Why does this relation fail symmetry and transitivity properties?
$begingroup$
The question states, let $S$ be the set of all humans.
Define $a ∼ b$ iff $a$ is a full-brother
of $b$.
Symmetry: Since $a$ shares both parents with $b$, then $b$ shares both parents with $a$. Would this be false because $b$ is not defined as a male, so $b$ is instead the full sister of $a$?
Transitivity: Since $a$ shares both parents with $b$, and $b$ shares both parents with $c$, then a shares both parents with $c$. What does the $c$ mean in this context? Is it simply another person introduced?
discrete-mathematics relations equivalence-relations
asked 4 hours ago
Michael RamageMichael Ramage
234
$endgroup$
add a comment |
$begingroup$
The question states, let $S$ be the set of all humans.
Define $a ∼ b$ iff $a$ is a full-brother
of $b$.
Symmetry: Since $a$ shares both parents with $b$, then $b$ shares both parents with $a$. Would this be false because $b$ is not defined as a male, so $b$ is instead the full sister of $a$?
Transitivity: Since $a$ shares both parents with $b$, and $b$ shares both parents with $c$, then a shares both parents with $c$. What does the $c$ mean in this context? Is it simply another person introduced?
discrete-mathematics relations equivalence-relations
asked 4 hours ago
Michael RamageMichael Ramage
234
$endgroup$
1
$begingroup$
$c$ is a variable for a third person. Transitivity would hold if $a,b,c $ were required to be different people. But it fails if $a $ and $c $ are the same person.
$endgroup$
– fleablood
3 hours ago
$begingroup$
That makes more sense. It never defines a,b, and c as being distinct. Thank you so much!
$endgroup$
– Michael Ramage
3 hours ago
add a comment |
$begingroup$
The question states, let $S$ be the set of all humans.
Define $a ∼ b$ iff $a$ is a full-brother
of $b$.
Symmetry: Since $a$ shares both parents with $b$, then $b$ shares both parents with $a$. Would this be false because $b$ is not defined as a male, so $b$ is instead the full sister of $a$?
Transitivity: Since $a$ shares both parents with $b$, and $b$ shares both parents with $c$, then a shares both parents with $c$. What does the $c$ mean in this context? Is it simply another person introduced?
discrete-mathematics relations equivalence-relations
asked 4 hours ago
Michael RamageMichael Ramage
234
$endgroup$
The question states, let $S$ be the set of all humans.
Define $a ∼ b$ iff $a$ is a full-brother
of $b$.
Symmetry: Since $a$ shares both parents with $b$, then $b$ shares both parents with $a$. Would this be false because $b$ is not defined as a male, so $b$ is instead the full sister of $a$?
Transitivity: Since $a$ shares both parents with $b$, and $b$ shares both parents with $c$, then a shares both parents with $c$. What does the $c$ mean in this context? Is it simply another person introduced?
discrete-mathematics relations equivalence-relations
discrete-mathematics relations equivalence-relations
asked 4 hours ago
Michael RamageMichael Ramage
234
asked 4 hours ago
Michael RamageMichael Ramage
234
edited 3 hours ago
Michael Ramage
asked 4 hours ago
Michael RamageMichael Ramage
234
asked 4 hours ago
Michael RamageMichael Ramage
234
asked 4 hours ago
Michael RamageMichael Ramage
234
234
1
$begingroup$
$c$ is a variable for a third person. Transitivity would hold if $a,b,c $ were required to be different people. But it fails if $a $ and $c $ are the same person.
$endgroup$
– fleablood
3 hours ago
$begingroup$
That makes more sense. It never defines a,b, and c as being distinct. Thank you so much!
$endgroup$
– Michael Ramage
3 hours ago
add a comment |
1
$begingroup$
$c$ is a variable for a third person. Transitivity would hold if $a,b,c $ were required to be different people. But it fails if $a $ and $c $ are the same person.
