Examples of solving for unknowns using equivalence relations that are not equality, inequality, or boolean truth?What are some good simple examples that getting the right result is not enough?What are some fun/nonstandard examples of arithmetic/geometric series?What are easy examples from daily life of constrained optimization?What are some good low-prerequisite examples for the heuristic advice “If you cannot prove it, prove something stronger.”?Examples for reasoning by analogy going wrongWhat are some good or neat examples of computing a function's Taylor series?Examples where roots are necessary for the solutionSimple examples that violate group axiomsExamples (for beginners) of real functions which are not given by elementary formulaeUsing discrete examples in the beginning of integration
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Examples of solving for unknowns using equivalence relations that are not equality, inequality, or boolean truth?
What are some good simple examples that getting the right result is not enough?What are some fun/nonstandard examples of arithmetic/geometric series?What are easy examples from daily life of constrained optimization?What are some good low-prerequisite examples for the heuristic advice “If you cannot prove it, prove something stronger.”?Examples for reasoning by analogy going wrongWhat are some good or neat examples of computing a function's Taylor series?Examples where roots are necessary for the solutionSimple examples that violate group axiomsExamples (for beginners) of real functions which are not given by elementary formulaeUsing discrete examples in the beginning of integration
.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty margin-bottom:0;
$begingroup$
In a book i'm writing, i want to introduce students to equations in slightly more general terms. Solving an equation with some unknown is just
one example of finding an object by fact that it is a part of
a certain equivalence class - specifically a class in the "equals"-relation. Same goes for inequalities too. In both scenarios we use various operations on both sides, make sure that none of these operations change the equivilance class of any member, and then isolate some desired expression. This is really intuitive and natural to work with for equality, but i think it could be a great exercise for the mind and maturity to try something very different. Does there exist any other nice similair problems and techniques for different relations? Maybe something using congruence or similarity of figures? We can obviously work with set equalities too, though that may be a little very abstract.
For information, my readers has not yet at this point in the book learned calculus. It's at just the point before. The target audience is math-interested high-schoolers.
examples
asked Jul 25 at 5:48
Buster BieBuster Bie
293 bronze badges
$endgroup$
|
show 5 more comments
$begingroup$
In a book i'm writing, i want to introduce students to equations in slightly more general terms. Solving an equation with some unknown is just
one example of finding an object by fact that it is a part of
a certain equivalence class - specifically a class in the "equals"-relation. Same goes for inequalities too. In both scenarios we use various operations on both sides, make sure that none of these operations change the equivilance class of any member, and then isolate some desired expression. This is really intuitive and natural to work with for equality, but i think it could be a great exercise for the mind and maturity to try something very different. Does there exist any other nice similair problems and techniques for different relations? Maybe something using congruence or similarity of figures? We can obviously work with set equalities too, though that may be a little very abstract.
For information, my readers has not yet at this point in the book learned calculus. It's at just the point before. The target audience is math-interested high-schoolers.
examples
asked Jul 25 at 5:48
Buster BieBuster Bie
293 bronze badges
$endgroup$
5
$begingroup$
An equation is a statement of equality. Using it to mean something else is only going to lead to confusion.
$endgroup$
– Peter Taylor
Jul 25 at 7:25
1
$begingroup$
An equation is a statement of equality in terms of an equivalence relation. So, a general question could be: given two objects, do they belong to the same equivalence class?
$endgroup$
– SCS
Jul 25 at 9:08
3
$begingroup$
Even after your edit, I still don't understand exactly what you're talking about. You write that solving inequalities involves "mak[ing] sure that none of these operations change the equivalance class of any member," but what is the equivalence relation you're talking about when you say this? The phrase "equivalence class" is completely meaningless outside of the context of an equivalence relation. (The title doesn't make sense either, since it says "equivalence relations that are not equality, inequality or boolean truth," but inequality and boolean truth are not equivalence relations.)
$endgroup$
– Tanner Swett
Jul 25 at 17:29
1
$begingroup$
In general terms, what you are asking is about different types of relations between entities, of which equations are a subgroup. Inequalities are not equations, and most of the suggestions below are not equations. Please correct your post accordingly?
