Definition of Order in real Analysis Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Ways to visualize the real numbers?Coding Forcing Notions by Ordinal Numbers: A Possible Approach to Shelah-Foreman-Magidor ConjectureJava comparator documentation: confused about the terminology “total order.”Need help proving this lexicographic relation is a partial orderHow to reconcile these three definitions of an order?Confused about the meaning of the term “Order Relation”Orders, Partial Orders, Strict Partial Orders, Total Orders, Strict Total Orders, and Strict OrdersRadii of Neighborhoods with non-Real Distance MetricInconsistency related to the definition of net/directed set in Kelley's General TopologyOrder relation and the same order type
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Definition of Order in real Analysis
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Ways to visualize the real numbers?Coding Forcing Notions by Ordinal Numbers: A Possible Approach to Shelah-Foreman-Magidor ConjectureJava comparator documentation: confused about the terminology “total order.”Need help proving this lexicographic relation is a partial orderHow to reconcile these three definitions of an order?Confused about the meaning of the term “Order Relation”Orders, Partial Orders, Strict Partial Orders, Total Orders, Strict Total Orders, and Strict OrdersRadii of Neighborhoods with non-Real Distance MetricInconsistency related to the definition of net/directed set in Kelley's General TopologyOrder relation and the same order type
$begingroup$
In Walter Rudin's Principle of Mathematical Analysis 3ed, what's the difference between $<$ sign, which is used to denote the order relation and $<$ sign, which is used to compare $x$ and $y$ in first property in the definition 1.5 (i.e order's definition) ? My main problem is that i think order is the definition of $<$ (i.e operator which is used to compare two numbers) but how can the same operator (or relation, I don't know) $<$ (bottom red circle in image i attached) is used to define the same thing (top red circle in the image i attached)
Definition 1.5?. What is the definition of $<$ sign used to compare two real numbers?
real-analysis order-theory
asked Apr 1 at 18:35
Soham GadhaveSoham Gadhave
83
$endgroup$
add a comment |
$begingroup$
In Walter Rudin's Principle of Mathematical Analysis 3ed, what's the difference between $<$ sign, which is used to denote the order relation and $<$ sign, which is used to compare $x$ and $y$ in first property in the definition 1.5 (i.e order's definition) ? My main problem is that i think order is the definition of $<$ (i.e operator which is used to compare two numbers) but how can the same operator (or relation, I don't know) $<$ (bottom red circle in image i attached) is used to define the same thing (top red circle in the image i attached)
Definition 1.5?. What is the definition of $<$ sign used to compare two real numbers?
real-analysis order-theory
asked Apr 1 at 18:35
Soham GadhaveSoham Gadhave
83
$endgroup$
1
$begingroup$
Could you give a bit more detail? Maybe quote the passages where these symbols come up?
$endgroup$
– Sambo
Apr 1 at 18:39
$begingroup$
We have $x<y$ for real numbers and $xprec y$ for order relations.
$endgroup$
– Dietrich Burde
Apr 1 at 18:40
1
$begingroup$
Where in Rudin do you see the $prec$ symbol? I can't find it.
$endgroup$
– Hans Lundmark
Apr 1 at 19:28
add a comment |
$begingroup$
In Walter Rudin's Principle of Mathematical Analysis 3ed, what's the difference between $<$ sign, which is used to denote the order relation and $<$ sign, which is used to compare $x$ and $y$ in first property in the definition 1.5 (i.e order's definition) ? My main problem is that i think order is the definition of $<$ (i.e operator which is used to compare two numbers) but how can the same operator (or relation, I don't know) $<$ (bottom red circle in image i attached) is used to define the same thing (top red circle in the image i attached)
Definition 1.5?. What is the definition of $<$ sign used to compare two real numbers?
real-analysis order-theory
asked Apr 1 at 18:35
Soham GadhaveSoham Gadhave
83
$endgroup$
In Walter Rudin's Principle of Mathematical Analysis 3ed, what's the difference between $<$ sign, which is used to denote the order relation and $<$ sign, which is used to compare $x$ and $y$ in first property in the definition 1.5 (i.e order's definition) ? My main problem is that i think order is the definition of $<$ (i.e operator which is used to compare two numbers) but how can the same operator (or relation, I don't know) $<$ (bottom red circle in image i attached) is used to define the same thing (top red circle in the image i attached)
Definition 1.5?. What is the definition of $<$ sign used to compare two real numbers?
real-analysis order-theory
real-analysis order-theory
asked Apr 1 at 18:35
Soham GadhaveSoham Gadhave
83
asked Apr 1 at 18:35
Soham GadhaveSoham Gadhave
83
edited Apr 3 at 15:13
Soham Gadhave
asked Apr 1 at 18:35
Soham GadhaveSoham Gadhave
83
asked Apr 1 at 18:35
Soham GadhaveSoham Gadhave
83
asked Apr 1 at 18:35
Soham GadhaveSoham Gadhave
83
83
1
$begingroup$
Could you give a bit more detail? Maybe quote the passages where these symbols come up?
$endgroup$
– Sambo
Apr 1 at 18:39
$begingroup$
We have $x<y$ for real numbers and $xprec y$ for order relations.
$endgroup$
– Dietrich Burde
Apr 1 at 18:40
1
$begingroup$
Where in Rudin do you see the $prec$ symbol? I can't find it.
$endgroup$
– Hans Lundmark
Apr 1 at 19:28
add a comment |
1
$begingroup$
Could you give a bit more detail? Maybe quote the passages where these symbols come up?