$endgroup$
– fleablood
3 hours ago
$begingroup$
That makes more sense. It never defines a,b, and c as being distinct. Thank you so much!
$endgroup$
– Michael Ramage
3 hours ago
1
1
$begingroup$
$c$ is a variable for a third person. Transitivity would hold if $a,b,c $ were required to be different people. But it fails if $a $ and $c $ are the same person.
$endgroup$
– fleablood
3 hours ago
$begingroup$
$c$ is a variable for a third person. Transitivity would hold if $a,b,c $ were required to be different people. But it fails if $a $ and $c $ are the same person.
$endgroup$
– fleablood
3 hours ago
$begingroup$
That makes more sense. It never defines a,b, and c as being distinct. Thank you so much!
$endgroup$
– Michael Ramage
3 hours ago
$begingroup$
That makes more sense. It never defines a,b, and c as being distinct. Thank you so much!
$endgroup$
– Michael Ramage
3 hours ago
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
Your title is inaccurate. An equivalence relationship can't fail symmetric and transitive properties, by definition, and this is not an equivalence relation because it does fail.
It fails reflexive because $a $~$a $ never happens. No-one is their own brother.
It fails symmetry for exactly the reason you state. If Allen, a boy, and Betty, girl, have the same parents than Allen is a full brother to Betty, but Betty is not a full brother to Allen.
Update!
Transitivity fails. If Allen is a full brother to Bob. And Bob is a full brother to Allen then Allen is not a full brother to Allen.
Transitivity fails.
answered 3 hours ago
fleabloodfleablood
71.4k22686
$endgroup$
$begingroup$
I have corrected it. Does it read correct now?
$endgroup$
– Michael Ramage
3 hours ago
$begingroup$
Yes. it's a relation. But it's not an equivalence relation. It's not an equivalence relation because it fails. BTW I don't know why your book says transitivity fails.
$endgroup$
– fleablood
3 hours ago
$begingroup$
I understand. It is a book by my professor Alex McCallister, "A Transition to Advanced Mathematics: A Survey Course." I have spent several hours attempting to understand why transitivity fails, when it does not, unfortunately.
$endgroup$
– Michael Ramage
3 hours ago
1
$begingroup$
Got it! If $a$ and $b $ are both boys then $a=b$ and $b=a$ but $ane a$.
$endgroup$
– fleablood
3 hours ago
add a comment |
$begingroup$
I am a full brother of my sister, but my sister is not a full brother of me. So this relation is not symmetric.
EDIT: Transitivity fails: see fleablood's comment.
answered 4 hours ago
Robert IsraelRobert Israel
325k23214468
$endgroup$
$begingroup$
... and yes, in this case $c$ is just a third person introduced.
$endgroup$
– Arthur
4 hours ago
$begingroup$
Since a is a full brother of b and b is a full brother of c means that both a and b are males, so c can be a sister to a and b. Even in that case, a is the brother of c. However, the back of my book says that all properties fail. I do not know if this is a type or not. Thank you both!
$endgroup$
– Michael Ramage
4 hours ago
2
$begingroup$
" Transitivity is true though. If a is a full brother of b and b is a full brother of c , then is a full brother of a". Not if a and c are the same person! Transitivity fails.
$endgroup$
– fleablood
3 hours ago
$begingroup$
Oops: you're right! Editing.
$endgroup$
– Robert Israel
31 mins ago
add a comment |
$begingroup$
For a relation to be equvalence relation you also need reflexivity that is
$$
asim a, qquad forall a in S.
$$
which would mean that $a$ is a full brother of himself which is absurd.
Reflecting on your other questions if you define $sim$ to be brothership then you definitely run into trouble with different sexes. So in case of $a$ and $b$ has different sex it is not holding up.
For the question about transitivity you would read, "a is a full brother of b" and "b is a full brother of c". I would say that this holds up.
I hope I could help
answered 4 hours ago
Vinyl_coat_jawaVinyl_coat_jawa
2,9501130
$endgroup$
$begingroup$
This helps. Thank you! I am trying to figure out why the back of the book states that all properties fail for this relation. Perhaps it is a type. I have seen a few along my journey through it.
