Clarify the steps: what happened in this mathematical modelling of TSP?












0














Source: http://examples.gurobi.com/traveling-salesman-problem



I don't get this part: (look at the source)




$$sum_{i,jin{1,2,3},ineq j} x_{ij}=3>2=|{1,2,3}|-1$$




I get that $x_{ij}$ is equal to 3, but why the "> 2" ?



And what is the deal with subtracting 1 from a set? How do you even do that?



How come $|{1,2,3}|-1 = 3 > 2$ ?!?



Okay so:
$$|{1,2,3}|-1 = 2$$



So how is he allowed to write:
$$|{1,2,3}|-1 = 3 > 2$$



?



That is basically the same as writing: (which is incorrect right?) $$2 = 3 > 2$$



I don't get this part at all, please elaborate on what happened in as simple language as possible. My level is high school final math level.










share|cite|improve this question
























  • $3 gt 2$ is what they imply I think. The remaining part subtracts 1 from the cardinality of the set and not the set itself. The cardinality of the set is the number of elements in it. Here, the set has 3 elements so you get 2 if you subtract 1 from the cardinality
    – Kaustabha Ray
    Dec 9 '18 at 10:01










  • This is the very common practice of chaining (in)equalities. Think of it as $3 > 2$ and $2 = 3 - 1$ being chained together as $3 > 2 = 3 - 1$.
    – Alex Vong
    Dec 9 '18 at 18:43


















0














Source: http://examples.gurobi.com/traveling-salesman-problem



I don't get this part: (look at the source)




$$sum_{i,jin{1,2,3},ineq j} x_{ij}=3>2=|{1,2,3}|-1$$




I get that $x_{ij}$ is equal to 3, but why the "> 2" ?



And what is the deal with subtracting 1 from a set? How do you even do that?



How come $|{1,2,3}|-1 = 3 > 2$ ?!?



Okay so:
$$|{1,2,3}|-1 = 2$$



So how is he allowed to write:
$$|{1,2,3}|-1 = 3 > 2$$



?



That is basically the same as writing: (which is incorrect right?) $$2 = 3 > 2$$



I don't get this part at all, please elaborate on what happened in as simple language as possible. My level is high school final math level.










share|cite|improve this question
























  • $3 gt 2$ is what they imply I think. The remaining part subtracts 1 from the cardinality of the set and not the set itself. The cardinality of the set is the number of elements in it. Here, the set has 3 elements so you get 2 if you subtract 1 from the cardinality
    – Kaustabha Ray
    Dec 9 '18 at 10:01










  • This is the very common practice of chaining (in)equalities. Think of it as $3 > 2$ and $2 = 3 - 1$ being chained together as $3 > 2 = 3 - 1$.
    – Alex Vong
    Dec 9 '18 at 18:43
















0












0








0







Source: http://examples.gurobi.com/traveling-salesman-problem



I don't get this part: (look at the source)




$$sum_{i,jin{1,2,3},ineq j} x_{ij}=3>2=|{1,2,3}|-1$$




I get that $x_{ij}$ is equal to 3, but why the "> 2" ?



And what is the deal with subtracting 1 from a set? How do you even do that?



How come $|{1,2,3}|-1 = 3 > 2$ ?!?



Okay so:
$$|{1,2,3}|-1 = 2$$



So how is he allowed to write:
$$|{1,2,3}|-1 = 3 > 2$$



?



That is basically the same as writing: (which is incorrect right?) $$2 = 3 > 2$$



I don't get this part at all, please elaborate on what happened in as simple language as possible. My level is high school final math level.










share|cite|improve this question















Source: http://examples.gurobi.com/traveling-salesman-problem



I don't get this part: (look at the source)




$$sum_{i,jin{1,2,3},ineq j} x_{ij}=3>2=|{1,2,3}|-1$$




I get that $x_{ij}$ is equal to 3, but why the "> 2" ?



And what is the deal with subtracting 1 from a set? How do you even do that?



