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Science Forum Index » Mathematics Forum » partial order and total order
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| Helen |
 Posted: Thu Dec 18, 2003 1:23 pm |
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Hi I am reading ttm's algebra book
it defines particial order as:
Given a set T and a relation >= on T, if the relation >= satisfies
the following conditions, then it is called partial ordering
a) t1>=t1 for all t1 \in T
b) t1 >= t2 and t2>=t3 => t1>=t3
c) t1>=t2 and t2>=t1 => t1=t2
and defines the total order by
Given a set T and a partial ordering >= on T, if the relation >= satisfies
the following conditions (d), then it is called total ordering
d) for any t1,t2 \in T we always have either t1>=t2 or t2>=t1
I don't understand the difference between those two order
thanks a lot |
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| A N Niel |
| Posted: Thu Dec 18, 2003 1:47 pm |
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In article <ba3374ff.0312181023.8b3663e@posting.google.com>, Helen
<junciu@yahoo.com> wrote:
Quote: Hi I am reading ttm's algebra book
it defines particial order as:
Given a set T and a relation >= on T, if the relation >= satisfies
the following conditions, then it is called partial ordering
a) t1>=t1 for all t1 \in T
b) t1 >= t2 and t2>=t3 => t1>=t3
c) t1>=t2 and t2>=t1 => t1=t2
and defines the total order by
Given a set T and a partial ordering >= on T, if the relation >= satisfies
the following conditions (d), then it is called total ordering
d) for any t1,t2 \in T we always have either t1>=t2 or t2>=t1
I don't understand the difference between those two order
thanks a lot
Take the set {1,2}, let T be the set of all subsets of T. So
T itself has 4 elements. Let >= be set inclusion. Then >= is
a partial order on T but not a total order. |
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| Jeremy Boden |
| Posted: Thu Dec 18, 2003 2:13 pm |
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In message <ba3374ff.0312181023.8b3663e@posting.google.com>, Helen
<junciu@yahoo.com> writes
Quote: Hi I am reading ttm's algebra book
it defines particial order as:
Given a set T and a relation >= on T, if the relation >= satisfies
the following conditions, then it is called partial ordering
a) t1>=t1 for all t1 \in T
b) t1 >= t2 and t2>=t3 => t1>=t3
c) t1>=t2 and t2>=t1 => t1=t2
and defines the total order by
Given a set T and a partial ordering >= on T, if the relation >= satisfies
the following conditions (d), then it is called total ordering
d) for any t1,t2 \in T we always have either t1>=t2 or t2>=t1
I don't understand the difference between those two order
thanks a lot
You might like to think of a partial ordering as being:-
a) Reflexive x R x,
b) Antisymmetric - if x R y and y R x then x = y
c) Transitive - if x R y and y R z then x R z
Note that (b) and (c) say IF x >= y (etc...)
However, elements are not necessarily comparable in a partial order.
A total order guarantees that any pair of elements are comparable.
--
Jeremy Boden |
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| Dean the Berzerker |
| Posted: Thu Dec 18, 2003 3:49 pm |
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The last condition seems to state that all of the elements of the set
"Helen" <junciu@yahoo.com> wrote in message
news:ba3374ff.0312181023.8b3663e@posting.google.com...
Quote: Hi I am reading ttm's algebra book
it defines particial order as:
Given a set T and a relation >= on T, if the relation >= satisfies
the following conditions, then it is called partial ordering
a) t1>=t1 for all t1 \in T
b) t1 >= t2 and t2>=t3 => t1>=t3
c) t1>=t2 and t2>=t1 => t1=t2
and defines the total order by
Given a set T and a partial ordering >= on T, if the relation >=
satisfies
the following conditions (d), then it is called total ordering
d) for any t1,t2 \in T we always have either t1>=t2 or t2>=t1
I don't understand the difference between those two order
thanks a lot
If I understand correctly, then if you consider only the 2nd and 3rd
conditions for a moment and observe that they are propositions; i.e.
they are more fully stated as being:
b) IF t1 >= t2 and IF t2>=t3 THEN t1>=t3
c) IF t1 >= t2 and IF t2>=t1 THEN t1=t2
These conditions do not pre-suppose that the relation >= exists between
any two elements of the set, only that IF the relationship >= holds in
the manner described above then the rules hold.