$endgroup$
– Michael Ramage
4 hours ago
$begingroup$
" For the question about transitivity you would read, "a is a full brother of b" and "b is a full brother of c". I would say that this holds up" it fails if a and c are the same person.
$endgroup$
– fleablood
3 hours ago
$begingroup$
@fleablood but if $a$ and $b$ are the same person then this is not transitivity rather symmetry, or am I missing something?
$endgroup$
– Vinyl_coat_jawa
3 hours ago
$begingroup$
If A and B are different people but full brothers then A~B and B~A. Transitivity would imply that A~A. Which is false. I didn't say it failed if A and B were the same person; I said it fails if A and C were the same person.
$endgroup$
– fleablood
3 hours ago
$begingroup$
I understand but I still consider transitivity having to deal with $3$ different elements of the set
$endgroup$
– Vinyl_coat_jawa
3 hours ago
|
show 2 more comments
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Your title is inaccurate. An equivalence relationship can't fail symmetric and transitive properties, by definition, and this is not an equivalence relation because it does fail.
It fails reflexive because $a $~$a $ never happens. No-one is their own brother.
It fails symmetry for exactly the reason you state. If Allen, a boy, and Betty, girl, have the same parents than Allen is a full brother to Betty, but Betty is not a full brother to Allen.
Update!
Transitivity fails. If Allen is a full brother to Bob. And Bob is a full brother to Allen then Allen is not a full brother to Allen.
Transitivity fails.
answered 3 hours ago
fleabloodfleablood
71.4k22686
$endgroup$
$begingroup$
I have corrected it. Does it read correct now?
$endgroup$
– Michael Ramage
3 hours ago
$begingroup$
Yes. it's a relation. But it's not an equivalence relation. It's not an equivalence relation because it fails. BTW I don't know why your book says transitivity fails.
$endgroup$
– fleablood
3 hours ago
$begingroup$
I understand. It is a book by my professor Alex McCallister, "A Transition to Advanced Mathematics: A Survey Course." I have spent several hours attempting to understand why transitivity fails, when it does not, unfortunately.
$endgroup$
– Michael Ramage
3 hours ago
1
$begingroup$
Got it! If $a$ and $b $ are both boys then $a=b$ and $b=a$ but $ane a$.
$endgroup$
– fleablood
3 hours ago
add a comment |
$begingroup$
Your title is inaccurate. An equivalence relationship can't fail symmetric and transitive properties, by definition, and this is not an equivalence relation because it does fail.
It fails reflexive because $a $~$a $ never happens. No-one is their own brother.
It fails symmetry for exactly the reason you state. If Allen, a boy, and Betty, girl, have the same parents than Allen is a full brother to Betty, but Betty is not a full brother to Allen.
Update!
Transitivity fails. If Allen is a full brother to Bob. And Bob is a full brother to Allen then Allen is not a full brother to Allen.
Transitivity fails.
answered 3 hours ago
fleabloodfleablood
71.4k22686
$endgroup$
$begingroup$
I have corrected it. Does it read correct now?
$endgroup$
– Michael Ramage
3 hours ago
$begingroup$
Yes. it's a relation. But it's not an equivalence relation. It's not an equivalence relation because it fails. BTW I don't know why your book says transitivity fails.
$endgroup$
– fleablood
3 hours ago
$begingroup$
I understand. It is a book by my professor Alex McCallister, "A Transition to Advanced Mathematics: A Survey Course." I have spent several hours attempting to understand why transitivity fails, when it does not, unfortunately.
$endgroup$
– Michael Ramage
3 hours ago
1
$begingroup$
Got it! If $a$ and $b $ are both boys then $a=b$ and $b=a$ but $ane a$.
$endgroup$
– fleablood
3 hours ago
add a comment |
$begingroup$
Your title is inaccurate. An equivalence relationship can't fail symmetric and transitive properties, by definition, and this is not an equivalence relation because it does fail.