How come $|{1,2,3}|-1 = 3 > 2$ ?!?



Okay so:
$$|{1,2,3}|-1 = 2$$



So how is he allowed to write:
$$|{1,2,3}|-1 = 3 > 2$$



?



That is basically the same as writing: (which is incorrect right?) $$2 = 3 > 2$$



I don't get this part at all, please elaborate on what happened in as simple language as possible. My level is high school final math level.







traveling-salesman notation






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 9 '18 at 11:42









David Richerby

65.9k15100190




65.9k15100190










asked Dec 9 '18 at 9:51









Ryan Cameron

296




296












  • $3 gt 2$ is what they imply I think. The remaining part subtracts 1 from the cardinality of the set and not the set itself. The cardinality of the set is the number of elements in it. Here, the set has 3 elements so you get 2 if you subtract 1 from the cardinality
    – Kaustabha Ray
    Dec 9 '18 at 10:01










  • This is the very common practice of chaining (in)equalities. Think of it as $3 > 2$ and $2 = 3 - 1$ being chained together as $3 > 2 = 3 - 1$.
    – Alex Vong
    Dec 9 '18 at 18:43




















  • $3 gt 2$ is what they imply I think. The remaining part subtracts 1 from the cardinality of the set and not the set itself. The cardinality of the set is the number of elements in it. Here, the set has 3 elements so you get 2 if you subtract 1 from the cardinality
    – Kaustabha Ray
    Dec 9 '18 at 10:01










  • This is the very common practice of chaining (in)equalities. Think of it as $3 > 2$ and $2 = 3 - 1$ being chained together as $3 > 2 = 3 - 1$.
    – Alex Vong
    Dec 9 '18 at 18:43


















$3 gt 2$ is what they imply I think. The remaining part subtracts 1 from the cardinality of the set and not the set itself. The cardinality of the set is the number of elements in it. Here, the set has 3 elements so you get 2 if you subtract 1 from the cardinality
– Kaustabha Ray
Dec 9 '18 at 10:01




$3 gt 2$ is what they imply I think. The remaining part subtracts 1 from the cardinality of the set and not the set itself. The cardinality of the set is the number of elements in it. Here, the set has 3 elements so you get 2 if you subtract 1 from the cardinality
– Kaustabha Ray
Dec 9 '18 at 10:01












This is the very common practice of chaining (in)equalities. Think of it as $3 > 2$ and $2 = 3 - 1$ being chained together as $3 > 2 = 3 - 1$.
– Alex Vong
Dec 9 '18 at 18:43






This is the very common practice of chaining (in)equalities. Think of it as $3 > 2$ and $2 = 3 - 1$ being chained together as $3 > 2 = 3 - 1$.
– Alex Vong
Dec 9 '18 at 18:43












2 Answers
2






active

oldest

votes


















2














The point of these constraints is eliminating subtours, which the source explains quite clearly. So for every subset $S$ of the nodes, such as ${1,2,3}$, they add a constraint which says $Sigma_{i,j in S, i neq j} x_{ij} leq |S| - 1$. So when this constraint is satisfied, there is no way to form a cycle on the vertices in $S$.



Now, if this constraint was not satisfied (i.e., the number of edges was at least $|S|$), then a cycle could be formed like they show in their figures. For example, on ${1,2,3}$, you can form a triangle (which is a cycle) if you use 3 edges.



Particularly regarding your confusion, note that they have written $|S|-1$ (and not $S-1$). Here, $|S|$ refers to the size of the set $S$ (also known as the cardinality of $S$), so $|{1,2,3}| = 3$. Further, notice that they don't write $2 = 3 > 2$, but instead $3 > 2 = 3 - 1$. If it's clearer, you can also assume the constraint just says $3 > |{1,2,3}| - 1$.






share|cite|improve this answer























  • Question, What does "S≠∅" Mean? That the subset should not be none/empty?
    – Ryan Cameron
    Dec 9 '18 at 10:51












  • @Ryan $emptyset$ stands for the set with no elements, i.e., the empty set. So you are exactly right.
    – Juho
    Dec 9 '18 at 10:56



















3














You seem to have misunderstood pretty much every part of the statement



$$sum_{i,jin{1,2,3},ineq j} x_{ij}=3>2=|{1,2,3}|-1,.$$




I get that $x_{ij}$ is equal to 3,




No, the sum of all values $x_{ij}$ where $i$ and $j$ are distinct values from ${1,2,3}$ is equal to $3$.




but why the "> 2" ?




