Function/Operation Definitions
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If I want to define a function/operation for equivalence classes, is it permissible to stipulate contingencies based upon element characteristics or does dependency on the choice of element characteristics invalidate the definition? A possible example:
if b > a, then (a,b)+(c,d) = ab+cd
and
if b < a, then (a,b)+(c,d) = abc+d, etc.
functions elementary-set-theory binary-operations
asked Nov 15 at 17:15
bblohowiak
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show 1 more comment
up vote
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down vote
favorite
If I want to define a function/operation for equivalence classes, is it permissible to stipulate contingencies based upon element characteristics or does dependency on the choice of element characteristics invalidate the definition? A possible example:
if b > a, then (a,b)+(c,d) = ab+cd
and
if b < a, then (a,b)+(c,d) = abc+d, etc.
functions elementary-set-theory binary-operations
asked Nov 15 at 17:15
bblohowiak
517
1
No, this will (in general) not yield a well-defined function. Also: The first sentence of your post makes my head hurt. That kind of convoluted language isn't in anybody's best interest when communicating about mathematics...
– Stefan Mesken
Nov 15 at 17:38
Your possible example has nothing do to with equivalence realations.
– William Elliot
Nov 16 at 4:25
An example of equivalence classes based upon element characteristics is sets of things with the same color, or classes of sets with the same cardinality.
– William Elliot
Nov 16 at 4:30
1
Your example is completely valid, it's an example of definition by cases (though the $+$ function here is undefined when $a=b$). However this works because this is not about equivalence classes. I'm not sure what you are trying to do with equivalence classes, depending on it you may or not be able to define functions for them.
– Ryunaq
Nov 16 at 15:37
@Ryunaq I'm approaching this backwards because I have been active in coding in imperative languages and have recently been studying functional programming. As I have focused more on the basic construction of integers and rationals, I have wondered about how operations with equivalence classes in those instances have been defined and to what extent other options are possible. I want to know what would determine the boundaries of possibility in this context, what things are off-limits in math that could be easy to do and not throw errors when done in code.
– bblohowiak
Nov 16 at 18:49
|
show 1 more comment
up vote
0
down vote
favorite
up vote
0
down vote
favorite
If I want to define a function/operation for equivalence classes, is it permissible to stipulate contingencies based upon element characteristics or does dependency on the choice of element characteristics invalidate the definition? A possible example:
if b > a, then (a,b)+(c,d) = ab+cd
and
if b < a, then (a,b)+(c,d) = abc+d, etc.
functions elementary-set-theory binary-operations
asked Nov 15 at 17:15
bblohowiak
517
If I want to define a function/operation for equivalence classes, is it permissible to stipulate contingencies based upon element characteristics or does dependency on the choice of element characteristics invalidate the definition? A possible example:
if b > a, then (a,b)+(c,d) = ab+cd
and
if b < a, then (a,b)+(c,d) = abc+d, etc.
functions elementary-set-theory binary-operations
functions elementary-set-theory binary-operations
asked Nov 15 at 17:15
bblohowiak
517
asked Nov 15 at 17:15
bblohowiak
517
asked Nov 15 at 17:15
bblohowiak
517
asked Nov 15 at 17:15
bblohowiak
517
asked Nov 15 at 17:15
bblohowiak
517
517
1
No, this will (in general) not yield a well-defined function. Also: The first sentence of your post makes my head hurt. That kind of convoluted language isn't in anybody's best interest when communicating about mathematics...
– Stefan Mesken
Nov 15 at 17:38
Your possible example has nothing do to with equivalence realations.
– William Elliot
Nov 16 at 4:25
An example of equivalence classes based upon element characteristics is sets of things with the same color, or classes of sets with the same cardinality.
– William Elliot
Nov 16 at 4:30
1
Your example is completely valid, it's an example of definition by cases (though the $+$ function here is undefined when $a=b$). However this works because this is not about equivalence classes. I'm not sure what you are trying to do with equivalence classes, depending on it you may or not be able to define functions for them.
– Ryunaq
Nov 16 at 15:37
@Ryunaq I'm approaching this backwards because I have been active in coding in imperative languages and have recently been studying functional programming. As I have focused more on the basic construction of integers and rationals, I have wondered about how operations with equivalence classes in those instances have been defined and to what extent other options are possible. I want to know what would determine the boundaries of possibility in this context, what things are off-limits in math that could be easy to do and not throw errors when done in code.
– bblohowiak
Nov 16 at 18:49
|
show 1 more comment
1
No, this will (in general) not yield a well-defined function. Also: The first sentence of your post makes my head hurt. That kind of convoluted language isn't in anybody's best interest when communicating about mathematics...
