Relations and understanding anti-symmetry
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The below picture is a problem involving relations from a review set. I have only seen problems with relations involving matrices and solving for the properties of reflexive, symmetric, and transitive.
So determining if a relation is anti-symmetric and quite simply approaching parts B and C where you're given the relation not already in a matrix is problematic.
Is there a way to transform the given relation in B and C to a matrix? And how, from a matrix, can we determine if the relation is anti-symmetric?
Here is the matrix setup for B.
begin{bmatrix}X&1&2&3&4&5&6\1&?&?&?&?&?&?\2&?&?&?&?&?&?\3&?&?&?&?&?&?\4&?&?&?&?&?&?\5&?&?&?&?&?&?\6&?&?&?&?&?&?end{bmatrix}
relations
asked Dec 9 '18 at 3:32
Mr.MipsMr.Mips
185
$endgroup$
|
show 4 more comments
$begingroup$
The below picture is a problem involving relations from a review set. I have only seen problems with relations involving matrices and solving for the properties of reflexive, symmetric, and transitive.
So determining if a relation is anti-symmetric and quite simply approaching parts B and C where you're given the relation not already in a matrix is problematic.
Is there a way to transform the given relation in B and C to a matrix? And how, from a matrix, can we determine if the relation is anti-symmetric?
Here is the matrix setup for B.
begin{bmatrix}X&1&2&3&4&5&6\1&?&?&?&?&?&?\2&?&?&?&?&?&?\3&?&?&?&?&?&?\4&?&?&?&?&?&?\5&?&?&?&?&?&?\6&?&?&?&?&?&?end{bmatrix}
relations
asked Dec 9 '18 at 3:32
Mr.MipsMr.Mips
185
$endgroup$
1
$begingroup$
An example of a symmetric relation: $aRb$ iff $gcd(a,b)=1$; here, for example, $2R3$ implies $3R2$, so both hold, yet $2neq 3$, so $R$ is not antisymmetric.
$endgroup$
– Shaun
Dec 9 '18 at 3:44
1
$begingroup$
An example of an antisymmetric relation: $aRb$ iff $ale b$; here, if $xle y$ and $yle x$, then $x=y$, yet $1le 2$ does not imply $2le 1$, so $R$ is not symmetric.
$endgroup$
– Shaun
Dec 9 '18 at 3:48
1
$begingroup$
NB: For my first example, I assumed that $R$ is a relation on the natural numbers; for the second, $R$ is a relation on the real numbers. Stating what a relation is on is important (and I forgot to include the information due to it being late here at the time of writing).
$endgroup$
– Shaun
Dec 9 '18 at 3:56
1
$begingroup$
@Shaun Great. I won't bother you anymore since it's late for you. Thank you for the clarification
$endgroup$
– Mr.Mips
Dec 9 '18 at 3:59
1
$begingroup$
You're welcome :)
$endgroup$
– Shaun
Dec 9 '18 at 4:00
|
show 4 more comments
$begingroup$
The below picture is a problem involving relations from a review set. I have only seen problems with relations involving matrices and solving for the properties of reflexive, symmetric, and transitive.
So determining if a relation is anti-symmetric and quite simply approaching parts B and C where you're given the relation not already in a matrix is problematic.
Is there a way to transform the given relation in B and C to a matrix? And how, from a matrix, can we determine if the relation is anti-symmetric?
Here is the matrix setup for B.
begin{bmatrix}X&1&2&3&4&5&6\1&?&?&?&?&?&?\2&?&?&?&?&?&?\3&?&?&?&?&?&?\4&?&?&?&?&?&?\5&?&?&?&?&?&?\6&?&?&?&?&?&?end{bmatrix}
relations
asked Dec 9 '18 at 3:32
Mr.MipsMr.Mips
185
$endgroup$
The below picture is a problem involving relations from a review set. I have only seen problems with relations involving matrices and solving for the properties of reflexive, symmetric, and transitive.