Because three is bigger than two.




And what is the deal with subtracting 1 from a set? How do you even do that?




No, it's subtracting one from the cardinality of the set. Notice the $|dots|$.




How come $|{1,2,3}|-1 = 3 > 2$ ?!?




It isn't. When we write something like $A=B>C=D$, it means that $A=B$, $B>C$ and $C=D$. You can't just re-order the terms and expect the statement to remain true, just as you can't reorder $3>2$ as $2<3$ and expect it to remain true.



So, the statement as a whole means:




  • The sum of the values $x_{ij}$ is equal to $3$.

  • Also, $3>2$.

  • Also, $2=|{1,2,3}|-1$.



So how is he allowed to write:
$$|{1,2,3}|-1 = 3 > 2,?$$




He isn't and he doesn't.






share|cite|improve this answer





















  • Sorry, I am confused. Regarding the sum of all values xij where i and j are distinct values from {1,2,3} is equal to 3. So I get distinct values 1 and 3 from the set where sum is not 3. So I am not understanding, can you help please? What do you mean sum of all values xij where i and j are distinct?
    – Koray Tugay
    Dec 9 '18 at 16:01






  • 1




    @KorayTugay In this context, $x_{i,j}$ exists only when $i<j$, so the sum above is $x_{1,2}+x_{1,3}+x_{2,3}$ and the result of this is 3 (since it's $1+1+1$, in the considered example). You don't sum the indices, you sum the values of all the numbers $x_{i,j}$.
    – chi
    Dec 9 '18 at 16:05






  • 2




    @KorayTugay I think the underlying problem here is that you don't understand mathematical notation. I suggest you talk to your school maths teacher about that, because you need more interactive help than we can really give on this site.
    – David Richerby
    Dec 9 '18 at 16:06










  • @chi I see thanks I understand. You count the possible combinations. Thanks.
    – Koray Tugay
    Dec 9 '18 at 17:46










  • @RyanCameron My suggestion to understanding $sum$ notation is to expand it out. By definition, we can write $$sum_{i, j in {1, 2, 3}, i neq j} x_{ij} = x_{12} + x_{13} + x_{21} + x_{23} + x_{31} + x_{32}$$ I remember my number theory professor used to write $$sum_{substack{1 le i, j le 3 \ i neq j}} x_{ij}$$ which I think is less formal and more readable.
    – Alex Vong
    Dec 9 '18 at 19:04













Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "419"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: false,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: null,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fcs.stackexchange.com%2fquestions%2f101270%2fclarify-the-steps-what-happened-in-this-mathematical-modelling-of-tsp%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























2 Answers
2






active

oldest

votes








2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes









2














The point of these constraints is eliminating subtours, which the source explains quite clearly. So for every subset $S$ of the nodes, such as ${1,2,3}$, they add a constraint which says $Sigma_{i,j in S, i neq j} x_{ij} leq |S| - 1$. So when this constraint is satisfied, there is no way to form a cycle on the vertices in $S$.



Now, if this constraint was not satisfied (i.e., the number of edges was at least $|S|$), then a cycle could be formed like they show in their figures. For example, on ${1,2,3}$, you can form a triangle (which is a cycle) if you use 3 edges.



