biological mother of P)". The first says that each person has a biological mother, and the second says that someone is the biological mother of every person.
$endgroup$
– Dave L. Renfro
Jan 20 at 10:07
$begingroup$
Thank you for your responses. It seems that the first sentence has different N for each epsilon but the second sentence has one N that satisfies for all epsilon. Is this correct?
$endgroup$
– toughshj
Jan 21 at 1:35
$begingroup$
Yes. That is correct.
$endgroup$
– John Douma
Jan 21 at 1:43
$begingroup$
Thank you so much. Can I ask one more question? The book says the first sentence is the definition of limit and the second sentence is "the sequence becomes constant and equal to L when you go far enough". I don't understand what the difference is between those two meaning. For second sentence, since it says the sequence becomes equal to L, it seems very similar with the definition of limit of a sequence.
$endgroup$
– toughshj
Jan 21 at 2:00
|
show 1 more comment
$begingroup$
I am reading Appendix B of "Introduction to Analysis by Arthur Mattuck 1st edition" It says that the following two sentences have different meaning.
The book says that if the epsilon is introduced in the first clause, everything coming later is understood to depend to it. It says that N depends on the epsilon in the first sentence but does not depend on it in the second sentence. However, I don't understand what the difference is. Could you explain me difference? Also, this is my first question on stack exchange. If you think my attitude is not acceptible or doesn't follow the rules of stack exchange, please let me know so that I can fix it. Thank you
logic formal-languages
asked Jan 20 at 6:19
toughshjtoughshj
1261
$endgroup$
I am reading Appendix B of "Introduction to Analysis by Arthur Mattuck 1st edition" It says that the following two sentences have different meaning.
The book says that if the epsilon is introduced in the first clause, everything coming later is understood to depend to it. It says that N depends on the epsilon in the first sentence but does not depend on it in the second sentence. However, I don't understand what the difference is. Could you explain me difference? Also, this is my first question on stack exchange. If you think my attitude is not acceptible or doesn't follow the rules of stack exchange, please let me know so that I can fix it. Thank you
logic formal-languages
logic formal-languages
asked Jan 20 at 6:19
toughshjtoughshj
1261
asked Jan 20 at 6:19
toughshjtoughshj
1261
asked Jan 20 at 6:19
toughshjtoughshj
1261
asked Jan 20 at 6:19
toughshjtoughshj
1261
asked Jan 20 at 6:19
toughshjtoughshj
1261
1261
$begingroup$
In the first case, I may have to adjust $N$ to account for smaller values of $epsilon$. In the second case, I can find an $N$ that works for all values of $epsilon$.
$endgroup$
– John Douma
Jan 20 at 6:34
1
$begingroup$
Consider the difference between "(for each person P)(there exists a person M)(M is the biological mother of P)" AND "(there exists a person M)(for each person P)(M is the biological mother of P)". The first says that each person has a biological mother, and the second says that someone is the biological mother of every person.
$endgroup$
– Dave L. Renfro
Jan 20 at 10:07
$begingroup$
Thank you for your responses. It seems that the first sentence has different N for each epsilon but the second sentence has one N that satisfies for all epsilon. Is this correct?
$endgroup$
– toughshj
Jan 21 at 1:35
$begingroup$
Yes. That is correct.
$endgroup$
– John Douma
Jan 21 at 1:43
$begingroup$
Thank you so much. Can I ask one more question? The book says the first sentence is the definition of limit and the second sentence is "the sequence becomes constant and equal to L when you go far enough". I don't understand what the difference is between those two meaning. For second sentence, since it says the sequence becomes equal to L, it seems very similar with the definition of limit of a sequence.
$endgroup$
– toughshj
Jan 21 at 2:00
|
show 1 more comment
$begingroup$
In the first case, I may have to adjust $N$ to account for smaller values of $epsilon$. In the second case, I can find an $N$ that works for all values of $epsilon$.
$endgroup$
– John Douma
Jan 20 at 6:34
1
$begingroup$
Consider the difference between "(for each person P)(there exists a person M)(M is the biological mother of P)" AND "(there exists a person M)(for each person P)(M is the biological mother of P)". The first says that each person has a biological mother, and the second says that someone is the biological mother of every person.
$endgroup$
– Dave L. Renfro
Jan 20 at 10:07
$begingroup$
Thank you for your responses. It seems that the first sentence has different N for each epsilon but the second sentence has one N that satisfies for all epsilon. Is this correct?
$endgroup$
– toughshj
Jan 21 at 1:35
$begingroup$
Yes. That is correct.
$endgroup$
– John Douma
Jan 21 at 1:43
$begingroup$
Thank you so much. Can I ask one more question? The book says the first sentence is the definition of limit and the second sentence is "the sequence becomes constant and equal to L when you go far enough". I don't understand what the difference is between those two meaning. For second sentence, since it says the sequence becomes equal to L, it seems very similar with the definition of limit of a sequence.
$endgroup$
– toughshj
Jan 21 at 2:00
$begingroup$
In the first case, I may have to adjust $N$ to account for smaller values of $epsilon$. In the second case, I can find an $N$ that works for all values of $epsilon$.
$endgroup$
– John Douma
Jan 20 at 6:34
$begingroup$
In the first case, I may have to adjust $N$ to account for smaller values of $epsilon$. In the second case, I can find an $N$ that works for all values of $epsilon$.
$endgroup$
– John Douma
Jan 20 at 6:34
1
1
$begingroup$
Consider the difference between "(for each person P)(there exists a person M)(M is the biological mother of P)" AND "(there exists a person M)(for each person P)(M is the biological mother of P)". The first says that each person has a biological mother, and the second says that someone is the biological mother of every person.