So determining if a relation is anti-symmetric and quite simply approaching parts B and C where you're given the relation not already in a matrix is problematic.
Is there a way to transform the given relation in B and C to a matrix? And how, from a matrix, can we determine if the relation is anti-symmetric?
Here is the matrix setup for B.
begin{bmatrix}X&1&2&3&4&5&6\1&?&?&?&?&?&?\2&?&?&?&?&?&?\3&?&?&?&?&?&?\4&?&?&?&?&?&?\5&?&?&?&?&?&?\6&?&?&?&?&?&?end{bmatrix}
relations
relations
asked Dec 9 '18 at 3:32
Mr.MipsMr.Mips
185
asked Dec 9 '18 at 3:32
Mr.MipsMr.Mips
185
edited Dec 9 '18 at 6:29
Mr.Mips
asked Dec 9 '18 at 3:32
Mr.MipsMr.Mips
185
asked Dec 9 '18 at 3:32
Mr.MipsMr.Mips
185
asked Dec 9 '18 at 3:32
Mr.MipsMr.Mips
185
185
1
$begingroup$
An example of a symmetric relation: $aRb$ iff $gcd(a,b)=1$; here, for example, $2R3$ implies $3R2$, so both hold, yet $2neq 3$, so $R$ is not antisymmetric.
$endgroup$
– Shaun
Dec 9 '18 at 3:44
1
$begingroup$
An example of an antisymmetric relation: $aRb$ iff $ale b$; here, if $xle y$ and $yle x$, then $x=y$, yet $1le 2$ does not imply $2le 1$, so $R$ is not symmetric.
$endgroup$
– Shaun
Dec 9 '18 at 3:48
1
$begingroup$
NB: For my first example, I assumed that $R$ is a relation on the natural numbers; for the second, $R$ is a relation on the real numbers. Stating what a relation is on is important (and I forgot to include the information due to it being late here at the time of writing).
$endgroup$
– Shaun
Dec 9 '18 at 3:56
1
$begingroup$
@Shaun Great. I won't bother you anymore since it's late for you. Thank you for the clarification
$endgroup$
– Mr.Mips
Dec 9 '18 at 3:59
1
$begingroup$
You're welcome :)
$endgroup$
– Shaun
Dec 9 '18 at 4:00
|
show 4 more comments
1
$begingroup$
An example of a symmetric relation: $aRb$ iff $gcd(a,b)=1$; here, for example, $2R3$ implies $3R2$, so both hold, yet $2neq 3$, so $R$ is not antisymmetric.
$endgroup$
– Shaun
Dec 9 '18 at 3:44
1
$begingroup$
An example of an antisymmetric relation: $aRb$ iff $ale b$; here, if $xle y$ and $yle x$, then $x=y$, yet $1le 2$ does not imply $2le 1$, so $R$ is not symmetric.
$endgroup$
– Shaun
Dec 9 '18 at 3:48
1
$begingroup$
NB: For my first example, I assumed that $R$ is a relation on the natural numbers; for the second, $R$ is a relation on the real numbers. Stating what a relation is on is important (and I forgot to include the information due to it being late here at the time of writing).
$endgroup$
– Shaun
Dec 9 '18 at 3:56
1
$begingroup$
@Shaun Great. I won't bother you anymore since it's late for you. Thank you for the clarification
$endgroup$
– Mr.Mips
Dec 9 '18 at 3:59
1
$begingroup$
You're welcome :)
$endgroup$
– Shaun
Dec 9 '18 at 4:00
1
1
$begingroup$
An example of a symmetric relation: $aRb$ iff $gcd(a,b)=1$; here, for example, $2R3$ implies $3R2$, so both hold, yet $2neq 3$, so $R$ is not antisymmetric.