It fails reflexive because $a $~$a $ never happens. No-one is their own brother.
It fails symmetry for exactly the reason you state. If Allen, a boy, and Betty, girl, have the same parents than Allen is a full brother to Betty, but Betty is not a full brother to Allen.
Update!
Transitivity fails. If Allen is a full brother to Bob. And Bob is a full brother to Allen then Allen is not a full brother to Allen.
Transitivity fails.
answered 3 hours ago
fleabloodfleablood
71.4k22686
$endgroup$
Your title is inaccurate. An equivalence relationship can't fail symmetric and transitive properties, by definition, and this is not an equivalence relation because it does fail.
It fails reflexive because $a $~$a $ never happens. No-one is their own brother.
It fails symmetry for exactly the reason you state. If Allen, a boy, and Betty, girl, have the same parents than Allen is a full brother to Betty, but Betty is not a full brother to Allen.
Update!
Transitivity fails. If Allen is a full brother to Bob. And Bob is a full brother to Allen then Allen is not a full brother to Allen.
Transitivity fails.
answered 3 hours ago
fleabloodfleablood
71.4k22686
edited 3 hours ago
answered 3 hours ago
fleabloodfleablood
71.4k22686
answered 3 hours ago
fleabloodfleablood
71.4k22686
answered 3 hours ago
fleabloodfleablood
71.4k22686
71.4k22686
$begingroup$
I have corrected it. Does it read correct now?
$endgroup$
– Michael Ramage
3 hours ago
$begingroup$
Yes. it's a relation. But it's not an equivalence relation. It's not an equivalence relation because it fails. BTW I don't know why your book says transitivity fails.
$endgroup$
– fleablood
3 hours ago
$begingroup$
I understand. It is a book by my professor Alex McCallister, "A Transition to Advanced Mathematics: A Survey Course." I have spent several hours attempting to understand why transitivity fails, when it does not, unfortunately.
$endgroup$
– Michael Ramage
3 hours ago
1
$begingroup$
Got it! If $a$ and $b $ are both boys then $a=b$ and $b=a$ but $ane a$.
$endgroup$
– fleablood
3 hours ago
add a comment |
$begingroup$
I have corrected it. Does it read correct now?
$endgroup$
– Michael Ramage
3 hours ago
$begingroup$
Yes. it's a relation. But it's not an equivalence relation. It's not an equivalence relation because it fails. BTW I don't know why your book says transitivity fails.
$endgroup$
– fleablood
3 hours ago
$begingroup$
I understand. It is a book by my professor Alex McCallister, "A Transition to Advanced Mathematics: A Survey Course." I have spent several hours attempting to understand why transitivity fails, when it does not, unfortunately.
$endgroup$
– Michael Ramage
3 hours ago
1
$begingroup$
Got it! If $a$ and $b $ are both boys then $a=b$ and $b=a$ but $ane a$.
$endgroup$
– fleablood
3 hours ago
$begingroup$
I have corrected it. Does it read correct now?
$endgroup$
– Michael Ramage
3 hours ago
$begingroup$
I have corrected it. Does it read correct now?
$endgroup$
– Michael Ramage
3 hours ago
$begingroup$
Yes. it's a relation. But it's not an equivalence relation. It's not an equivalence relation because it fails. BTW I don't know why your book says transitivity fails.
$endgroup$
– fleablood
3 hours ago
$begingroup$
Yes. it's a relation. But it's not an equivalence relation. It's not an equivalence relation because it fails. BTW I don't know why your book says transitivity fails.
$endgroup$
– fleablood
3 hours ago
$begingroup$
I understand. It is a book by my professor Alex McCallister, "A Transition to Advanced Mathematics: A Survey Course." I have spent several hours attempting to understand why transitivity fails, when it does not, unfortunately.