$endgroup$
– Namaste
Jul 25 at 17:42
4
$begingroup$
Also, I can't think of how solving an equation can be construed as "finding an object by fact that it is a part of a certain equivalence class". If I solve the equation $x^2 = 4$, I'm finding some objects by the fact that they're elements of a certain set, but that set is not an equivalence class.
$endgroup$
– Tanner Swett
Jul 25 at 17:43
|
show 5 more comments
$begingroup$
In a book i'm writing, i want to introduce students to equations in slightly more general terms. Solving an equation with some unknown is just
one example of finding an object by fact that it is a part of
a certain equivalence class - specifically a class in the "equals"-relation. Same goes for inequalities too. In both scenarios we use various operations on both sides, make sure that none of these operations change the equivilance class of any member, and then isolate some desired expression. This is really intuitive and natural to work with for equality, but i think it could be a great exercise for the mind and maturity to try something very different. Does there exist any other nice similair problems and techniques for different relations? Maybe something using congruence or similarity of figures? We can obviously work with set equalities too, though that may be a little very abstract.
For information, my readers has not yet at this point in the book learned calculus. It's at just the point before. The target audience is math-interested high-schoolers.
examples
asked Jul 25 at 5:48
Buster BieBuster Bie
293 bronze badges
$endgroup$
In a book i'm writing, i want to introduce students to equations in slightly more general terms. Solving an equation with some unknown is just
one example of finding an object by fact that it is a part of
a certain equivalence class - specifically a class in the "equals"-relation. Same goes for inequalities too. In both scenarios we use various operations on both sides, make sure that none of these operations change the equivilance class of any member, and then isolate some desired expression. This is really intuitive and natural to work with for equality, but i think it could be a great exercise for the mind and maturity to try something very different. Does there exist any other nice similair problems and techniques for different relations? Maybe something using congruence or similarity of figures? We can obviously work with set equalities too, though that may be a little very abstract.
For information, my readers has not yet at this point in the book learned calculus. It's at just the point before. The target audience is math-interested high-schoolers.
examples
examples
asked Jul 25 at 5:48
Buster BieBuster Bie
293 bronze badges
asked Jul 25 at 5:48
Buster BieBuster Bie
293 bronze badges
edited Jul 25 at 10:06
Buster Bie
asked Jul 25 at 5:48
Buster BieBuster Bie
293 bronze badges
asked Jul 25 at 5:48
Buster BieBuster Bie
293 bronze badges
asked Jul 25 at 5:48
Buster BieBuster Bie
293 bronze badges
293 bronze badges
5
$begingroup$
An equation is a statement of equality. Using it to mean something else is only going to lead to confusion.
$endgroup$
– Peter Taylor
Jul 25 at 7:25
1
$begingroup$
An equation is a statement of equality in terms of an equivalence relation. So, a general question could be: given two objects, do they belong to the same equivalence class?
$endgroup$
– SCS
Jul 25 at 9:08
3
$begingroup$
Even after your edit, I still don't understand exactly what you're talking about. You write that solving inequalities involves "mak[ing] sure that none of these operations change the equivalance class of any member," but what is the equivalence relation you're talking about when you say this? The phrase "equivalence class" is completely meaningless outside of the context of an equivalence relation. (The title doesn't make sense either, since it says "equivalence relations that are not equality, inequality or boolean truth," but inequality and boolean truth are not equivalence relations.)
$endgroup$
– Tanner Swett
Jul 25 at 17:29
1
$begingroup$
In general terms, what you are asking is about different types of relations between entities, of which equations are a subgroup. Inequalities are not equations, and most of the suggestions below are not equations. Please correct your post accordingly?
$endgroup$
– Namaste
Jul 25 at 17:42
4
$begingroup$
Also, I can't think of how solving an equation can be construed as "finding an object by fact that it is a part of a certain equivalence class". If I solve the equation $x^2 = 4$, I'm finding some objects by the fact that they're elements of a certain set, but that set is not an equivalence class.