$endgroup$
– Sambo
Apr 1 at 18:39
$begingroup$
We have $x<y$ for real numbers and $xprec y$ for order relations.
$endgroup$
– Dietrich Burde
Apr 1 at 18:40
1
$begingroup$
Where in Rudin do you see the $prec$ symbol? I can't find it.
$endgroup$
– Hans Lundmark
Apr 1 at 19:28
1
1
$begingroup$
Could you give a bit more detail? Maybe quote the passages where these symbols come up?
$endgroup$
– Sambo
Apr 1 at 18:39
$begingroup$
Could you give a bit more detail? Maybe quote the passages where these symbols come up?
$endgroup$
– Sambo
Apr 1 at 18:39
$begingroup$
We have $x<y$ for real numbers and $xprec y$ for order relations.
$endgroup$
– Dietrich Burde
Apr 1 at 18:40
$begingroup$
We have $x<y$ for real numbers and $xprec y$ for order relations.
$endgroup$
– Dietrich Burde
Apr 1 at 18:40
1
1
$begingroup$
Where in Rudin do you see the $prec$ symbol? I can't find it.
$endgroup$
– Hans Lundmark
Apr 1 at 19:28
$begingroup$
Where in Rudin do you see the $prec$ symbol? I can't find it.
$endgroup$
– Hans Lundmark
Apr 1 at 19:28
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
In that passage Rudin is defining what it means for a relation to be called an order relation. Any symbol there will do. When you read that definition, imagine replace the "$<$" by "$R$".
There are many relations that satisfy those properties. For example, the set $S$ might be the set of words in the English alphabet, and $R$ the relation "comes earlier in the dictionary".
The example that will be of the most use to Rudin is the one where $S$ is the set of real numbers and $R$ is the ordinary numerical relation "is smaller than".
Rudin provides this abstract definition because he may want to reason about order relations in general, not just the one you know about for numbers.
answered Apr 3 at 15:22
Ethan BolkerEthan Bolker
46.3k555121
$endgroup$
add a comment |
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$begingroup$
In that passage Rudin is defining what it means for a relation to be called an order relation. Any symbol there will do. When you read that definition, imagine replace the "$<$" by "$R$".
There are many relations that satisfy those properties. For example, the set $S$ might be the set of words in the English alphabet, and $R$ the relation "comes earlier in the dictionary".
The example that will be of the most use to Rudin is the one where $S$ is the set of real numbers and $R$ is the ordinary numerical relation "is smaller than".
Rudin provides this abstract definition because he may want to reason about order relations in general, not just the one you know about for numbers.
answered Apr 3 at 15:22
Ethan BolkerEthan Bolker
46.3k555121
$endgroup$
add a comment |
$begingroup$
In that passage Rudin is defining what it means for a relation to be called an order relation. Any symbol there will do. When you read that definition, imagine replace the "$<$" by "$R$".
There are many relations that satisfy those properties. For example, the set $S$ might be the set of words in the English alphabet, and $R$ the relation "comes earlier in the dictionary".
The example that will be of the most use to Rudin is the one where $S$ is the set of real numbers and $R$ is the ordinary numerical relation "is smaller than".
Rudin provides this abstract definition because he may want to reason about order relations in general, not just the one you know about for numbers.
answered Apr 3 at 15:22
Ethan BolkerEthan Bolker
46.3k555121
$endgroup$
add a comment |
$begingroup$
In that passage Rudin is defining what it means for a relation to be called an order relation. Any symbol there will do. When you read that definition, imagine replace the "$<$" by "$R$".
There are many relations that satisfy those properties. For example, the set $S$ might be the set of words in the English alphabet, and $R$ the relation "comes earlier in the dictionary".
The example that will be of the most use to Rudin is the one where $S$ is the set of real numbers and $R$ is the ordinary numerical relation "is smaller than".
Rudin provides this abstract definition because he may want to reason about order relations in general, not just the one you know about for numbers.
answered Apr 3 at 15:22
Ethan BolkerEthan Bolker
46.3k555121
$endgroup$
In that passage Rudin is defining what it means for a relation to be called an order relation. Any symbol there will do. When you read that definition, imagine replace the "$<$" by "$R$".
There are many relations that satisfy those properties. For example, the set $S$ might be the set of words in the English alphabet, and $R$ the relation "comes earlier in the dictionary".
The example that will be of the most use to Rudin is the one where $S$ is the set of real numbers and $R$ is the ordinary numerical relation "is smaller than".
Rudin provides this abstract definition because he may want to reason about order relations in general, not just the one you know about for numbers.
answered Apr 3 at 15:22
Ethan BolkerEthan Bolker
46.3k555121
answered Apr 3 at 15:22
Ethan BolkerEthan Bolker
46.3k555121
answered Apr 3 at 15:22
Ethan BolkerEthan Bolker
46.3k555121
answered Apr 3 at 15:22
Ethan BolkerEthan Bolker
46.3k555121
46.3k555121
add a comment |
add a comment |
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1
$begingroup$
Could you give a bit more detail? Maybe quote the passages where these symbols come up?
$endgroup$
– Sambo
Apr 1 at 18:39
$begingroup$
We have $x<y$ for real numbers and $xprec y$ for order relations.
$endgroup$
– Dietrich Burde
Apr 1 at 18:40
1
$begingroup$
Where in Rudin do you see the $prec$ symbol? I can't find it.
$endgroup$
– Hans Lundmark
Apr 1 at 19:28