$endgroup$
– Michael Ramage
3 hours ago
$begingroup$
I understand. It is a book by my professor Alex McCallister, "A Transition to Advanced Mathematics: A Survey Course." I have spent several hours attempting to understand why transitivity fails, when it does not, unfortunately.
$endgroup$
– Michael Ramage
3 hours ago
1
1
$begingroup$
Got it! If $a$ and $b $ are both boys then $a=b$ and $b=a$ but $ane a$.
$endgroup$
– fleablood
3 hours ago
$begingroup$
Got it! If $a$ and $b $ are both boys then $a=b$ and $b=a$ but $ane a$.
$endgroup$
– fleablood
3 hours ago
add a comment |
$begingroup$
I am a full brother of my sister, but my sister is not a full brother of me. So this relation is not symmetric.
EDIT: Transitivity fails: see fleablood's comment.
answered 4 hours ago
Robert IsraelRobert Israel
325k23214468
$endgroup$
$begingroup$
... and yes, in this case $c$ is just a third person introduced.
$endgroup$
– Arthur
4 hours ago
$begingroup$
Since a is a full brother of b and b is a full brother of c means that both a and b are males, so c can be a sister to a and b. Even in that case, a is the brother of c. However, the back of my book says that all properties fail. I do not know if this is a type or not. Thank you both!
$endgroup$
– Michael Ramage
4 hours ago
2
$begingroup$
" Transitivity is true though. If a is a full brother of b and b is a full brother of c , then is a full brother of a". Not if a and c are the same person! Transitivity fails.
$endgroup$
– fleablood
3 hours ago
$begingroup$
Oops: you're right! Editing.
$endgroup$
– Robert Israel
31 mins ago
add a comment |
$begingroup$
I am a full brother of my sister, but my sister is not a full brother of me. So this relation is not symmetric.
EDIT: Transitivity fails: see fleablood's comment.
answered 4 hours ago
Robert IsraelRobert Israel
325k23214468
$endgroup$
$begingroup$
... and yes, in this case $c$ is just a third person introduced.
$endgroup$
– Arthur
4 hours ago
$begingroup$
Since a is a full brother of b and b is a full brother of c means that both a and b are males, so c can be a sister to a and b. Even in that case, a is the brother of c. However, the back of my book says that all properties fail. I do not know if this is a type or not. Thank you both!
$endgroup$
– Michael Ramage
4 hours ago
2
$begingroup$
" Transitivity is true though. If a is a full brother of b and b is a full brother of c , then is a full brother of a". Not if a and c are the same person! Transitivity fails.
$endgroup$
– fleablood
3 hours ago
$begingroup$
Oops: you're right! Editing.
$endgroup$
– Robert Israel
31 mins ago
add a comment |
$begingroup$
I am a full brother of my sister, but my sister is not a full brother of me. So this relation is not symmetric.
EDIT: Transitivity fails: see fleablood's comment.
answered 4 hours ago
Robert IsraelRobert Israel
325k23214468
$endgroup$
I am a full brother of my sister, but my sister is not a full brother of me. So this relation is not symmetric.
EDIT: Transitivity fails: see fleablood's comment.
answered 4 hours ago
Robert IsraelRobert Israel
325k23214468
edited 31 mins ago
answered 4 hours ago
Robert IsraelRobert Israel
325k23214468
answered 4 hours ago
Robert IsraelRobert Israel
325k23214468
answered 4 hours ago
Robert IsraelRobert Israel
325k23214468
325k23214468
$begingroup$
... and yes, in this case $c$ is just a third person introduced.
$endgroup$
– Arthur
4 hours ago
$begingroup$
Since a is a full brother of b and b is a full brother of c means that both a and b are males, so c can be a sister to a and b. Even in that case, a is the brother of c. However, the back of my book says that all properties fail. I do not know if this is a type or not. Thank you both!
$endgroup$
– Michael Ramage
4 hours ago
2
$begingroup$
" Transitivity is true though. If a is a full brother of b and b is a full brother of c , then is a full brother of a". Not if a and c are the same person! Transitivity fails.