$endgroup$
– Tanner Swett
Jul 25 at 17:43
|
show 5 more comments
5
$begingroup$
An equation is a statement of equality. Using it to mean something else is only going to lead to confusion.
$endgroup$
– Peter Taylor
Jul 25 at 7:25
1
$begingroup$
An equation is a statement of equality in terms of an equivalence relation. So, a general question could be: given two objects, do they belong to the same equivalence class?
$endgroup$
– SCS
Jul 25 at 9:08
3
$begingroup$
Even after your edit, I still don't understand exactly what you're talking about. You write that solving inequalities involves "mak[ing] sure that none of these operations change the equivalance class of any member," but what is the equivalence relation you're talking about when you say this? The phrase "equivalence class" is completely meaningless outside of the context of an equivalence relation. (The title doesn't make sense either, since it says "equivalence relations that are not equality, inequality or boolean truth," but inequality and boolean truth are not equivalence relations.)
$endgroup$
– Tanner Swett
Jul 25 at 17:29
1
$begingroup$
In general terms, what you are asking is about different types of relations between entities, of which equations are a subgroup. Inequalities are not equations, and most of the suggestions below are not equations. Please correct your post accordingly?
$endgroup$
– Namaste
Jul 25 at 17:42
4
$begingroup$
Also, I can't think of how solving an equation can be construed as "finding an object by fact that it is a part of a certain equivalence class". If I solve the equation $x^2 = 4$, I'm finding some objects by the fact that they're elements of a certain set, but that set is not an equivalence class.
$endgroup$
– Tanner Swett
Jul 25 at 17:43
5
5
$begingroup$
An equation is a statement of equality. Using it to mean something else is only going to lead to confusion.
$endgroup$
– Peter Taylor
Jul 25 at 7:25
$begingroup$
An equation is a statement of equality. Using it to mean something else is only going to lead to confusion.
$endgroup$
– Peter Taylor
Jul 25 at 7:25
1
1
$begingroup$
An equation is a statement of equality in terms of an equivalence relation. So, a general question could be: given two objects, do they belong to the same equivalence class?
$endgroup$
– SCS
Jul 25 at 9:08
$begingroup$
An equation is a statement of equality in terms of an equivalence relation. So, a general question could be: given two objects, do they belong to the same equivalence class?
$endgroup$
– SCS
Jul 25 at 9:08
3
3
$begingroup$
Even after your edit, I still don't understand exactly what you're talking about. You write that solving inequalities involves "mak[ing] sure that none of these operations change the equivalance class of any member," but what is the equivalence relation you're talking about when you say this? The phrase "equivalence class" is completely meaningless outside of the context of an equivalence relation. (The title doesn't make sense either, since it says "equivalence relations that are not equality, inequality or boolean truth," but inequality and boolean truth are not equivalence relations.)
$endgroup$
– Tanner Swett
Jul 25 at 17:29
$begingroup$
Even after your edit, I still don't understand exactly what you're talking about. You write that solving inequalities involves "mak[ing] sure that none of these operations change the equivalance class of any member," but what is the equivalence relation you're talking about when you say this? The phrase "equivalence class" is completely meaningless outside of the context of an equivalence relation. (The title doesn't make sense either, since it says "equivalence relations that are not equality, inequality or boolean truth," but inequality and boolean truth are not equivalence relations.)
$endgroup$
– Tanner Swett
Jul 25 at 17:29
1
1
$begingroup$
In general terms, what you are asking is about different types of relations between entities, of which equations are a subgroup. Inequalities are not equations, and most of the suggestions below are not equations. Please correct your post accordingly?
$endgroup$
– Namaste
Jul 25 at 17:42
$begingroup$
In general terms, what you are asking is about different types of relations between entities, of which equations are a subgroup. Inequalities are not equations, and most of the suggestions below are not equations. Please correct your post accordingly?
$endgroup$
– Namaste
Jul 25 at 17:42
4
4
$begingroup$
Also, I can't think of how solving an equation can be construed as "finding an object by fact that it is a part of a certain equivalence class". If I solve the equation $x^2 = 4$, I'm finding some objects by the fact that they're elements of a certain set, but that set is not an equivalence class.
