Mixing
Bounds of defining operations for equivalence classes
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 '18 at 15:29
bblohowiak
918
add a comment |
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 '18 at 15:29
bblohowiak
918
add a comment |
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 '18 at 15:29
bblohowiak
918
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 '18 at 15:29
bblohowiak
918
asked Nov 20 '18 at 15:29
bblohowiak
918
edited Nov 24 '18 at 18:32
asked Nov 20 '18 at 15:29
bblohowiak
918
asked Nov 20 '18 at 15:29
bblohowiak
918
asked Nov 20 '18 at 15:29
bblohowiak
918
918
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
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 '18 at 18:44
Ethan Bolker
41.6k547110
add a comment |
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: "69"
};
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: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
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
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3006458%2fbounds-of-defining-operations-for-equivalence-classes%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
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 '18 at 18:44
Ethan Bolker
41.6k547110
add a comment |
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 '18 at 18:44
Ethan Bolker
41.6k547110
add a comment |
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 '18 at 18:44
Ethan Bolker
41.6k547110
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 '18 at 18:44
Ethan Bolker
41.6k547110
answered Nov 24 '18 at 18:44
Ethan Bolker
41.6k547110
answered Nov 24 '18 at 18:44
Ethan Bolker
41.6k547110
answered Nov 24 '18 at 18:44
Ethan Bolker
41.6k547110
41.6k547110
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics 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.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3006458%2fbounds-of-defining-operations-for-equivalence-classes%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
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