$endgroup$
– fleablood
3 hours ago
$begingroup$
Oops: you're right! Editing.
$endgroup$
– Robert Israel
31 mins ago
add a comment |
$begingroup$
... and yes, in this case $c$ is just a third person introduced.
$endgroup$
– Arthur
4 hours ago
$begingroup$
Since a is a full brother of b and b is a full brother of c means that both a and b are males, so c can be a sister to a and b. Even in that case, a is the brother of c. However, the back of my book says that all properties fail. I do not know if this is a type or not. Thank you both!
$endgroup$
– Michael Ramage
4 hours ago
2
$begingroup$
" Transitivity is true though. If a is a full brother of b and b is a full brother of c , then is a full brother of a". Not if a and c are the same person! Transitivity fails.
$endgroup$
– fleablood
3 hours ago
$begingroup$
Oops: you're right! Editing.
$endgroup$
– Robert Israel
31 mins ago
$begingroup$
... and yes, in this case $c$ is just a third person introduced.
$endgroup$
– Arthur
4 hours ago
$begingroup$
... and yes, in this case $c$ is just a third person introduced.
$endgroup$
– Arthur
4 hours ago
$begingroup$
Since a is a full brother of b and b is a full brother of c means that both a and b are males, so c can be a sister to a and b. Even in that case, a is the brother of c. However, the back of my book says that all properties fail. I do not know if this is a type or not. Thank you both!
$endgroup$
– Michael Ramage
4 hours ago
$begingroup$
Since a is a full brother of b and b is a full brother of c means that both a and b are males, so c can be a sister to a and b. Even in that case, a is the brother of c. However, the back of my book says that all properties fail. I do not know if this is a type or not. Thank you both!
$endgroup$
– Michael Ramage
4 hours ago
2
2
$begingroup$
" Transitivity is true though. If a is a full brother of b and b is a full brother of c , then is a full brother of a". Not if a and c are the same person! Transitivity fails.
$endgroup$
– fleablood
3 hours ago
$begingroup$
" Transitivity is true though. If a is a full brother of b and b is a full brother of c , then is a full brother of a". Not if a and c are the same person! Transitivity fails.
$endgroup$
– fleablood
3 hours ago
$begingroup$
Oops: you're right! Editing.
$endgroup$
– Robert Israel
31 mins ago
$begingroup$
Oops: you're right! Editing.
$endgroup$
– Robert Israel
31 mins ago
add a comment |
$begingroup$
For a relation to be equvalence relation you also need reflexivity that is
$$
asim a, qquad forall a in S.
$$
which would mean that $a$ is a full brother of himself which is absurd.
Reflecting on your other questions if you define $sim$ to be brothership then you definitely run into trouble with different sexes. So in case of $a$ and $b$ has different sex it is not holding up.
For the question about transitivity you would read, "a is a full brother of b" and "b is a full brother of c". I would say that this holds up.
I hope I could help
answered 4 hours ago
Vinyl_coat_jawaVinyl_coat_jawa
2,9501130
$endgroup$
$begingroup$
This helps. Thank you! I am trying to figure out why the back of the book states that all properties fail for this relation. Perhaps it is a type. I have seen a few along my journey through it.
$endgroup$
– Michael Ramage
4 hours ago
$begingroup$
" For the question about transitivity you would read, "a is a full brother of b" and "b is a full brother of c". I would say that this holds up" it fails if a and c are the same person.
$endgroup$
– fleablood
3 hours ago
$begingroup$
@fleablood but if $a$ and $b$ are the same person then this is not transitivity rather symmetry, or am I missing something?
$endgroup$
– Vinyl_coat_jawa
3 hours ago
$begingroup$
If A and B are different people but full brothers then A~B and B~A. Transitivity would imply that A~A. Which is false. I didn't say it failed if A and B were the same person; I said it fails if A and C were the same person.