$endgroup$
– Tanner Swett
Jul 25 at 17:43
$begingroup$
Also, I can't think of how solving an equation can be construed as "finding an object by fact that it is a part of a certain equivalence class". If I solve the equation $x^2 = 4$, I'm finding some objects by the fact that they're elements of a certain set, but that set is not an equivalence class.
$endgroup$
– Tanner Swett
Jul 25 at 17:43
|
show 5 more comments
3 Answers
3
active
oldest
votes
$begingroup$
Maybe proportions. Granted there's an equality in there but the emphasis is on proportions. You can even generalize the idea to SAT analogies.
Perhaps conversions or dimensional analysis would fit well in the book. Another idea is the piano tuner business case, estimating methods ala Fermi. Note these are not strictly relations. But might fit well into what you are trying to do and would work with the audience.
Maybe also some simple probability stuff with application to gambling and cards and dice and such. Things like 4:1 odds means 20% win chance.
A little bit of finance math nice also. Compound interest. Time value of money to convert between future and present values.
edited Jul 25 at 12:30
Chris Cunningham♦
11k5 gold badges42 silver badges103 bronze badges
answered Jul 25 at 7:07
guest2guest2
362 bronze badges
$endgroup$
$begingroup$
I am confused. In a gambling game, if I am give 4 to 1 odds, doesn't that mean my chances of winning are 25%? (assume zero to the house)
$endgroup$
– JoeTaxpayer
Jul 25 at 11:44
2
$begingroup$
4:1 odds mean your probability is $frac14+1$, not $frac14$. For example even odds are 1:1, which corresponds to a probability of $frac11+1$. Oh -- I see. The answer originally said 80%. I'll edit to 20%. @JoeTaxpayer
$endgroup$
– Chris Cunningham♦
Jul 25 at 12:27
$begingroup$
Ha. Ok, thanks. Now it makes sense. A $4 return on a $1 bet is not “4:1” odds. I think I understood the math but not the way of articulating it correctly. Thanks again.
$endgroup$
– JoeTaxpayer
Jul 25 at 12:34
add a comment |
$begingroup$
Consider the equivalence class of knot diagrams depicting the unknot.
Given a knot diagram, Reidemeister moves do not change the knot type.
So one can apply these moves to a knot diagram because
"none of these operations change the equivalence class."
If you reach the unknot, then you know your original diagram was
just a different drawing of the unknot.
Fig: Dominic Goulding,
"Knot Theory:
The Yang-Baxter Equation, Quantum Groups and
Computation of the Homfly Polynomial," 2010.
answered Jul 25 at 12:48
Joseph O'RourkeJoseph O'Rourke
16.2k3 gold badges35 silver badges86 bronze badges
$endgroup$
add a comment |
$begingroup$
Also, let's not forget isomorphisms in group theory,
nor congruence relations in geometry, and, e.g., similarity in geometry. Further, in geometry, relations between lines might include $overlineAB parallel, overlineCD$, or $overlineEF perp overlineGH$ (two lines being parallel, or two lines being perpendicular, respectively
There are also also congruence equations,$mod n: ;; 3 equiv 10 pmod 7$.
answered Jul 25 at 17:27
NamasteNamaste
6531 gold badge6 silver badges20 bronze badges
$endgroup$
1
$begingroup$
Isomorphisms are probably going to be too advanced for the intended audience.
$endgroup$
– Jessica B
Jul 25 at 20:59
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Maybe proportions. Granted there's an equality in there but the emphasis is on proportions. You can even generalize the idea to SAT analogies.
Perhaps conversions or dimensional analysis would fit well in the book. Another idea is the piano tuner business case, estimating methods ala Fermi. Note these are not strictly relations. But might fit well into what you are trying to do and would work with the audience.
Maybe also some simple probability stuff with application to gambling and cards and dice and such. Things like 4:1 odds means 20% win chance.
