In the topology,Let $X={1,2,3}$ and let $d:X times X to[0,+ ∞)$ be a function defined by formulas
$begingroup$
Let $X={1,2,3}$ and let $d:X times X to[0,+infty)$ be a function defined by formulas:
$$
d(1,1)=d(2,2)=d(3,3)=0,\
d(1,2)=d(2,1)=d(1,3)=d(3,1)=1,\
d(2,3)=d(3,2)=3.$$
Check if $d$ is a distance function on $X$.
general-topology
edited Dec 13 '18 at 22:33
Tianlalu
3,08421138
asked Dec 13 '18 at 18:08
arbadearbade
14
$endgroup$
add a comment |
$begingroup$
Let $X={1,2,3}$ and let $d:X times X to[0,+infty)$ be a function defined by formulas:
$$
d(1,1)=d(2,2)=d(3,3)=0,\
d(1,2)=d(2,1)=d(1,3)=d(3,1)=1,\
d(2,3)=d(3,2)=3.$$
Check if $d$ is a distance function on $X$.
general-topology
edited Dec 13 '18 at 22:33
Tianlalu
3,08421138
asked Dec 13 '18 at 18:08
arbadearbade
14
$endgroup$
$begingroup$
What have you tried so far? Do you remember the definition of distance functions?
$endgroup$
– Mindlack
Dec 13 '18 at 18:11
$begingroup$
Hello and welcome to Math.SE! In this community, when posting a question, you are encouraged to include in the body of the question some of your thoughts about the problem, some context about it. Check this link for some information about posting good questions.
$endgroup$
– Edmundo Martins
Dec 13 '18 at 18:16
$begingroup$
Yes I have 3 or 4 rules for this questions, but I can not be sure....Could you help me and explain the questions and answer ?
$endgroup$
– arbade
Dec 13 '18 at 18:16
add a comment |
$begingroup$
Let $X={1,2,3}$ and let $d:X times X to[0,+infty)$ be a function defined by formulas:
$$
d(1,1)=d(2,2)=d(3,3)=0,\
d(1,2)=d(2,1)=d(1,3)=d(3,1)=1,\
d(2,3)=d(3,2)=3.$$
Check if $d$ is a distance function on $X$.
general-topology
edited Dec 13 '18 at 22:33
Tianlalu
3,08421138
asked Dec 13 '18 at 18:08
arbadearbade
14
$endgroup$
Let $X={1,2,3}$ and let $d:X times X to[0,+infty)$ be a function defined by formulas:
$$
d(1,1)=d(2,2)=d(3,3)=0,\
d(1,2)=d(2,1)=d(1,3)=d(3,1)=1,\
d(2,3)=d(3,2)=3.$$
Check if $d$ is a distance function on $X$.
general-topology
general-topology
edited Dec 13 '18 at 22:33
Tianlalu
3,08421138
asked Dec 13 '18 at 18:08
arbadearbade
14
edited Dec 13 '18 at 22:33
Tianlalu
3,08421138
asked Dec 13 '18 at 18:08
arbadearbade
14
edited Dec 13 '18 at 22:33
Tianlalu
3,08421138
edited Dec 13 '18 at 22:33
Tianlalu
3,08421138
edited Dec 13 '18 at 22:33
Tianlalu
3,08421138
3,08421138
asked Dec 13 '18 at 18:08
arbadearbade
14
asked Dec 13 '18 at 18:08
arbadearbade
14
asked Dec 13 '18 at 18:08
arbadearbade
14
14
$begingroup$
What have you tried so far? Do you remember the definition of distance functions?
$endgroup$
– Mindlack
Dec 13 '18 at 18:11
$begingroup$
Hello and welcome to Math.SE! In this community, when posting a question, you are encouraged to include in the body of the question some of your thoughts about the problem, some context about it. Check this link for some information about posting good questions.
$endgroup$
– Edmundo Martins
Dec 13 '18 at 18:16
$begingroup$
Yes I have 3 or 4 rules for this questions, but I can not be sure....Could you help me and explain the questions and answer ?
$endgroup$
– arbade
Dec 13 '18 at 18:16
add a comment |
$begingroup$
What have you tried so far? Do you remember the definition of distance functions?
$endgroup$
– Mindlack
Dec 13 '18 at 18:11
$begingroup$
Hello and welcome to Math.SE! In this community, when posting a question, you are encouraged to include in the body of the question some of your thoughts about the problem, some context about it. Check this link for some information about posting good questions.
$endgroup$
– Edmundo Martins
Dec 13 '18 at 18:16
$begingroup$
Yes I have 3 or 4 rules for this questions, but I can not be sure....Could you help me and explain the questions and answer ?
$endgroup$
– arbade
Dec 13 '18 at 18:16
$begingroup$
What have you tried so far? Do you remember the definition of distance functions?
$endgroup$
– Mindlack
Dec 13 '18 at 18:11
$begingroup$
What have you tried so far? Do you remember the definition of distance functions?
$endgroup$
– Mindlack
Dec 13 '18 at 18:11
$begingroup$
Hello and welcome to Math.SE! In this community, when posting a question, you are encouraged to include in the body of the question some of your thoughts about the problem, some context about it. Check this link for some information about posting good questions.