$endgroup$
– Dave L. Renfro
Jan 20 at 10:07
$begingroup$
Consider the difference between "(for each person P)(there exists a person M)(M is the biological mother of P)" AND "(there exists a person M)(for each person P)(M is the biological mother of P)". The first says that each person has a biological mother, and the second says that someone is the biological mother of every person.
$endgroup$
– Dave L. Renfro
Jan 20 at 10:07
$begingroup$
Thank you for your responses. It seems that the first sentence has different N for each epsilon but the second sentence has one N that satisfies for all epsilon. Is this correct?
$endgroup$
– toughshj
Jan 21 at 1:35
$begingroup$
Thank you for your responses. It seems that the first sentence has different N for each epsilon but the second sentence has one N that satisfies for all epsilon. Is this correct?
$endgroup$
– toughshj
Jan 21 at 1:35
$begingroup$
Yes. That is correct.
$endgroup$
– John Douma
Jan 21 at 1:43
$begingroup$
Yes. That is correct.
$endgroup$
– John Douma
Jan 21 at 1:43
$begingroup$
Thank you so much. Can I ask one more question? The book says the first sentence is the definition of limit and the second sentence is "the sequence becomes constant and equal to L when you go far enough". I don't understand what the difference is between those two meaning. For second sentence, since it says the sequence becomes equal to L, it seems very similar with the definition of limit of a sequence.
$endgroup$
– toughshj
Jan 21 at 2:00
$begingroup$
Thank you so much. Can I ask one more question? The book says the first sentence is the definition of limit and the second sentence is "the sequence becomes constant and equal to L when you go far enough". I don't understand what the difference is between those two meaning. For second sentence, since it says the sequence becomes equal to L, it seems very similar with the definition of limit of a sequence.
$endgroup$
– toughshj
Jan 21 at 2:00
|
show 1 more comment
1 Answer
1
active
oldest
votes
$begingroup$
Hint
why position of “for all” makes difference ?
Consider a simple example with natural numbers and their ordering, i.e. the structure $(mathbb N, <)$.
Can you see the difference between :
$forall n exists m (n < m)$
and :
$exists m forall n (n < m)$ ?
answered Jan 20 at 7:55
Mauro ALLEGRANZAMauro ALLEGRANZA
66.8k449115
$endgroup$
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint
why position of “for all” makes difference ?
Consider a simple example with natural numbers and their ordering, i.e. the structure $(mathbb N, <)$.
Can you see the difference between :
$forall n exists m (n < m)$
and :
$exists m forall n (n < m)$ ?
answered Jan 20 at 7:55
Mauro ALLEGRANZAMauro ALLEGRANZA
66.8k449115
$endgroup$
add a comment |
$begingroup$
Hint
why position of “for all” makes difference ?
Consider a simple example with natural numbers and their ordering, i.e. the structure $(mathbb N, <)$.
Can you see the difference between :
$forall n exists m (n < m)$
and :
$exists m forall n (n < m)$ ?
answered Jan 20 at 7:55
Mauro ALLEGRANZAMauro ALLEGRANZA
66.8k449115
$endgroup$
add a comment |
$begingroup$
Hint
why position of “for all” makes difference ?
Consider a simple example with natural numbers and their ordering, i.e. the structure $(mathbb N, <)$.
Can you see the difference between :
$forall n exists m (n < m)$
and :
$exists m forall n (n < m)$ ?
answered Jan 20 at 7:55
Mauro ALLEGRANZAMauro ALLEGRANZA
66.8k449115
$endgroup$
Hint
why position of “for all” makes difference ?
Consider a simple example with natural numbers and their ordering, i.e. the structure $(mathbb N, <)$.
Can you see the difference between :
$forall n exists m (n < m)$
and :
$exists m forall n (n < m)$ ?
answered Jan 20 at 7:55
Mauro ALLEGRANZAMauro ALLEGRANZA
66.8k449115
answered Jan 20 at 7:55
Mauro ALLEGRANZAMauro ALLEGRANZA
66.8k449115
answered Jan 20 at 7:55
Mauro ALLEGRANZAMauro ALLEGRANZA
66.8k449115
answered Jan 20 at 7:55
Mauro ALLEGRANZAMauro ALLEGRANZA
66.8k449115
66.8k449115
add a comment |
add a comment |
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$begingroup$
In the first case, I may have to adjust $N$ to account for smaller values of $epsilon$. In the second case, I can find an $N$ that works for all values of $epsilon$.
$endgroup$
– John Douma
Jan 20 at 6:34
1
$begingroup$
Consider the difference between "(for each person P)(there exists a person M)(M is the biological mother of P)" AND "(there exists a person M)(for each person P)(M is the biological mother of P)". The first says that each person has a biological mother, and the second says that someone is the biological mother of every person.
$endgroup$
– Dave L. Renfro
Jan 20 at 10:07
$begingroup$
Thank you for your responses. It seems that the first sentence has different N for each epsilon but the second sentence has one N that satisfies for all epsilon. Is this correct?
$endgroup$
– toughshj
Jan 21 at 1:35
$begingroup$
Yes. That is correct.
$endgroup$
– John Douma
Jan 21 at 1:43
$begingroup$
Thank you so much. Can I ask one more question? The book says the first sentence is the definition of limit and the second sentence is "the sequence becomes constant and equal to L when you go far enough". I don't understand what the difference is between those two meaning. For second sentence, since it says the sequence becomes equal to L, it seems very similar with the definition of limit of a sequence.
$endgroup$
– toughshj
Jan 21 at 2:00