A little bit of finance math nice also. Compound interest. Time value of money to convert between future and present values.
edited Jul 25 at 12:30
Chris Cunningham♦
11k5 gold badges42 silver badges103 bronze badges
answered Jul 25 at 7:07
guest2guest2
362 bronze badges
$endgroup$
$begingroup$
I am confused. In a gambling game, if I am give 4 to 1 odds, doesn't that mean my chances of winning are 25%? (assume zero to the house)
$endgroup$
– JoeTaxpayer
Jul 25 at 11:44
2
$begingroup$
4:1 odds mean your probability is $frac14+1$, not $frac14$. For example even odds are 1:1, which corresponds to a probability of $frac11+1$. Oh -- I see. The answer originally said 80%. I'll edit to 20%. @JoeTaxpayer
$endgroup$
– Chris Cunningham♦
Jul 25 at 12:27
$begingroup$
Ha. Ok, thanks. Now it makes sense. A $4 return on a $1 bet is not “4:1” odds. I think I understood the math but not the way of articulating it correctly. Thanks again.
$endgroup$
– JoeTaxpayer
Jul 25 at 12:34
add a comment |
$begingroup$
Maybe proportions. Granted there's an equality in there but the emphasis is on proportions. You can even generalize the idea to SAT analogies.
Perhaps conversions or dimensional analysis would fit well in the book. Another idea is the piano tuner business case, estimating methods ala Fermi. Note these are not strictly relations. But might fit well into what you are trying to do and would work with the audience.
Maybe also some simple probability stuff with application to gambling and cards and dice and such. Things like 4:1 odds means 20% win chance.
A little bit of finance math nice also. Compound interest. Time value of money to convert between future and present values.
edited Jul 25 at 12:30
Chris Cunningham♦
11k5 gold badges42 silver badges103 bronze badges
answered Jul 25 at 7:07
guest2guest2
362 bronze badges
$endgroup$
$begingroup$
I am confused. In a gambling game, if I am give 4 to 1 odds, doesn't that mean my chances of winning are 25%? (assume zero to the house)
$endgroup$
– JoeTaxpayer
Jul 25 at 11:44
2
$begingroup$
4:1 odds mean your probability is $frac14+1$, not $frac14$. For example even odds are 1:1, which corresponds to a probability of $frac11+1$. Oh -- I see. The answer originally said 80%. I'll edit to 20%. @JoeTaxpayer
$endgroup$
– Chris Cunningham♦
Jul 25 at 12:27
$begingroup$
Ha. Ok, thanks. Now it makes sense. A $4 return on a $1 bet is not “4:1” odds. I think I understood the math but not the way of articulating it correctly. Thanks again.
$endgroup$
– JoeTaxpayer
Jul 25 at 12:34
add a comment |
$begingroup$
Maybe proportions. Granted there's an equality in there but the emphasis is on proportions. You can even generalize the idea to SAT analogies.
Perhaps conversions or dimensional analysis would fit well in the book. Another idea is the piano tuner business case, estimating methods ala Fermi. Note these are not strictly relations. But might fit well into what you are trying to do and would work with the audience.
Maybe also some simple probability stuff with application to gambling and cards and dice and such. Things like 4:1 odds means 20% win chance.
A little bit of finance math nice also. Compound interest. Time value of money to convert between future and present values.
edited Jul 25 at 12:30
Chris Cunningham♦
11k5 gold badges42 silver badges103 bronze badges
answered Jul 25 at 7:07
guest2guest2
362 bronze badges
$endgroup$
Maybe proportions. Granted there's an equality in there but the emphasis is on proportions. You can even generalize the idea to SAT analogies.
Perhaps conversions or dimensional analysis would fit well in the book. Another idea is the piano tuner business case, estimating methods ala Fermi. Note these are not strictly relations. But might fit well into what you are trying to do and would work with the audience.
Maybe also some simple probability stuff with application to gambling and cards and dice and such. Things like 4:1 odds means 20% win chance.
