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Stefan Ram
science forum beginner
Joined: 14 Sep 2005
Posts: 12
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 Posted: Mon Jun 19, 2006 5:03 am Post subject:
binary relations
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R is a subset of AxB, i.e., a binary relation.
For an element a of the set A, R(a) is { beB | (a,b)eR },
i.e., the set of all b, so that the pair (a,b) is an
element of the relation R.
Is there a common name for this set or its elements?
May be, it might help you to think of the pairs as
arrows of a graph. The question would be: Is there
a name for the set of all arrows (or their targets)
starting at a certain point? |
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Virgil
science forum Guru
Joined: 24 Mar 2005
Posts: 5536
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| Posted: Mon Jun 19, 2006 7:05 am Post subject:
Re: binary relations
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In article <relations-20060619065837@ram.dialup.fu-berlin.de>,
ram@zedat.fu-berlin.de (Stefan Ram) wrote:
| Quote: | R is a subset of AxB, i.e., a binary relation.
For an element a of the set A, R(a) is { beB | (a,b)eR },
i.e., the set of all b, so that the pair (a,b) is an
element of the relation R.
Is there a common name for this set or its elements?
May be, it might help you to think of the pairs as
arrows of a graph. The question would be: Is there
a name for the set of all arrows (or their targets)
starting at a certain point?
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One might, I suppose generalize, "domain" and "codomain" from functions
to relations so that
Domain(R) = {a e A | (a,b) e R}, and
Codomain(R) = {b e B | (a,b) e R} |
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cody.roux@gmail.com
science forum beginner
Joined: 30 Apr 2006
Posts: 34
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| Posted: Mon Jun 19, 2006 8:41 am Post subject:
Re: binary relations
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Stefan Ram a écrit :
| Quote: | R is a subset of AxB, i.e., a binary relation.
For an element a of the set A, R(a) is { beB | (a,b)eR },
i.e., the set of all b, so that the pair (a,b) is an
element of the relation R.
Is there a common name for this set or its elements?
May be, it might help you to think of the pairs as
arrows of a graph. The question would be: Is there
a name for the set of all arrows (or their targets)
starting at a certain point?
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You might also whant to look at:
http://en.wikipedia.org/wiki/Glossary_of_graph_theory#Direction |
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Arturo Magidin
science forum Guru
Joined: 25 Mar 2005
Posts: 1838
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| Posted: Mon Jun 19, 2006 3:30 pm Post subject:
Re: binary relations
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In article <vmhjr2-FC40CA.01055519062006@news.usenetmonster.com>,
Virgil <vmhjr2@comcast.net> wrote:
| Quote: | In article <relations-20060619065837@ram.dialup.fu-berlin.de>,
ram@zedat.fu-berlin.de (Stefan Ram) wrote:
R is a subset of AxB, i.e., a binary relation.
For an element a of the set A, R(a) is { beB | (a,b)eR },
i.e., the set of all b, so that the pair (a,b) is an
element of the relation R.
Is there a common name for this set or its elements?
May be, it might help you to think of the pairs as
arrows of a graph. The question would be: Is there
a name for the set of all arrows (or their targets)
starting at a certain point?
One might, I suppose generalize, "domain" and "codomain" from functions
to relations so that
Domain(R) = {a e A | (a,b) e R}, and
Codomain(R) = {b e B | (a,b) e R}
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You forgot your quantifiers; either you want to fix b in B and let
Domain(R,b) = {a in A | (a,b) in R}
and fix a in A and let
Codomain(R,a) = {b in B | (a,b) in R}.
Or else you want to say
Domain(R) = {a in A | there exists b in B such that (a,b) in R}
Codomain(R) = {b in B | there exists a in A such that (a,b) in R}.
The OP is asking whether there is a common name for what I called
"Codomain(R,b)". I for one do not know of any common name for these
sets.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu |
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Stefan Ram
science forum beginner
Joined: 14 Sep 2005
Posts: 12
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| Posted: Tue Jun 20, 2006 2:28 am Post subject:
Re: binary relations
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magidin@math.berkeley.edu (Arturo Magidin) writes:
| Quote: | Codomain(R) = {b e B | (a,b) e R}
Codomain(R,a) = {b in B | (a,b) in R}. (...)
The OP is asking whether there is a common name for what I called
"Codomain(R,b)". I for one do not know of any common name for these
sets.
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In the German newsgroup, someone gave a reply similar to
Virgil's and I answered similar to your answer with the
consequence that the person I was answering to started to
insult me.
Now, I at least have learned a German term from other
postings: Translating German word-by-word, the elements of
what you call »Codomain(R,a)« would be called the (immediate)
/successors/ of a (implying that the relation R is given by
the context).
In English text, however, the term »successor« might often be
used with some other meanings related to orders. However, I
found a proof where the word "successor" is used for a
relation (actually, a digraph) also in an English text, here:
Predecessor - for v in a digraph, a vertex u with u -> v
Predecessor set - for v in a digraph, the set of precedessors
(...)
Successor - for u in a digraph, a vertex v with u -> v
Successor set - for u in a digraph, the set of successors
http://www.math.uiuc.edu/~west/openp/gloss.html |
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Arturo Magidin
science forum Guru
Joined: 25 Mar 2005
Posts: 1838
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| Posted: Tue Jun 20, 2006 2:50 am Post subject:
Re: binary relations
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In article <successor-20060620042659@ram.dialup.fu-berlin.de>,
Stefan Ram <ram@zedat.fu-berlin.de> wrote:
| Quote: | magidin@math.berkeley.edu (Arturo Magidin) writes:
Codomain(R) = {b e B | (a,b) e R}
Codomain(R,a) = {b in B | (a,b) in R}. (...)
The OP is asking whether there is a common name for what I called
"Codomain(R,b)". I for one do not know of any common name for these
sets.
In the German newsgroup, someone gave a reply similar to
Virgil's and I answered similar to your answer with the
consequence that the person I was answering to started to
insult me.
Now, I at least have learned a German term from other
postings: Translating German word-by-word, the elements of
what you call »Codomain(R,a)« would be called the (immediate)
/successors/ of a (implying that the relation R is given by
the context).
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This makes perfect sense in English if you envision the relation as a
list of directed edges of a bipartite digraph, which the original
poster suggested.
| Quote: | In English text, however, the term »successor« might often be
used with some other meanings related to orders.
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Only certain kinds of orders...
There is nothing wrong with a term being used in more than one context
to mean two different things (in this case, a particular kind of
orders and digraphs). I mean, just look at the word "normal". Off the
top of my head, I can come up with four different meanings, and if I
scratch my head a bit more I can probably come up with at least that
many more.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu |
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