$endgroup$
– Edmundo Martins
Dec 13 '18 at 18:16
$begingroup$
Hello and welcome to Math.SE! In this community, when posting a question, you are encouraged to include in the body of the question some of your thoughts about the problem, some context about it. Check this link for some information about posting good questions.
$endgroup$
– Edmundo Martins
Dec 13 '18 at 18:16
$begingroup$
Yes I have 3 or 4 rules for this questions, but I can not be sure....Could you help me and explain the questions and answer ?
$endgroup$
– arbade
Dec 13 '18 at 18:16
$begingroup$
Yes I have 3 or 4 rules for this questions, but I can not be sure....Could you help me and explain the questions and answer ?
$endgroup$
– arbade
Dec 13 '18 at 18:16
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
To check that $d$ is a distance on $X$ you need:
1) $d(x,y)geq 0$ $forall x,y in X$ and $d(x,y)=0$ iff $x=y$
2) $d(x,y)=d(y,x)$ $forall x,y in X$
3) $d(x,y) leq d(x,z)+d(z,y)$ $forall x,y,z in X$
The first line of formulas gives you 1), the second line gives you 2), but then
$3=d(2,3) leq d(2,1)+d(1,3)=1+1=2$ and that is a contradiction. Therefore $d$ is not a distance
answered Dec 13 '18 at 18:24
user289143user289143
1,002313
$endgroup$
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%2f3038393%2fin-the-topology-let-x-1-2-3-and-let-dx-times-x-to0-%25e2%2588%259e-be-a-functio%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
$begingroup$
To check that $d$ is a distance on $X$ you need:
1) $d(x,y)geq 0$ $forall x,y in X$ and $d(x,y)=0$ iff $x=y$
2) $d(x,y)=d(y,x)$ $forall x,y in X$
3) $d(x,y) leq d(x,z)+d(z,y)$ $forall x,y,z in X$
The first line of formulas gives you 1), the second line gives you 2), but then
$3=d(2,3) leq d(2,1)+d(1,3)=1+1=2$ and that is a contradiction. Therefore $d$ is not a distance
answered Dec 13 '18 at 18:24
user289143user289143
1,002313
$endgroup$
add a comment |
$begingroup$
To check that $d$ is a distance on $X$ you need:
1) $d(x,y)geq 0$ $forall x,y in X$ and $d(x,y)=0$ iff $x=y$
2) $d(x,y)=d(y,x)$ $forall x,y in X$
3) $d(x,y) leq d(x,z)+d(z,y)$ $forall x,y,z in X$
The first line of formulas gives you 1), the second line gives you 2), but then
$3=d(2,3) leq d(2,1)+d(1,3)=1+1=2$ and that is a contradiction. Therefore $d$ is not a distance
answered Dec 13 '18 at 18:24
user289143user289143
1,002313
$endgroup$
add a comment |
$begingroup$
To check that $d$ is a distance on $X$ you need:
1) $d(x,y)geq 0$ $forall x,y in X$ and $d(x,y)=0$ iff $x=y$
2) $d(x,y)=d(y,x)$ $forall x,y in X$
3) $d(x,y) leq d(x,z)+d(z,y)$ $forall x,y,z in X$
The first line of formulas gives you 1), the second line gives you 2), but then
$3=d(2,3) leq d(2,1)+d(1,3)=1+1=2$ and that is a contradiction. Therefore $d$ is not a distance
answered Dec 13 '18 at 18:24
user289143user289143
1,002313
$endgroup$
To check that $d$ is a distance on $X$ you need:
1) $d(x,y)geq 0$ $forall x,y in X$ and $d(x,y)=0$ iff $x=y$
2) $d(x,y)=d(y,x)$ $forall x,y in X$
3) $d(x,y) leq d(x,z)+d(z,y)$ $forall x,y,z in X$
The first line of formulas gives you 1), the second line gives you 2), but then
$3=d(2,3) leq d(2,1)+d(1,3)=1+1=2$ and that is a contradiction. Therefore $d$ is not a distance
answered Dec 13 '18 at 18:24
user289143user289143
1,002313
answered Dec 13 '18 at 18:24
user289143user289143
1,002313
answered Dec 13 '18 at 18:24
user289143user289143
1,002313
answered Dec 13 '18 at 18:24
user289143user289143
1,002313
1,002313
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.
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%2f3038393%2fin-the-topology-let-x-1-2-3-and-let-dx-times-x-to0-%25e2%2588%259e-be-a-functio%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
$begingroup$
What have you tried so far? Do you remember the definition of distance functions?
$endgroup$
– Mindlack
Dec 13 '18 at 18:11
$begingroup$
Hello and welcome to Math.SE! In this community, when posting a question, you are encouraged to include in the body of the question some of your thoughts about the problem, some context about it. Check this link for some information about posting good questions.
$endgroup$
– Edmundo Martins
Dec 13 '18 at 18:16
$begingroup$
Yes I have 3 or 4 rules for this questions, but I can not be sure....Could you help me and explain the questions and answer ?
$endgroup$
– arbade
Dec 13 '18 at 18:16