A little bit of finance math nice also. Compound interest. Time value of money to convert between future and present values.
edited Jul 25 at 12:30
Chris Cunningham♦
11k5 gold badges42 silver badges103 bronze badges
answered Jul 25 at 7:07
guest2guest2
362 bronze badges
edited Jul 25 at 12:30
Chris Cunningham♦
11k5 gold badges42 silver badges103 bronze badges
edited Jul 25 at 12:30
Chris Cunningham♦
11k5 gold badges42 silver badges103 bronze badges
edited Jul 25 at 12:30
Chris Cunningham♦
11k5 gold badges42 silver badges103 bronze badges
11k5 gold badges42 silver badges103 bronze badges
answered Jul 25 at 7:07
guest2guest2
362 bronze badges
answered Jul 25 at 7:07
guest2guest2
362 bronze badges
answered Jul 25 at 7:07
guest2guest2
362 bronze badges
362 bronze badges
$begingroup$
I am confused. In a gambling game, if I am give 4 to 1 odds, doesn't that mean my chances of winning are 25%? (assume zero to the house)
$endgroup$
– JoeTaxpayer
Jul 25 at 11:44
2
$begingroup$
4:1 odds mean your probability is $frac14+1$, not $frac14$. For example even odds are 1:1, which corresponds to a probability of $frac11+1$. Oh -- I see. The answer originally said 80%. I'll edit to 20%. @JoeTaxpayer
$endgroup$
– Chris Cunningham♦
Jul 25 at 12:27
$begingroup$
Ha. Ok, thanks. Now it makes sense. A $4 return on a $1 bet is not “4:1” odds. I think I understood the math but not the way of articulating it correctly. Thanks again.
$endgroup$
– JoeTaxpayer
Jul 25 at 12:34
add a comment |
$begingroup$
I am confused. In a gambling game, if I am give 4 to 1 odds, doesn't that mean my chances of winning are 25%? (assume zero to the house)
$endgroup$
– JoeTaxpayer
Jul 25 at 11:44
2
$begingroup$
4:1 odds mean your probability is $frac14+1$, not $frac14$. For example even odds are 1:1, which corresponds to a probability of $frac11+1$. Oh -- I see. The answer originally said 80%. I'll edit to 20%. @JoeTaxpayer
$endgroup$
– Chris Cunningham♦
Jul 25 at 12:27
$begingroup$
Ha. Ok, thanks. Now it makes sense. A $4 return on a $1 bet is not “4:1” odds. I think I understood the math but not the way of articulating it correctly. Thanks again.
$endgroup$
– JoeTaxpayer
Jul 25 at 12:34
$begingroup$
I am confused. In a gambling game, if I am give 4 to 1 odds, doesn't that mean my chances of winning are 25%? (assume zero to the house)
$endgroup$
– JoeTaxpayer
Jul 25 at 11:44
$begingroup$
I am confused. In a gambling game, if I am give 4 to 1 odds, doesn't that mean my chances of winning are 25%? (assume zero to the house)
$endgroup$
– JoeTaxpayer
Jul 25 at 11:44
2
2
$begingroup$
4:1 odds mean your probability is $frac14+1$, not $frac14$. For example even odds are 1:1, which corresponds to a probability of $frac11+1$. Oh -- I see. The answer originally said 80%. I'll edit to 20%. @JoeTaxpayer
$endgroup$
– Chris Cunningham♦
Jul 25 at 12:27
$begingroup$
4:1 odds mean your probability is $frac14+1$, not $frac14$. For example even odds are 1:1, which corresponds to a probability of $frac11+1$. Oh -- I see. The answer originally said 80%. I'll edit to 20%. @JoeTaxpayer
$endgroup$
– Chris Cunningham♦
Jul 25 at 12:27
$begingroup$
Ha. Ok, thanks. Now it makes sense. A $4 return on a $1 bet is not “4:1” odds. I think I understood the math but not the way of articulating it correctly. Thanks again.
$endgroup$
– JoeTaxpayer
Jul 25 at 12:34
$begingroup$
Ha. Ok, thanks. Now it makes sense. A $4 return on a $1 bet is not “4:1” odds. I think I understood the math but not the way of articulating it correctly. Thanks again.