$endgroup$
– Shaun
Dec 9 '18 at 3:44
$begingroup$
An example of a symmetric relation: $aRb$ iff $gcd(a,b)=1$; here, for example, $2R3$ implies $3R2$, so both hold, yet $2neq 3$, so $R$ is not antisymmetric.
$endgroup$
– Shaun
Dec 9 '18 at 3:44
1
1
$begingroup$
An example of an antisymmetric relation: $aRb$ iff $ale b$; here, if $xle y$ and $yle x$, then $x=y$, yet $1le 2$ does not imply $2le 1$, so $R$ is not symmetric.
$endgroup$
– Shaun
Dec 9 '18 at 3:48
$begingroup$
An example of an antisymmetric relation: $aRb$ iff $ale b$; here, if $xle y$ and $yle x$, then $x=y$, yet $1le 2$ does not imply $2le 1$, so $R$ is not symmetric.
$endgroup$
– Shaun
Dec 9 '18 at 3:48
1
1
$begingroup$
NB: For my first example, I assumed that $R$ is a relation on the natural numbers; for the second, $R$ is a relation on the real numbers. Stating what a relation is on is important (and I forgot to include the information due to it being late here at the time of writing).
$endgroup$
– Shaun
Dec 9 '18 at 3:56
$begingroup$
NB: For my first example, I assumed that $R$ is a relation on the natural numbers; for the second, $R$ is a relation on the real numbers. Stating what a relation is on is important (and I forgot to include the information due to it being late here at the time of writing).
$endgroup$
– Shaun
Dec 9 '18 at 3:56
1
1
$begingroup$
@Shaun Great. I won't bother you anymore since it's late for you. Thank you for the clarification
$endgroup$
– Mr.Mips
Dec 9 '18 at 3:59
$begingroup$
@Shaun Great. I won't bother you anymore since it's late for you. Thank you for the clarification
$endgroup$
– Mr.Mips
Dec 9 '18 at 3:59
1
1
$begingroup$
You're welcome :)
$endgroup$
– Shaun
Dec 9 '18 at 4:00
$begingroup$
You're welcome :)
$endgroup$
– Shaun
Dec 9 '18 at 4:00
|
show 4 more comments
1 Answer
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See my comments on the question for examples demonstrating the difference between symmetric and antisymmetric relations.
Given a relation $R$ on a set $X$ with $|X|=n<infty$, say, then $R$ is equivalent to an $ntimes n$ matrix $mathcal{R}$ with entries in ${0, 1}$ (or ${text{false, true}}$ if you prefer), where, if one labels the rows & columns according to the elements of $X$, the entry
$$mathcal{R}_{ij}:=begin{cases}
0,text{(false)} & text{if not } quad iRj, \
1,text{(true)} & text{if }quad iRj.
end{cases}$$
As far as I am aware, there is no easy way to see if $R$ is (anti)symmetric on $X$, given only $mathcal{R}$, but I could be wrong.
answered Dec 9 '18 at 4:15
ShaunShaun
9,060113682
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$begingroup$
See my comments on the question for examples demonstrating the difference between symmetric and antisymmetric relations.
Given a relation $R$ on a set $X$ with $|X|=n<infty$, say, then $R$ is equivalent to an $ntimes n$ matrix $mathcal{R}$ with entries in ${0, 1}$ (or ${text{false, true}}$ if you prefer), where, if one labels the rows & columns according to the elements of $X$, the entry
$$mathcal{R}_{ij}:=begin{cases}
0,text{(false)} & text{if not } quad iRj, \
1,text{(true)} & text{if }quad iRj.
end{cases}$$
As far as I am aware, there is no easy way to see if $R$ is (anti)symmetric on $X$, given only $mathcal{R}$, but I could be wrong.
answered Dec 9 '18 at 4:15
ShaunShaun
9,060113682
$endgroup$
add a comment |
$begingroup$
See my comments on the question for examples demonstrating the difference between symmetric and antisymmetric relations.