$endgroup$
– fleablood
3 hours ago
$begingroup$
I understand but I still consider transitivity having to deal with $3$ different elements of the set
$endgroup$
– Vinyl_coat_jawa
3 hours ago
|
show 2 more comments
$begingroup$
For a relation to be equvalence relation you also need reflexivity that is
$$
asim a, qquad forall a in S.
$$
which would mean that $a$ is a full brother of himself which is absurd.
Reflecting on your other questions if you define $sim$ to be brothership then you definitely run into trouble with different sexes. So in case of $a$ and $b$ has different sex it is not holding up.
For the question about transitivity you would read, "a is a full brother of b" and "b is a full brother of c". I would say that this holds up.
I hope I could help
answered 4 hours ago
Vinyl_coat_jawaVinyl_coat_jawa
2,9501130
$endgroup$
$begingroup$
This helps. Thank you! I am trying to figure out why the back of the book states that all properties fail for this relation. Perhaps it is a type. I have seen a few along my journey through it.
$endgroup$
– Michael Ramage
4 hours ago
$begingroup$
" For the question about transitivity you would read, "a is a full brother of b" and "b is a full brother of c". I would say that this holds up" it fails if a and c are the same person.
$endgroup$
– fleablood
3 hours ago
$begingroup$
@fleablood but if $a$ and $b$ are the same person then this is not transitivity rather symmetry, or am I missing something?
$endgroup$
– Vinyl_coat_jawa
3 hours ago
$begingroup$
If A and B are different people but full brothers then A~B and B~A. Transitivity would imply that A~A. Which is false. I didn't say it failed if A and B were the same person; I said it fails if A and C were the same person.
$endgroup$
– fleablood
3 hours ago
$begingroup$
I understand but I still consider transitivity having to deal with $3$ different elements of the set
$endgroup$
– Vinyl_coat_jawa
3 hours ago
|
show 2 more comments
$begingroup$
For a relation to be equvalence relation you also need reflexivity that is
$$
asim a, qquad forall a in S.
$$
which would mean that $a$ is a full brother of himself which is absurd.
Reflecting on your other questions if you define $sim$ to be brothership then you definitely run into trouble with different sexes. So in case of $a$ and $b$ has different sex it is not holding up.
For the question about transitivity you would read, "a is a full brother of b" and "b is a full brother of c". I would say that this holds up.
I hope I could help
answered 4 hours ago
Vinyl_coat_jawaVinyl_coat_jawa
2,9501130
$endgroup$
For a relation to be equvalence relation you also need reflexivity that is
$$
asim a, qquad forall a in S.
$$
which would mean that $a$ is a full brother of himself which is absurd.
Reflecting on your other questions if you define $sim$ to be brothership then you definitely run into trouble with different sexes. So in case of $a$ and $b$ has different sex it is not holding up.
For the question about transitivity you would read, "a is a full brother of b" and "b is a full brother of c". I would say that this holds up.
I hope I could help
answered 4 hours ago
Vinyl_coat_jawaVinyl_coat_jawa
2,9501130
answered 4 hours ago
Vinyl_coat_jawaVinyl_coat_jawa
2,9501130
answered 4 hours ago
Vinyl_coat_jawaVinyl_coat_jawa
2,9501130
answered 4 hours ago
Vinyl_coat_jawaVinyl_coat_jawa
2,9501130
2,9501130
$begingroup$
This helps. Thank you! I am trying to figure out why the back of the book states that all properties fail for this relation. Perhaps it is a type. I have seen a few along my journey through it.
$endgroup$
– Michael Ramage
4 hours ago
$begingroup$
" For the question about transitivity you would read, "a is a full brother of b" and "b is a full brother of c". I would say that this holds up" it fails if a and c are the same person.
$endgroup$
– fleablood
3 hours ago
$begingroup$
@fleablood but if $a$ and $b$ are the same person then this is not transitivity rather symmetry, or am I missing something?