Particularly regarding your confusion, note that they have written $|S|-1$ (and not $S-1$). Here, $|S|$ refers to the size of the set $S$ (also known as the cardinality of $S$), so $|{1,2,3}| = 3$. Further, notice that they don't write $2 = 3 > 2$, but instead $3 > 2 = 3 - 1$. If it's clearer, you can also assume the constraint just says $3 > |{1,2,3}| - 1$.






share|cite|improve this answer























  • Question, What does "S≠∅" Mean? That the subset should not be none/empty?
    – Ryan Cameron
    Dec 9 '18 at 10:51












  • @Ryan $emptyset$ stands for the set with no elements, i.e., the empty set. So you are exactly right.
    – Juho
    Dec 9 '18 at 10:56
















2














The point of these constraints is eliminating subtours, which the source explains quite clearly. So for every subset $S$ of the nodes, such as ${1,2,3}$, they add a constraint which says $Sigma_{i,j in S, i neq j} x_{ij} leq |S| - 1$. So when this constraint is satisfied, there is no way to form a cycle on the vertices in $S$.



Now, if this constraint was not satisfied (i.e., the number of edges was at least $|S|$), then a cycle could be formed like they show in their figures. For example, on ${1,2,3}$, you can form a triangle (which is a cycle) if you use 3 edges.



Particularly regarding your confusion, note that they have written $|S|-1$ (and not $S-1$). Here, $|S|$ refers to the size of the set $S$ (also known as the cardinality of $S$), so $|{1,2,3}| = 3$. Further, notice that they don't write $2 = 3 > 2$, but instead $3 > 2 = 3 - 1$. If it's clearer, you can also assume the constraint just says $3 > |{1,2,3}| - 1$.






share|cite|improve this answer























  • Question, What does "S≠∅" Mean? That the subset should not be none/empty?
    – Ryan Cameron
    Dec 9 '18 at 10:51












  • @Ryan $emptyset$ stands for the set with no elements, i.e., the empty set. So you are exactly right.
    – Juho
    Dec 9 '18 at 10:56














2












2








2






The point of these constraints is eliminating subtours, which the source explains quite clearly. So for every subset $S$ of the nodes, such as ${1,2,3}$, they add a constraint which says $Sigma_{i,j in S, i neq j} x_{ij} leq |S| - 1$. So when this constraint is satisfied, there is no way to form a cycle on the vertices in $S$.



Now, if this constraint was not satisfied (i.e., the number of edges was at least $|S|$), then a cycle could be formed like they show in their figures. For example, on ${1,2,3}$, you can form a triangle (which is a cycle) if you use 3 edges.



Particularly regarding your confusion, note that they have written $|S|-1$ (and not $S-1$). Here, $|S|$ refers to the size of the set $S$ (also known as the cardinality of $S$), so $|{1,2,3}| = 3$. Further, notice that they don't write $2 = 3 > 2$, but instead $3 > 2 = 3 - 1$. If it's clearer, you can also assume the constraint just says $3 > |{1,2,3}| - 1$.






share|cite|improve this answer














The point of these constraints is eliminating subtours, which the source explains quite clearly. So for every subset $S$ of the nodes, such as ${1,2,3}$, they add a constraint which says $Sigma_{i,j in S, i neq j} x_{ij} leq |S| - 1$. So when this constraint is satisfied, there is no way to form a cycle on the vertices in $S$.



Now, if this constraint was not satisfied (i.e., the number of edges was at least $|S|$), then a cycle could be formed like they show in their figures. For example, on ${1,2,3}$, you can form a triangle (which is a cycle) if you use 3 edges.



Particularly regarding your confusion, note that they have written $|S|-1$ (and not $S-1$). Here, $|S|$ refers to the size of the set $S$ (also known as the cardinality of $S$), so $|{1,2,3}| = 3$. Further, notice that they don't write $2 = 3 > 2$, but instead $3 > 2 = 3 - 1$. If it's clearer, you can also assume the constraint just says $3 > |{1,2,3}| - 1$.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 9 '18 at 10:35

























answered Dec 9 '18 at 10:23









Juho

15.2k54089




15.2k54089












  • Question, What does "S≠∅" Mean? That the subset should not be none/empty?
    – Ryan Cameron
    Dec 9 '18 at 10:51