– Stefan Mesken
Nov 15 at 17:38
Your possible example has nothing do to with equivalence realations.
– William Elliot
Nov 16 at 4:25
An example of equivalence classes based upon element characteristics is sets of things with the same color, or classes of sets with the same cardinality.
– William Elliot
Nov 16 at 4:30
1
Your example is completely valid, it's an example of definition by cases (though the $+$ function here is undefined when $a=b$). However this works because this is not about equivalence classes. I'm not sure what you are trying to do with equivalence classes, depending on it you may or not be able to define functions for them.
– Ryunaq
Nov 16 at 15:37
@Ryunaq I'm approaching this backwards because I have been active in coding in imperative languages and have recently been studying functional programming. As I have focused more on the basic construction of integers and rationals, I have wondered about how operations with equivalence classes in those instances have been defined and to what extent other options are possible. I want to know what would determine the boundaries of possibility in this context, what things are off-limits in math that could be easy to do and not throw errors when done in code.
– bblohowiak
Nov 16 at 18:49
1
1
No, this will (in general) not yield a well-defined function. Also: The first sentence of your post makes my head hurt. That kind of convoluted language isn't in anybody's best interest when communicating about mathematics...
– Stefan Mesken
Nov 15 at 17:38
No, this will (in general) not yield a well-defined function. Also: The first sentence of your post makes my head hurt. That kind of convoluted language isn't in anybody's best interest when communicating about mathematics...
– Stefan Mesken
Nov 15 at 17:38
Your possible example has nothing do to with equivalence realations.
– William Elliot
Nov 16 at 4:25
Your possible example has nothing do to with equivalence realations.
– William Elliot
Nov 16 at 4:25
An example of equivalence classes based upon element characteristics is sets of things with the same color, or classes of sets with the same cardinality.
– William Elliot
Nov 16 at 4:30
An example of equivalence classes based upon element characteristics is sets of things with the same color, or classes of sets with the same cardinality.
– William Elliot
Nov 16 at 4:30
1
1
Your example is completely valid, it's an example of definition by cases (though the $+$ function here is undefined when $a=b$). However this works because this is not about equivalence classes. I'm not sure what you are trying to do with equivalence classes, depending on it you may or not be able to define functions for them.
– Ryunaq
Nov 16 at 15:37
Your example is completely valid, it's an example of definition by cases (though the $+$ function here is undefined when $a=b$). However this works because this is not about equivalence classes. I'm not sure what you are trying to do with equivalence classes, depending on it you may or not be able to define functions for them.
– Ryunaq
Nov 16 at 15:37
@Ryunaq I'm approaching this backwards because I have been active in coding in imperative languages and have recently been studying functional programming. As I have focused more on the basic construction of integers and rationals, I have wondered about how operations with equivalence classes in those instances have been defined and to what extent other options are possible. I want to know what would determine the boundaries of possibility in this context, what things are off-limits in math that could be easy to do and not throw errors when done in code.
– bblohowiak
Nov 16 at 18:49
@Ryunaq I'm approaching this backwards because I have been active in coding in imperative languages and have recently been studying functional programming. As I have focused more on the basic construction of integers and rationals, I have wondered about how operations with equivalence classes in those instances have been defined and to what extent other options are possible. I want to know what would determine the boundaries of possibility in this context, what things are off-limits in math that could be easy to do and not throw errors when done in code.
– bblohowiak
Nov 16 at 18:49
|
show 1 more comment
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No, this will (in general) not yield a well-defined function. Also: The first sentence of your post makes my head hurt. That kind of convoluted language isn't in anybody's best interest when communicating about mathematics...
– Stefan Mesken
Nov 15 at 17:38
Your possible example has nothing do to with equivalence realations.
– William Elliot
Nov 16 at 4:25
An example of equivalence classes based upon element characteristics is sets of things with the same color, or classes of sets with the same cardinality.
– William Elliot
Nov 16 at 4:30
1
Your example is completely valid, it's an example of definition by cases (though the $+$ function here is undefined when $a=b$). However this works because this is not about equivalence classes. I'm not sure what you are trying to do with equivalence classes, depending on it you may or not be able to define functions for them.
– Ryunaq
Nov 16 at 15:37
@Ryunaq I'm approaching this backwards because I have been active in coding in imperative languages and have recently been studying functional programming. As I have focused more on the basic construction of integers and rationals, I have wondered about how operations with equivalence classes in those instances have been defined and to what extent other options are possible. I want to know what would determine the boundaries of possibility in this context, what things are off-limits in math that could be easy to do and not throw errors when done in code.
– bblohowiak
Nov 16 at 18:49