Given a relation $R$ on a set $X$ with $|X|=n<infty$, say, then $R$ is equivalent to an $ntimes n$ matrix $mathcal{R}$ with entries in ${0, 1}$ (or ${text{false, true}}$ if you prefer), where, if one labels the rows & columns according to the elements of $X$, the entry
$$mathcal{R}_{ij}:=begin{cases}
0,text{(false)} & text{if not } quad iRj, \
1,text{(true)} & text{if }quad iRj.
end{cases}$$
As far as I am aware, there is no easy way to see if $R$ is (anti)symmetric on $X$, given only $mathcal{R}$, but I could be wrong.
answered Dec 9 '18 at 4:15
ShaunShaun
9,060113682
$endgroup$
add a comment |
$begingroup$
See my comments on the question for examples demonstrating the difference between symmetric and antisymmetric relations.
Given a relation $R$ on a set $X$ with $|X|=n<infty$, say, then $R$ is equivalent to an $ntimes n$ matrix $mathcal{R}$ with entries in ${0, 1}$ (or ${text{false, true}}$ if you prefer), where, if one labels the rows & columns according to the elements of $X$, the entry
$$mathcal{R}_{ij}:=begin{cases}
0,text{(false)} & text{if not } quad iRj, \
1,text{(true)} & text{if }quad iRj.
end{cases}$$
As far as I am aware, there is no easy way to see if $R$ is (anti)symmetric on $X$, given only $mathcal{R}$, but I could be wrong.
answered Dec 9 '18 at 4:15
ShaunShaun
9,060113682
$endgroup$
See my comments on the question for examples demonstrating the difference between symmetric and antisymmetric relations.
Given a relation $R$ on a set $X$ with $|X|=n<infty$, say, then $R$ is equivalent to an $ntimes n$ matrix $mathcal{R}$ with entries in ${0, 1}$ (or ${text{false, true}}$ if you prefer), where, if one labels the rows & columns according to the elements of $X$, the entry
$$mathcal{R}_{ij}:=begin{cases}
0,text{(false)} & text{if not } quad iRj, \
1,text{(true)} & text{if }quad iRj.
end{cases}$$
As far as I am aware, there is no easy way to see if $R$ is (anti)symmetric on $X$, given only $mathcal{R}$, but I could be wrong.
answered Dec 9 '18 at 4:15
ShaunShaun
9,060113682
edited Dec 9 '18 at 4:21
answered Dec 9 '18 at 4:15
ShaunShaun
9,060113682
answered Dec 9 '18 at 4:15
ShaunShaun
9,060113682
answered Dec 9 '18 at 4:15
ShaunShaun
9,060113682
9,060113682
add a comment |
add a comment |
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$begingroup$
An example of a symmetric relation: $aRb$ iff $gcd(a,b)=1$; here, for example, $2R3$ implies $3R2$, so both hold, yet $2neq 3$, so $R$ is not antisymmetric.
$endgroup$
– Shaun
Dec 9 '18 at 3:44
1
$begingroup$
An example of an antisymmetric relation: $aRb$ iff $ale b$; here, if $xle y$ and $yle x$, then $x=y$, yet $1le 2$ does not imply $2le 1$, so $R$ is not symmetric.
$endgroup$
– Shaun
Dec 9 '18 at 3:48
1
$begingroup$
NB: For my first example, I assumed that $R$ is a relation on the natural numbers; for the second, $R$ is a relation on the real numbers. Stating what a relation is on is important (and I forgot to include the information due to it being late here at the time of writing).
$endgroup$
– Shaun
Dec 9 '18 at 3:56
1
$begingroup$
@Shaun Great. I won't bother you anymore since it's late for you. Thank you for the clarification
$endgroup$
– Mr.Mips
Dec 9 '18 at 3:59
1
$begingroup$
You're welcome :)
$endgroup$
– Shaun
Dec 9 '18 at 4:00