Bounds of defining operations for equivalence classes
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When operations for number systems are defined in terms of representatives of equivalence classes, can those operations meet the criteria for being well defined if the definition includes specific values of the number system from which a system is constructed?
As an example, if I want to define addition for a number system constructed from nonnegative reals:
(a,b) + (c,d) = [((a+c)/2), ((b+d)/3)]
Is it permissible to include digits that already belong to the reals? The operation of division? What if I wanted to use a signum function?
I recognize that there may be prohibitions against such things when it comes to operation definitions; if they are off-limits, I would appreciate direction to resources or explanations that indicate the basis for such prohibitions.
elementary-set-theory definition binary-operations
asked Nov 20 at 15:29
bblohowiak
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When operations for number systems are defined in terms of representatives of equivalence classes, can those operations meet the criteria for being well defined if the definition includes specific values of the number system from which a system is constructed?
As an example, if I want to define addition for a number system constructed from nonnegative reals:
(a,b) + (c,d) = [((a+c)/2), ((b+d)/3)]
Is it permissible to include digits that already belong to the reals? The operation of division? What if I wanted to use a signum function?
I recognize that there may be prohibitions against such things when it comes to operation definitions; if they are off-limits, I would appreciate direction to resources or explanations that indicate the basis for such prohibitions.
elementary-set-theory definition binary-operations
asked Nov 20 at 15:29
bblohowiak
808
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
When operations for number systems are defined in terms of representatives of equivalence classes, can those operations meet the criteria for being well defined if the definition includes specific values of the number system from which a system is constructed?
As an example, if I want to define addition for a number system constructed from nonnegative reals:
(a,b) + (c,d) = [((a+c)/2), ((b+d)/3)]
Is it permissible to include digits that already belong to the reals? The operation of division? What if I wanted to use a signum function?
I recognize that there may be prohibitions against such things when it comes to operation definitions; if they are off-limits, I would appreciate direction to resources or explanations that indicate the basis for such prohibitions.
elementary-set-theory definition binary-operations
asked Nov 20 at 15:29
bblohowiak
808
When operations for number systems are defined in terms of representatives of equivalence classes, can those operations meet the criteria for being well defined if the definition includes specific values of the number system from which a system is constructed?
As an example, if I want to define addition for a number system constructed from nonnegative reals:
(a,b) + (c,d) = [((a+c)/2), ((b+d)/3)]
Is it permissible to include digits that already belong to the reals? The operation of division? What if I wanted to use a signum function?
I recognize that there may be prohibitions against such things when it comes to operation definitions; if they are off-limits, I would appreciate direction to resources or explanations that indicate the basis for such prohibitions.
elementary-set-theory definition binary-operations
elementary-set-theory definition binary-operations
asked Nov 20 at 15:29
bblohowiak
808
asked Nov 20 at 15:29
bblohowiak
808
edited Nov 24 at 18:32
asked Nov 20 at 15:29
bblohowiak
808
asked Nov 20 at 15:29
bblohowiak
808
asked Nov 20 at 15:29
bblohowiak
808
808
add a comment |
add a comment |
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I am not sure I understand your question, but I will try to answer what I think you are asking.
The operation you have defined that maps two ordered pairs of real numbers (not necessarily positive) to another pair of real numbers is well defined. That fact that it mentions $2$ and $3$ is not a problem.
I see nothing in your example that addresses representatives of equivalence classes.
I hope you chose that just as an example. It's probably not particularly nice, or useful.
answered Nov 24 at 18:44
Ethan Bolker
40.6k545107
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1 Answer
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1 Answer
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active
oldest
votes
active
oldest
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active
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up vote
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accepted
I am not sure I understand your question, but I will try to answer what I think you are asking.
The operation you have defined that maps two ordered pairs of real numbers (not necessarily positive) to another pair of real numbers is well defined. That fact that it mentions $2$ and $3$ is not a problem.
I see nothing in your example that addresses representatives of equivalence classes.
I hope you chose that just as an example. It's probably not particularly nice, or useful.
answered Nov 24 at 18:44
Ethan Bolker
40.6k545107
add a comment |
up vote
1
down vote
accepted
I am not sure I understand your question, but I will try to answer what I think you are asking.
The operation you have defined that maps two ordered pairs of real numbers (not necessarily positive) to another pair of real numbers is well defined. That fact that it mentions $2$ and $3$ is not a problem.
I see nothing in your example that addresses representatives of equivalence classes.
I hope you chose that just as an example. It's probably not particularly nice, or useful.
answered Nov 24 at 18:44
Ethan Bolker
40.6k545107
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
I am not sure I understand your question, but I will try to answer what I think you are asking.
The operation you have defined that maps two ordered pairs of real numbers (not necessarily positive) to another pair of real numbers is well defined. That fact that it mentions $2$ and $3$ is not a problem.
I see nothing in your example that addresses representatives of equivalence classes.
I hope you chose that just as an example. It's probably not particularly nice, or useful.
answered Nov 24 at 18:44
Ethan Bolker
40.6k545107
I am not sure I understand your question, but I will try to answer what I think you are asking.
The operation you have defined that maps two ordered pairs of real numbers (not necessarily positive) to another pair of real numbers is well defined. That fact that it mentions $2$ and $3$ is not a problem.
I see nothing in your example that addresses representatives of equivalence classes.
I hope you chose that just as an example. It's probably not particularly nice, or useful.
answered Nov 24 at 18:44
Ethan Bolker
40.6k545107
answered Nov 24 at 18:44
Ethan Bolker
40.6k545107
answered Nov 24 at 18:44
Ethan Bolker
40.6k545107
answered Nov 24 at 18:44
Ethan Bolker
40.6k545107
40.6k545107
add a comment |
add a comment |
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