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











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0
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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
    19 hours ago










  • 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
    10 hours ago

















up vote
0
down vote

favorite












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
    19 hours ago










  • 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
    10 hours ago















up vote
0
down vote

favorite









up vote
0
down vote

favorite











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 17 hours ago









David Richerby

65.2k1597186




65.2k1597186










asked 19 hours ago









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
    19 hours ago










  • 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
    10 hours ago




















  • $3 gt 2$ 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
    19 hours ago










  • 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
    10 hours ago


















$3 gt 2$ 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
19 hours ago




$3 gt 2$ 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
19 hours ago












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
10 hours ago






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
10 hours ago












2 Answers
2






active

oldest

votes

















up vote
2
down vote



accepted










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
    18 hours ago












  • @Ryan $emptyset$ stands for the set with no elements, i.e., the empty set. So you are exactly right.
    – Juho
    18 hours ago


















up vote
3
down vote













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
    13 hours ago






  • 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
    13 hours ago






  • 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
    13 hours ago










  • @chi I see thanks I understand. You count the possible combinations. Thanks.
    – Koray Tugay
    11 hours ago










  • @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
    10 hours ago













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2 Answers
2






active

oldest

votes








2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
2
down vote



accepted










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
    18 hours ago












  • @Ryan $emptyset$ stands for the set with no elements, i.e., the empty set. So you are exactly right.
    – Juho
    18 hours ago















up vote
2
down vote



accepted










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
    18 hours ago












  • @Ryan $emptyset$ stands for the set with no elements, i.e., the empty set. So you are exactly right.
    – Juho
    18 hours ago













up vote
2
down vote



accepted







up vote
2
down vote



accepted






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 18 hours ago

























answered 18 hours ago









Juho

15.2k54089




15.2k54089












  • Question, What does "S≠∅" Mean? That the subset should not be none/empty?
    – Ryan Cameron
    18 hours ago












  • @Ryan $emptyset$ stands for the set with no elements, i.e., the empty set. So you are exactly right.
    – Juho
    18 hours ago


















  • Question, What does "S≠∅" Mean? That the subset should not be none/empty?
    – Ryan Cameron
    18 hours ago












  • @Ryan $emptyset$ stands for the set with no elements, i.e., the empty set. So you are exactly right.
    – Juho
    18 hours ago
















Question, What does "S≠∅" Mean? That the subset should not be none/empty?
– Ryan Cameron
18 hours ago






Question, What does "S≠∅" Mean? That the subset should not be none/empty?
– Ryan Cameron
18 hours ago














@Ryan $emptyset$ stands for the set with no elements, i.e., the empty set. So you are exactly right.
– Juho
18 hours ago




@Ryan $emptyset$ stands for the set with no elements, i.e., the empty set. So you are exactly right.
– Juho
18 hours ago










up vote
3
down vote













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
    13 hours ago






  • 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
    13 hours ago






  • 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
    13 hours ago










  • @chi I see thanks I understand. You count the possible combinations. Thanks.
    – Koray Tugay
    11 hours ago










  • @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
    10 hours ago

















up vote
3
down vote













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
    13 hours ago






  • 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
    13 hours ago






  • 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
    13 hours ago










  • @chi I see thanks I understand. You count the possible combinations. Thanks.
    – Koray Tugay
    11 hours ago










  • @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
    10 hours ago















up vote
3
down vote










up vote
3
down vote









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.







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answered 17 hours ago









David Richerby

65.2k1597186




65.2k1597186












  • 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
    13 hours ago






  • 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
    13 hours ago






  • 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
    13 hours ago










  • @chi I see thanks I understand. You count the possible combinations. Thanks.
    – Koray Tugay
    11 hours ago










  • @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
    10 hours ago




















  • 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
    13 hours ago






  • 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
    13 hours ago






  • 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
    13 hours ago










  • @chi I see thanks I understand. You count the possible combinations. Thanks.
    – Koray Tugay
    11 hours ago










  • @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
    10 hours ago


















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
13 hours ago




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
13 hours ago




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
13 hours ago




@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
13 hours ago




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
13 hours ago




@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
13 hours ago












@chi I see thanks I understand. You count the possible combinations. Thanks.
– Koray Tugay
11 hours ago




@chi I see thanks I understand. You count the possible combinations. Thanks.
– Koray Tugay
11 hours ago












@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
10 hours ago






@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
10 hours ago




















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