  • @Ryan $emptyset$ stands for the set with no elements, i.e., the empty set. So you are exactly right.
    – Juho
    Dec 9 '18 at 10:56


















  • Question, What does "S≠∅" Mean? That the subset should not be none/empty?
    – Ryan Cameron
    Dec 9 '18 at 10:51












  • @Ryan $emptyset$ stands for the set with no elements, i.e., the empty set. So you are exactly right.
    – Juho
    Dec 9 '18 at 10:56
















Question, What does "S≠∅" Mean? That the subset should not be none/empty?
– Ryan Cameron
Dec 9 '18 at 10:51






Question, What does "S≠∅" Mean? That the subset should not be none/empty?
– Ryan Cameron
Dec 9 '18 at 10:51














@Ryan $emptyset$ stands for the set with no elements, i.e., the empty set. So you are exactly right.
– Juho
Dec 9 '18 at 10:56




@Ryan $emptyset$ stands for the set with no elements, i.e., the empty set. So you are exactly right.
– Juho
Dec 9 '18 at 10:56











3














You seem to have misunderstood pretty much every part of the statement



$$sum_{i,jin{1,2,3},ineq j} x_{ij}=3>2=|{1,2,3}|-1,.$$




I get that $x_{ij}$ is equal to 3,




No, the sum of all values $x_{ij}$ where $i$ and $j$ are distinct values from ${1,2,3}$ is equal to $3$.




but why the "> 2" ?




Because three is bigger than two.




And what is the deal with subtracting 1 from a set? How do you even do that?




No, it's subtracting one from the cardinality of the set. Notice the $|dots|$.




How come $|{1,2,3}|-1 = 3 > 2$ ?!?




It isn't. When we write something like $A=B>C=D$, it means that $A=B$, $B>C$ and $C=D$. You can't just re-order the terms and expect the statement to remain true, just as you can't reorder $3>2$ as $2<3$ and expect it to remain true.



So, the statement as a whole means:




  • The sum of the values $x_{ij}$ is equal to $3$.

  • Also, $3>2$.

  • Also, $2=|{1,2,3}|-1$.



So how is he allowed to write:
$$|{1,2,3}|-1 = 3 > 2,?$$




He isn't and he doesn't.






share|cite|improve this answer





















  • Sorry, I am confused. Regarding the sum of all values xij where i and j are distinct values from {1,2,3} is equal to 3. So I get distinct values 1 and 3 from the set where sum is not 3. So I am not understanding, can you help please? What do you mean sum of all values xij where i and j are distinct?
    – Koray Tugay
    Dec 9 '18 at 16:01






  • 1




    @KorayTugay In this context, $x_{i,j}$ exists only when $i<j$, so the sum above is $x_{1,2}+x_{1,3}+x_{2,3}$ and the result of this is 3 (since it's $1+1+1$, in the considered example). You don't sum the indices, you sum the values of all the numbers $x_{i,j}$.
    – chi
    Dec 9 '18 at 16:05






  • 2




    @KorayTugay I think the underlying problem here is that you don't understand mathematical notation. I suggest you talk to your school maths teacher about that, because you need more interactive help than we can really give on this site.
    – David Richerby
    Dec 9 '18 at 16:06










  • @chi I see thanks I understand. You count the possible combinations. Thanks.
    – Koray Tugay
    Dec 9 '18 at 17:46










  • @RyanCameron My suggestion to understanding $sum$ notation is to expand it out. By definition, we can write $$sum_{i, j in {1, 2, 3}, i neq j} x_{ij} = x_{12} + x_{13} + x_{21} + x_{23} + x_{31} + x_{32}$$ I remember my number theory professor used to write $$sum_{substack{1 le i, j le 3 \ i neq j}} x_{ij}$$ which I think is less formal and more readable.
    – Alex Vong
    Dec 9 '18 at 19:04


















3














You seem to have misunderstood pretty much every part of the statement



$$sum_{i,jin{1,2,3},ineq j} x_{ij}=3>2=|{1,2,3}|-1,.$$




I get that $x_{ij}$ is equal to 3,




No, the sum of all values $x_{ij}$ where $i$ and $j$ are distinct values from ${1,2,3}$ is equal to $3$.




but why the "> 2" ?