$endgroup$
– JoeTaxpayer
Jul 25 at 12:34
add a comment |
$begingroup$
Consider the equivalence class of knot diagrams depicting the unknot.
Given a knot diagram, Reidemeister moves do not change the knot type.
So one can apply these moves to a knot diagram because
"none of these operations change the equivalence class."
If you reach the unknot, then you know your original diagram was
just a different drawing of the unknot.
Fig: Dominic Goulding,
"Knot Theory:
The Yang-Baxter Equation, Quantum Groups and
Computation of the Homfly Polynomial," 2010.
answered Jul 25 at 12:48
Joseph O'RourkeJoseph O'Rourke
16.2k3 gold badges35 silver badges86 bronze badges
$endgroup$
add a comment |
$begingroup$
Consider the equivalence class of knot diagrams depicting the unknot.
Given a knot diagram, Reidemeister moves do not change the knot type.
So one can apply these moves to a knot diagram because
"none of these operations change the equivalence class."
If you reach the unknot, then you know your original diagram was
just a different drawing of the unknot.
Fig: Dominic Goulding,
"Knot Theory:
The Yang-Baxter Equation, Quantum Groups and
Computation of the Homfly Polynomial," 2010.
answered Jul 25 at 12:48
Joseph O'RourkeJoseph O'Rourke
16.2k3 gold badges35 silver badges86 bronze badges
$endgroup$
add a comment |
$begingroup$
Consider the equivalence class of knot diagrams depicting the unknot.
Given a knot diagram, Reidemeister moves do not change the knot type.
So one can apply these moves to a knot diagram because
"none of these operations change the equivalence class."
If you reach the unknot, then you know your original diagram was
just a different drawing of the unknot.
Fig: Dominic Goulding,
"Knot Theory:
The Yang-Baxter Equation, Quantum Groups and
Computation of the Homfly Polynomial," 2010.
answered Jul 25 at 12:48
Joseph O'RourkeJoseph O'Rourke
16.2k3 gold badges35 silver badges86 bronze badges
$endgroup$
Consider the equivalence class of knot diagrams depicting the unknot.
Given a knot diagram, Reidemeister moves do not change the knot type.
So one can apply these moves to a knot diagram because
"none of these operations change the equivalence class."
If you reach the unknot, then you know your original diagram was
just a different drawing of the unknot.
Fig: Dominic Goulding,
"Knot Theory:
The Yang-Baxter Equation, Quantum Groups and
Computation of the Homfly Polynomial," 2010.
answered Jul 25 at 12:48
Joseph O'RourkeJoseph O'Rourke
16.2k3 gold badges35 silver badges86 bronze badges
answered Jul 25 at 12:48
Joseph O'RourkeJoseph O'Rourke
16.2k3 gold badges35 silver badges86 bronze badges
answered Jul 25 at 12:48
Joseph O'RourkeJoseph O'Rourke
16.2k3 gold badges35 silver badges86 bronze badges
answered Jul 25 at 12:48
Joseph O'RourkeJoseph O'Rourke
16.2k3 gold badges35 silver badges86 bronze badges
16.2k3 gold badges35 silver badges86 bronze badges
add a comment |
add a comment |
$begingroup$
Also, let's not forget isomorphisms in group theory,
nor congruence relations in geometry, and, e.g., similarity in geometry. Further, in geometry, relations between lines might include $overlineAB parallel, overlineCD$, or $overlineEF perp overlineGH$ (two lines being parallel, or two lines being perpendicular, respectively
There are also also congruence equations,$mod n: ;; 3 equiv 10 pmod 7$.
answered Jul 25 at 17:27
NamasteNamaste
6531 gold badge6 silver badges20 bronze badges
$endgroup$
1
$begingroup$
Isomorphisms are probably going to be too advanced for the intended audience.
