Points distinguishable by set of functions
$begingroup$
Let $F$ be the set of monotone functions from $[0, 1]^d$ to $mathbb{R}^+$, with $d > 1$.
There is a well-known result [1] according to which no total order on $[0, 1]^d$, $d > 1$, can be represented by a function in $F$.
This means that, no matter what function $f in F$ I choose, there will always be two points $P$ and $Q$ in $[0, 1]^d$ that are indistinguishable by $f$, i.e., $f(P)=f(Q)$.
Example: $f(x,y)=x+y$, $P=(1,0)$ and $Q=(0,1)$.
My question: what if I consider a set of functions $F'subseteq F$. Are there known conditions under which I reach distinguishability of all points in $[0, 1]^d$ (i.e., any two points in $[0, 1]^d$ are distinguishable by at least one function in $F'$)?
I guess, e.g., that having less than $d$ functions implies indistinguishability, for the same reasons as the mentioned result (although proving it is a different story). I wonder, though, if there are any known necessary and sufficient conditions for (in)distinguishability. Any references and/or naming conventions perhaps different from what I have used here would be highly appreciated.
(I include the vector-spaces tag because this notion reminds me of linear (in)dependence in vector-spaces.)
[1] J. C. Candeal and E. Indurain. Utility functions on chains. Journal of Mathematical Economics, 22(2):161 – 168, 1993
functions vector-spaces
asked Dec 3 '18 at 12:42
MaiauxMaiaux
1135
$endgroup$
add a comment |
$begingroup$
Let $F$ be the set of monotone functions from $[0, 1]^d$ to $mathbb{R}^+$, with $d > 1$.
There is a well-known result [1] according to which no total order on $[0, 1]^d$, $d > 1$, can be represented by a function in $F$.
This means that, no matter what function $f in F$ I choose, there will always be two points $P$ and $Q$ in $[0, 1]^d$ that are indistinguishable by $f$, i.e., $f(P)=f(Q)$.
Example: $f(x,y)=x+y$, $P=(1,0)$ and $Q=(0,1)$.
My question: what if I consider a set of functions $F'subseteq F$. Are there known conditions under which I reach distinguishability of all points in $[0, 1]^d$ (i.e., any two points in $[0, 1]^d$ are distinguishable by at least one function in $F'$)?
I guess, e.g., that having less than $d$ functions implies indistinguishability, for the same reasons as the mentioned result (although proving it is a different story). I wonder, though, if there are any known necessary and sufficient conditions for (in)distinguishability. Any references and/or naming conventions perhaps different from what I have used here would be highly appreciated.
(I include the vector-spaces tag because this notion reminds me of linear (in)dependence in vector-spaces.)
[1] J. C. Candeal and E. Indurain. Utility functions on chains. Journal of Mathematical Economics, 22(2):161 – 168, 1993
functions vector-spaces
asked Dec 3 '18 at 12:42
MaiauxMaiaux
1135
$endgroup$
add a comment |
$begingroup$
Let $F$ be the set of monotone functions from $[0, 1]^d$ to $mathbb{R}^+$, with $d > 1$.
There is a well-known result [1] according to which no total order on $[0, 1]^d$, $d > 1$, can be represented by a function in $F$.
This means that, no matter what function $f in F$ I choose, there will always be two points $P$ and $Q$ in $[0, 1]^d$ that are indistinguishable by $f$, i.e., $f(P)=f(Q)$.
Example: $f(x,y)=x+y$, $P=(1,0)$ and $Q=(0,1)$.
My question: what if I consider a set of functions $F'subseteq F$. Are there known conditions under which I reach distinguishability of all points in $[0, 1]^d$ (i.e., any two points in $[0, 1]^d$ are distinguishable by at least one function in $F'$)?
I guess, e.g., that having less than $d$ functions implies indistinguishability, for the same reasons as the mentioned result (although proving it is a different story). I wonder, though, if there are any known necessary and sufficient conditions for (in)distinguishability. Any references and/or naming conventions perhaps different from what I have used here would be highly appreciated.
(I include the vector-spaces tag because this notion reminds me of linear (in)dependence in vector-spaces.)
[1] J. C. Candeal and E. Indurain. Utility functions on chains. Journal of Mathematical Economics, 22(2):161 – 168, 1993
functions vector-spaces
asked Dec 3 '18 at 12:42
MaiauxMaiaux
1135
$endgroup$
Let $F$ be the set of monotone functions from $[0, 1]^d$ to $mathbb{R}^+$, with $d > 1$.
There is a well-known result [1] according to which no total order on $[0, 1]^d$, $d > 1$, can be represented by a function in $F$.
This means that, no matter what function $f in F$ I choose, there will always be two points $P$ and $Q$ in $[0, 1]^d$ that are indistinguishable by $f$, i.e., $f(P)=f(Q)$.
Example: $f(x,y)=x+y$, $P=(1,0)$ and $Q=(0,1)$.
My question: what if I consider a set of functions $F'subseteq F$. Are there known conditions under which I reach distinguishability of all points in $[0, 1]^d$ (i.e., any two points in $[0, 1]^d$ are distinguishable by at least one function in $F'$)?
I guess, e.g., that having less than $d$ functions implies indistinguishability, for the same reasons as the mentioned result (although proving it is a different story). I wonder, though, if there are any known necessary and sufficient conditions for (in)distinguishability. Any references and/or naming conventions perhaps different from what I have used here would be highly appreciated.
(I include the vector-spaces tag because this notion reminds me of linear (in)dependence in vector-spaces.)
[1] J. C. Candeal and E. Indurain. Utility functions on chains. Journal of Mathematical Economics, 22(2):161 – 168, 1993
functions vector-spaces
functions vector-spaces
asked Dec 3 '18 at 12:42
MaiauxMaiaux
1135
asked Dec 3 '18 at 12:42
MaiauxMaiaux
1135
asked Dec 3 '18 at 12:42
MaiauxMaiaux
1135
asked Dec 3 '18 at 12:42
MaiauxMaiaux
1135
asked Dec 3 '18 at 12:42
MaiauxMaiaux
1135
1135
add a comment |
add a comment |
0
active
oldest
votes
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%2f3024008%2fpoints-distinguishable-by-set-of-functions%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
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%2f3024008%2fpoints-distinguishable-by-set-of-functions%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