Because three is bigger than two.




And what is the deal with subtracting 1 from a set? How do you even do that?




No, it's subtracting one from the cardinality of the set. Notice the $|dots|$.




How come $|{1,2,3}|-1 = 3 > 2$ ?!?




It isn't. When we write something like $A=B>C=D$, it means that $A=B$, $B>C$ and $C=D$. You can't just re-order the terms and expect the statement to remain true, just as you can't reorder $3>2$ as $2<3$ and expect it to remain true.



So, the statement as a whole means:




  • The sum of the values $x_{ij}$ is equal to $3$.

  • Also, $3>2$.

  • Also, $2=|{1,2,3}|-1$.



So how is he allowed to write:
$$|{1,2,3}|-1 = 3 > 2,?$$




He isn't and he doesn't.






share|cite|improve this answer





















  • Sorry, I am confused. Regarding the sum of all values xij where i and j are distinct values from {1,2,3} is equal to 3. So I get distinct values 1 and 3 from the set where sum is not 3. So I am not understanding, can you help please? What do you mean sum of all values xij where i and j are distinct?
    – Koray Tugay
    Dec 9 '18 at 16:01






  • 1




    @KorayTugay In this context, $x_{i,j}$ exists only when $i<j$, so the sum above is $x_{1,2}+x_{1,3}+x_{2,3}$ and the result of this is 3 (since it's $1+1+1$, in the considered example). You don't sum the indices, you sum the values of all the numbers $x_{i,j}$.
    – chi
    Dec 9 '18 at 16:05






  • 2




    @KorayTugay I think the underlying problem here is that you don't understand mathematical notation. I suggest you talk to your school maths teacher about that, because you need more interactive help than we can really give on this site.
    – David Richerby
    Dec 9 '18 at 16:06










  • @chi I see thanks I understand. You count the possible combinations. Thanks.
    – Koray Tugay
    Dec 9 '18 at 17:46










  • @RyanCameron My suggestion to understanding $sum$ notation is to expand it out. By definition, we can write $$sum_{i, j in {1, 2, 3}, i neq j} x_{ij} = x_{12} + x_{13} + x_{21} + x_{23} + x_{31} + x_{32}$$ I remember my number theory professor used to write $$sum_{substack{1 le i, j le 3 \ i neq j}} x_{ij}$$ which I think is less formal and more readable.
    – Alex Vong
    Dec 9 '18 at 19:04
















3












3








3






You seem to have misunderstood pretty much every part of the statement



$$sum_{i,jin{1,2,3},ineq j} x_{ij}=3>2=|{1,2,3}|-1,.$$




I get that $x_{ij}$ is equal to 3,




No, the sum of all values $x_{ij}$ where $i$ and $j$ are distinct values from ${1,2,3}$ is equal to $3$.




but why the "> 2" ?




Because three is bigger than two.




And what is the deal with subtracting 1 from a set? How do you even do that?




No, it's subtracting one from the cardinality of the set. Notice the $|dots|$.




How come $|{1,2,3}|-1 = 3 > 2$ ?!?




It isn't. When we write something like $A=B>C=D$, it means that $A=B$, $B>C$ and $C=D$. You can't just re-order the terms and expect the statement to remain true, just as you can't reorder $3>2$ as $2<3$ and expect it to remain true.



So, the statement as a whole means:




  • The sum of the values $x_{ij}$ is equal to $3$.

  • Also, $3>2$.

  • Also, $2=|{1,2,3}|-1$.