$endgroup$
– Vinyl_coat_jawa
3 hours ago
$begingroup$
If A and B are different people but full brothers then A~B and B~A. Transitivity would imply that A~A. Which is false. I didn't say it failed if A and B were the same person; I said it fails if A and C were the same person.
$endgroup$
– fleablood
3 hours ago
$begingroup$
I understand but I still consider transitivity having to deal with $3$ different elements of the set
$endgroup$
– Vinyl_coat_jawa
3 hours ago
|
show 2 more comments
$begingroup$
This helps. Thank you! I am trying to figure out why the back of the book states that all properties fail for this relation. Perhaps it is a type. I have seen a few along my journey through it.
$endgroup$
– Michael Ramage
4 hours ago
$begingroup$
" For the question about transitivity you would read, "a is a full brother of b" and "b is a full brother of c". I would say that this holds up" it fails if a and c are the same person.
$endgroup$
– fleablood
3 hours ago
$begingroup$
@fleablood but if $a$ and $b$ are the same person then this is not transitivity rather symmetry, or am I missing something?
$endgroup$
– Vinyl_coat_jawa
3 hours ago
$begingroup$
If A and B are different people but full brothers then A~B and B~A. Transitivity would imply that A~A. Which is false. I didn't say it failed if A and B were the same person; I said it fails if A and C were the same person.
$endgroup$
– fleablood
3 hours ago
$begingroup$
I understand but I still consider transitivity having to deal with $3$ different elements of the set
$endgroup$
– Vinyl_coat_jawa
3 hours ago
$begingroup$
This helps. Thank you! I am trying to figure out why the back of the book states that all properties fail for this relation. Perhaps it is a type. I have seen a few along my journey through it.
$endgroup$
– Michael Ramage
4 hours ago
$begingroup$
This helps. Thank you! I am trying to figure out why the back of the book states that all properties fail for this relation. Perhaps it is a type. I have seen a few along my journey through it.
$endgroup$
– Michael Ramage
4 hours ago
$begingroup$
" For the question about transitivity you would read, "a is a full brother of b" and "b is a full brother of c". I would say that this holds up" it fails if a and c are the same person.
$endgroup$
– fleablood
3 hours ago
$begingroup$
" For the question about transitivity you would read, "a is a full brother of b" and "b is a full brother of c". I would say that this holds up" it fails if a and c are the same person.
$endgroup$
– fleablood
3 hours ago
$begingroup$
@fleablood but if $a$ and $b$ are the same person then this is not transitivity rather symmetry, or am I missing something?
$endgroup$
– Vinyl_coat_jawa
3 hours ago
$begingroup$
@fleablood but if $a$ and $b$ are the same person then this is not transitivity rather symmetry, or am I missing something?
$endgroup$
– Vinyl_coat_jawa
3 hours ago
$begingroup$
If A and B are different people but full brothers then A~B and B~A. Transitivity would imply that A~A. Which is false. I didn't say it failed if A and B were the same person; I said it fails if A and C were the same person.
$endgroup$
– fleablood
3 hours ago
$begingroup$
If A and B are different people but full brothers then A~B and B~A. Transitivity would imply that A~A. Which is false. I didn't say it failed if A and B were the same person; I said it fails if A and C were the same person.
$endgroup$
– fleablood
3 hours ago
$begingroup$
I understand but I still consider transitivity having to deal with $3$ different elements of the set
$endgroup$
– Vinyl_coat_jawa
3 hours ago
$begingroup$
I understand but I still consider transitivity having to deal with $3$ different elements of the set
$endgroup$
– Vinyl_coat_jawa
3 hours ago
|
show 2 more comments
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$begingroup$
$c$ is a variable for a third person. Transitivity would hold if $a,b,c $ were required to be different people. But it fails if $a $ and $c $ are the same person.
$endgroup$
– fleablood
3 hours ago
$begingroup$
That makes more sense. It never defines a,b, and c as being distinct. Thank you so much!
$endgroup$
– Michael Ramage
3 hours ago