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
3,08621038
asked Dec 13 '18 at 18:08
arbadearbade
14
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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
3,08621038
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
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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.
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– 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
3,08621038
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
3,08621038
asked Dec 13 '18 at 18:08
arbadearbade
14
edited Dec 13 '18 at 22:33
Tianlalu
3,08621038
asked Dec 13 '18 at 18:08
arbadearbade
14
edited Dec 13 '18 at 22:33
Tianlalu
3,08621038
edited Dec 13 '18 at 22:33
Tianlalu
3,08621038
edited Dec 13 '18 at 22:33
Tianlalu
3,08621038
3,08621038
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
913313
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add a comment |
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1 Answer
1
active
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1 Answer
1
active
oldest
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active
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$begingroup$
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
913313
$endgroup$
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
913313
$endgroup$
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
913313
$endgroup$
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
913313
answered Dec 13 '18 at 18:24
user289143user289143
913313
answered Dec 13 '18 at 18:24
user289143user289143
913313
answered Dec 13 '18 at 18:24
user289143user289143
913313
913313
add a comment |
add a comment |
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$begingroup$
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