So how is he allowed to write:
$$|{1,2,3}|-1 = 3 > 2,?$$




He isn't and he doesn't.






share|cite|improve this answer












You seem to have misunderstood pretty much every part of the statement



$$sum_{i,jin{1,2,3},ineq j} x_{ij}=3>2=|{1,2,3}|-1,.$$




I get that $x_{ij}$ is equal to 3,




No, the sum of all values $x_{ij}$ where $i$ and $j$ are distinct values from ${1,2,3}$ is equal to $3$.




but why the "> 2" ?




Because three is bigger than two.




And what is the deal with subtracting 1 from a set? How do you even do that?




No, it's subtracting one from the cardinality of the set. Notice the $|dots|$.




How come $|{1,2,3}|-1 = 3 > 2$ ?!?




It isn't. When we write something like $A=B>C=D$, it means that $A=B$, $B>C$ and $C=D$. You can't just re-order the terms and expect the statement to remain true, just as you can't reorder $3>2$ as $2<3$ and expect it to remain true.



So, the statement as a whole means:




  • The sum of the values $x_{ij}$ is equal to $3$.

  • Also, $3>2$.

  • Also, $2=|{1,2,3}|-1$.



So how is he allowed to write:
$$|{1,2,3}|-1 = 3 > 2,?$$




He isn't and he doesn't.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 9 '18 at 11:51









David Richerby

65.9k15100190




65.9k15100190












  • Sorry, I am confused. Regarding the sum of all values xij where i and j are distinct values from {1,2,3} is equal to 3. So I get distinct values 1 and 3 from the set where sum is not 3. So I am not understanding, can you help please? What do you mean sum of all values xij where i and j are distinct?
    – Koray Tugay
    Dec 9 '18 at 16:01






  • 1




    @KorayTugay In this context, $x_{i,j}$ exists only when $i<j$, so the sum above is $x_{1,2}+x_{1,3}+x_{2,3}$ and the result of this is 3 (since it's $1+1+1$, in the considered example). You don't sum the indices, you sum the values of all the numbers $x_{i,j}$.
    – chi
    Dec 9 '18 at 16:05






  • 2




    @KorayTugay I think the underlying problem here is that you don't understand mathematical notation. I suggest you talk to your school maths teacher about that, because you need more interactive help than we can really give on this site.
    – David Richerby
    Dec 9 '18 at 16:06










  • @chi I see thanks I understand. You count the possible combinations. Thanks.
    – Koray Tugay
    Dec 9 '18 at 17:46










  • @RyanCameron My suggestion to understanding $sum$ notation is to expand it out. By definition, we can write $$sum_{i, j in {1, 2, 3}, i neq j} x_{ij} = x_{12} + x_{13} + x_{21} + x_{23} + x_{31} + x_{32}$$ I remember my number theory professor used to write $$sum_{substack{1 le i, j le 3 \ i neq j}} x_{ij}$$ which I think is less formal and more readable.
    – Alex Vong
    Dec 9 '18 at 19:04




















  • Sorry, I am confused. Regarding the sum of all values xij where i and j are distinct values from {1,2,3} is equal to 3. So I get distinct values 1 and 3 from the set where sum is not 3. So I am not understanding, can you help please? What do you mean sum of all values xij where i and j are distinct?
    – Koray Tugay
    Dec 9 '18 at 16:01






  • 1




    @KorayTugay In this context, $x_{i,j}$ exists only when $i<j$, so the sum above is $x_{1,2}+x_{1,3}+x_{2,3}$ and the result of this is 3 (since it's $1+1+1$, in the considered example). You don't sum the indices, you sum the values of all the numbers $x_{i,j}$.
    – chi
    Dec 9 '18 at 16:05






  • 2




    @KorayTugay I think the underlying problem here is that you don't understand mathematical notation. I suggest you talk to your school maths teacher about that, because you need more interactive help than we can really give on this site.
    – David Richerby
    Dec 9 '18 at 16:06










  • @chi I see thanks I understand. You count the possible combinations. Thanks.
    – Koray Tugay
    Dec 9 '18 at 17:46










