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Please explain the empty relation
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TOMERDR
science forum beginner


Joined: 09 May 2006
Posts: 26

PostPosted: Wed Jul 05, 2006 7:46 am    Post subject: Re: Please explain the empty relation Reply with quote

Thanks.exactly what i needed.


Arturo Magidin wrote:
Quote:
In article <1151919897.763001.121120@m73g2000cwd.googlegroups.com>,
TOMERDR <tomerdr@hotmail.com> wrote:


1.What exactly is an empty relation?
(no one in relation with other,or a relation on empty set?)

A relation between two sets A and B is a subset of A x B. One such
subset is the empty set. This is called the empty relation.

In the empty relation, no element of the first set, A, is in the
relation with any element of the second set, B.

When A or B are themselves empty, the only possible relation from A to
B is the empty relation.


2.Why is it both symetric and anti symetric according to the definition
of symetric and anti symetric

For the relation to be symmetric and/or anti-symmetric, it must be a
relation from a set to itself (the concept is only defined in that
context). As such, we are assuming A = B.

That said, the definition says: in order for the relation R to be
symmtric, it is necessary that every time you have a pair (x,y) in R,
the corresponding pair (y,x) musty also be in R. This is true for the
empty relation, because the antecedent of the implication is never
true. An implication is always true when the antecedent is false.

Viewed another way: in order for the empty relation not to be
symmetric, it is necessary that there be elements x and y in A for
which the pair (x,y) is in R, but the pair (y,x) is not in R. This can
never happen in the empty relation, because you can never find a pair
of elements x and y in A for which (x,y) is in R, and you can forget
the second clause. (This may be because A is empty, in which case R is
necessarily empty; or it could be because R is empty, whether or not A
is empty).

The same argument works for anti-symmetry. Anti-symmetry requires that
for all elements x and y in A, if both (x,y) and (y,x) are elements of
R, then x must be equal to y. Since the antecedent of the implication
is always false for the empty relation, the implication itself is
always true and so R is anti-symmetric.

--
======================================================================
"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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Arturo Magidin
science forum Guru


Joined: 25 Mar 2005
Posts: 1838

PostPosted: Mon Jul 03, 2006 4:22 pm    Post subject: Re: Please explain the empty relation Reply with quote

In article <1151919897.763001.121120@m73g2000cwd.googlegroups.com>,
TOMERDR <tomerdr@hotmail.com> wrote:
Quote:


1.What exactly is an empty relation?
(no one in relation with other,or a relation on empty set?)

A relation between two sets A and B is a subset of A x B. One such
subset is the empty set. This is called the empty relation.

In the empty relation, no element of the first set, A, is in the
relation with any element of the second set, B.

When A or B are themselves empty, the only possible relation from A to
B is the empty relation.


Quote:
2.Why is it both symetric and anti symetric according to the definition
of symetric and anti symetric

For the relation to be symmetric and/or anti-symmetric, it must be a
relation from a set to itself (the concept is only defined in that
context). As such, we are assuming A = B.

That said, the definition says: in order for the relation R to be
symmtric, it is necessary that every time you have a pair (x,y) in R,
the corresponding pair (y,x) musty also be in R. This is true for the
empty relation, because the antecedent of the implication is never
true. An implication is always true when the antecedent is false.

Viewed another way: in order for the empty relation not to be
symmetric, it is necessary that there be elements x and y in A for
which the pair (x,y) is in R, but the pair (y,x) is not in R. This can
never happen in the empty relation, because you can never find a pair
of elements x and y in A for which (x,y) is in R, and you can forget
the second clause. (This may be because A is empty, in which case R is
necessarily empty; or it could be because R is empty, whether or not A
is empty).

The same argument works for anti-symmetry. Anti-symmetry requires that
for all elements x and y in A, if both (x,y) and (y,x) are elements of
R, then x must be equal to y. Since the antecedent of the implication
is always false for the empty relation, the implication itself is
always true and so R is anti-symmetric.

--
======================================================================
"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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G.E. Ivey
science forum Guru


Joined: 29 Apr 2005
Posts: 308

PostPosted: Mon Jul 03, 2006 4:07 pm    Post subject: Re: Please explain the empty relation Reply with quote

Quote:


1.What exactly is an empty relation?
(no one in relation with other,or a relation on
on empty set?)
2.Why is it both symetric and anti symetric according
to the definition
of symetric and anti symetric


Thanks in advance.

I'm going to give a slightly different answer than William Elliot (using a variant definition of "relation"). A relation on a set, A, is "a set of ordered pairs of elements of A". An empty relation then, would be the empty set.

Using the definition of relation William Elliot is (a rule linking two things- something like "x is related to y if and only if x- y= 6"), my "set of ordered pairs" would contain (x,y) if and only if the rule is true. That's why he says the empty relation corresponds to "always false".

The definition of "symmetric" for a relation, using my definition of relation, is
"if (x,y) is in the set, then (y,x) is also".
For the empty set, since "(x,y) is in the set" is never true, the statement is vacuously true.

The definition of "anti-symmetric" for a relation is
"if (x,y) is in the set, then (y,x) is NOT in the set".
Again, this is vacuously true.

"Vacuously true": The implication statement "P implies Q" or "if P then Q" is TRUE by definition if P is FALSE.
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William Elliot
science forum Guru


Joined: 24 Mar 2005
Posts: 1906

PostPosted: Mon Jul 03, 2006 12:21 pm    Post subject: Re: Please explain the empty relation Reply with quote

On Mon, 3 Jul 2006, TOMERDR wrote:

Quote:
1.What exactly is an empty relation?
(no one in relation with other,or a relation on empty set?)

One that is always false.

Quote:
2.Why is it both symetric and anti symetric according to the definition
of symetric and anti symetric

Those definitions apply only to binary relations and as they are of the

form for all a,b, if aRb, then ... Since aRb is false, the ... can be
anything, even: Bush is intelligent.
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TOMERDR
science forum beginner


Joined: 09 May 2006
Posts: 26

PostPosted: Mon Jul 03, 2006 9:44 am    Post subject: Please explain the empty relation Reply with quote

1.What exactly is an empty relation?
(no one in relation with other,or a relation on empty set?)
2.Why is it both symetric and anti symetric according to the definition
of symetric and anti symetric


Thanks in advance.
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