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Stefan Ram
science forum beginner

Joined: 14 Sep 2005
Posts: 12

Posted: Mon Jun 19, 2006 5:03 am    Post subject: binary relations

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?
Virgil
science forum Guru

Joined: 24 Mar 2005
Posts: 5536

Posted: Mon Jun 19, 2006 7:05 am    Post subject: Re: binary relations

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?

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}
cody.roux@gmail.com
science forum beginner

Joined: 30 Apr 2006
Posts: 34

Posted: Mon Jun 19, 2006 8:41 am    Post subject: Re: binary relations

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?

You might also whant to look at:
http://en.wikipedia.org/wiki/Glossary_of_graph_theory#Direction
Arturo Magidin
science forum Guru

Joined: 25 Mar 2005
Posts: 1838

Posted: Mon Jun 19, 2006 3:30 pm    Post subject: Re: binary relations

In article <vmhjr2-FC40CA.01055519062006@news.usenetmonster.com>,
Virgil <vmhjr2@comcast.net> wrote:
 Quote: In article , 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}

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
Stefan Ram
science forum beginner

Joined: 14 Sep 2005
Posts: 12

Posted: Tue Jun 20, 2006 2:28 am    Post subject: Re: binary relations

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.

In the German newsgroup, someone gave a reply similar to
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
Arturo Magidin
science forum Guru

Joined: 25 Mar 2005
Posts: 1838

Posted: Tue Jun 20, 2006 2:50 am    Post subject: Re: binary relations

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).

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.

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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