  • @RyanCameron My suggestion to understanding $sum$ notation is to expand it out. By definition, we can write $$sum_{i, j in {1, 2, 3}, i neq j} x_{ij} = x_{12} + x_{13} + x_{21} + x_{23} + x_{31} + x_{32}$$ I remember my number theory professor used to write $$sum_{substack{1 le i, j le 3 \ i neq j}} x_{ij}$$ which I think is less formal and more readable.
    – Alex Vong
    Dec 9 '18 at 19:04


















Sorry, I am confused. Regarding the sum of all values xij where i and j are distinct values from {1,2,3} is equal to 3. So I get distinct values 1 and 3 from the set where sum is not 3. So I am not understanding, can you help please? What do you mean sum of all values xij where i and j are distinct?
– Koray Tugay
Dec 9 '18 at 16:01




Sorry, I am confused. Regarding the sum of all values xij where i and j are distinct values from {1,2,3} is equal to 3. So I get distinct values 1 and 3 from the set where sum is not 3. So I am not understanding, can you help please? What do you mean sum of all values xij where i and j are distinct?
– Koray Tugay
Dec 9 '18 at 16:01




1




1




@KorayTugay In this context, $x_{i,j}$ exists only when $i<j$, so the sum above is $x_{1,2}+x_{1,3}+x_{2,3}$ and the result of this is 3 (since it's $1+1+1$, in the considered example). You don't sum the indices, you sum the values of all the numbers $x_{i,j}$.
– chi
Dec 9 '18 at 16:05




@KorayTugay In this context, $x_{i,j}$ exists only when $i<j$, so the sum above is $x_{1,2}+x_{1,3}+x_{2,3}$ and the result of this is 3 (since it's $1+1+1$, in the considered example). You don't sum the indices, you sum the values of all the numbers $x_{i,j}$.
– chi
Dec 9 '18 at 16:05




2




2




@KorayTugay I think the underlying problem here is that you don't understand mathematical notation. I suggest you talk to your school maths teacher about that, because you need more interactive help than we can really give on this site.
– David Richerby
Dec 9 '18 at 16:06




@KorayTugay I think the underlying problem here is that you don't understand mathematical notation. I suggest you talk to your school maths teacher about that, because you need more interactive help than we can really give on this site.
– David Richerby
Dec 9 '18 at 16:06












@chi I see thanks I understand. You count the possible combinations. Thanks.
– Koray Tugay
Dec 9 '18 at 17:46




@chi I see thanks I understand. You count the possible combinations. Thanks.
– Koray Tugay
Dec 9 '18 at 17:46












@RyanCameron My suggestion to understanding $sum$ notation is to expand it out. By definition, we can write $$sum_{i, j in {1, 2, 3}, i neq j} x_{ij} = x_{12} + x_{13} + x_{21} + x_{23} + x_{31} + x_{32}$$ I remember my number theory professor used to write $$sum_{substack{1 le i, j le 3 \ i neq j}} x_{ij}$$ which I think is less formal and more readable.
– Alex Vong
Dec 9 '18 at 19:04






@RyanCameron My suggestion to understanding $sum$ notation is to expand it out. By definition, we can write $$sum_{i, j in {1, 2, 3}, i neq j} x_{ij} = x_{12} + x_{13} + x_{21} + x_{23} + x_{31} + x_{32}$$ I remember my number theory professor used to write $$sum_{substack{1 le i, j le 3 \ i neq j}} x_{ij}$$ which I think is less formal and more readable.
– Alex Vong
Dec 9 '18 at 19:04




















draft saved

draft discarded




















































Thanks for contributing an answer to Computer Science Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.





Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


Please pay close attention to the following guidance:


  • Please be sure to answer the question. Provide details and share your research!

But avoid



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fcs.stackexchange.com%2fquestions%2f101270%2fclarify-the-steps-what-happened-in-this-mathematical-modelling-of-tsp%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







Popular posts from this blog

Plaza Victoria

Puebla de Zaragoza

Musa