$endgroup$
– Jessica B
Jul 25 at 20:59
add a comment |
$begingroup$
Also, let's not forget isomorphisms in group theory,
nor congruence relations in geometry, and, e.g., similarity in geometry. Further, in geometry, relations between lines might include $overlineAB parallel, overlineCD$, or $overlineEF perp overlineGH$ (two lines being parallel, or two lines being perpendicular, respectively
There are also also congruence equations,$mod n: ;; 3 equiv 10 pmod 7$.
answered Jul 25 at 17:27
NamasteNamaste
6531 gold badge6 silver badges20 bronze badges
$endgroup$
1
$begingroup$
Isomorphisms are probably going to be too advanced for the intended audience.
$endgroup$
– Jessica B
Jul 25 at 20:59
add a comment |
$begingroup$
Also, let's not forget isomorphisms in group theory,
nor congruence relations in geometry, and, e.g., similarity in geometry. Further, in geometry, relations between lines might include $overlineAB parallel, overlineCD$, or $overlineEF perp overlineGH$ (two lines being parallel, or two lines being perpendicular, respectively
There are also also congruence equations,$mod n: ;; 3 equiv 10 pmod 7$.
answered Jul 25 at 17:27
NamasteNamaste
6531 gold badge6 silver badges20 bronze badges
$endgroup$
Also, let's not forget isomorphisms in group theory,
nor congruence relations in geometry, and, e.g., similarity in geometry. Further, in geometry, relations between lines might include $overlineAB parallel, overlineCD$, or $overlineEF perp overlineGH$ (two lines being parallel, or two lines being perpendicular, respectively
There are also also congruence equations,$mod n: ;; 3 equiv 10 pmod 7$.
answered Jul 25 at 17:27
NamasteNamaste
6531 gold badge6 silver badges20 bronze badges
edited Jul 25 at 17:36
answered Jul 25 at 17:27
NamasteNamaste
6531 gold badge6 silver badges20 bronze badges
answered Jul 25 at 17:27
NamasteNamaste
6531 gold badge6 silver badges20 bronze badges
answered Jul 25 at 17:27
NamasteNamaste
6531 gold badge6 silver badges20 bronze badges
6531 gold badge6 silver badges20 bronze badges
1
$begingroup$
Isomorphisms are probably going to be too advanced for the intended audience.
$endgroup$
– Jessica B
Jul 25 at 20:59
add a comment |
1
$begingroup$
Isomorphisms are probably going to be too advanced for the intended audience.
$endgroup$
– Jessica B
Jul 25 at 20:59
1
1
$begingroup$
Isomorphisms are probably going to be too advanced for the intended audience.
$endgroup$
– Jessica B
Jul 25 at 20:59
$begingroup$
Isomorphisms are probably going to be too advanced for the intended audience.
$endgroup$
– Jessica B
Jul 25 at 20:59
add a comment |
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$begingroup$
An equation is a statement of equality. Using it to mean something else is only going to lead to confusion.
$endgroup$
– Peter Taylor
Jul 25 at 7:25
1
$begingroup$
An equation is a statement of equality in terms of an equivalence relation. So, a general question could be: given two objects, do they belong to the same equivalence class?
$endgroup$
– SCS
Jul 25 at 9:08
3
$begingroup$
Even after your edit, I still don't understand exactly what you're talking about. You write that solving inequalities involves "mak[ing] sure that none of these operations change the equivalance class of any member," but what is the equivalence relation you're talking about when you say this? The phrase "equivalence class" is completely meaningless outside of the context of an equivalence relation. (The title doesn't make sense either, since it says "equivalence relations that are not equality, inequality or boolean truth," but inequality and boolean truth are not equivalence relations.)
$endgroup$
– Tanner Swett
Jul 25 at 17:29
1
$begingroup$
In general terms, what you are asking is about different types of relations between entities, of which equations are a subgroup. Inequalities are not equations, and most of the suggestions below are not equations. Please correct your post accordingly?
$endgroup$
– Namaste
Jul 25 at 17:42
4
$begingroup$
Also, I can't think of how solving an equation can be construed as "finding an object by fact that it is a part of a certain equivalence class". If I solve the equation $x^2 = 4$, I'm finding some objects by the fact that they're elements of a certain set, but that set is not an equivalence class.
$endgroup$
– Tanner Swett
Jul 25 at 17:43