Condtion d states that the relation holds between EVERY combination of
members in the set in the manner it (condition d) sets forth.
Condtion a holds for all members of the set but only in for each member
in relation to itself. i.e. for each t1, t1 >= t1, but it doesn't say
that for each t2 (every other element of the set other than t1) t1 >= t2
or t2>= t1 (which is condition d).
I think that one example of what it calls a partial ordering is within
the set of complex numbers. While one can order real numers such that 5
Quote: = 4, and one can order the imaginary numbers such that 5i >= 4i, this
ordering does not state that 5i >= 4. (While you can say that the
Magnitude of 5i is greater than or equal to the Magnitude of 4, you can
also say the the Magnitude of -8 is is greater than or equal to the
Magnitude of 5; But the way that the integers are normally ordered, -8
is less than or equal to 5 (it is to the left of 5 on the number line).
A simpler example might be the set of the integers between 1 and 10, and
a dog. One can define by statute (that is without explanation) the >=
relation on a dog such that condition a holds, that is that a dog >= a
dog. But if you don't define the full orderintg for the integers 1
through 10 that naturally exists to include a dog, then the set is only
partially ordered.
???? |
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| Virgil |
| Posted: Thu Dec 18, 2003 4:38 pm |
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In article <ba3374ff.0312181023.8b3663e@posting.google.com>,
junciu@yahoo.com (Helen) wrote:
Quote: Hi I am reading ttm's algebra book
it defines particial order as:
Given a set T and a relation >= on T, if the relation >= satisfies
the following conditions, then it is called partial ordering
a) t1>=t1 for all t1 \in T
b) t1 >= t2 and t2>=t3 => t1>=t3
c) t1>=t2 and t2>=t1 => t1=t2
and defines the total order by
Given a set T and a partial ordering >= on T, if the relation >= satisfies
the following conditions (d), then it is called total ordering
d) for any t1,t2 \in T we always have either t1>=t2 or t2>=t1
I don't understand the difference between those two order
thanks a lot
EXAMPLE: Sets are partially ordered, but not totally ordered, by
inclusion.
If I use 'se' to mean "is a subset of or is equal to" then
{} se {} and
{1} se {1} and
{2} se {2} and
{1,2} se {1,2} and
{} se {1} and
{} se {2} and
{1} se {1,2} and
{2} se {1,2} and
{} se {1,2}
so that 'se' is a partial order on { {}, {1}, {2}, {1,2} }.
You should verify for yourself that all the requirements of a
partial order are satisfied.
But neither
{1} se {2} nor
{2} se {1}, so
'se' is not a total order on { {}, {1}, {2}, {1,2} } |
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| |-|erc |
| Posted: Fri Dec 19, 2003 1:47 am |
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--------------------------------- <^> <(·¿·)> <^> -----------------------------------
"Helen" <junciu@yahoo.com> wrote in
Quote: Hi I am reading ttm's algebra book
it defines particial order as:
Given a set T and a relation >= on T, if the relation >= satisfies
the following conditions, then it is called partial ordering
a) t1>=t1 for all t1 \in T
b) t1 >= t2 and t2>=t3 => t1>=t3
c) t1>=t2 and t2>=t1 => t1=t2
and defines the total order by
Given a set T and a partial ordering >= on T, if the relation >= satisfies
the following conditions (d), then it is called total ordering
d) for any t1,t2 \in T we always have either t1>=t2 or t2>=t1
I don't understand the difference between those two order
thanks a lot
a partial order is a hierarchy of items, or a set of hierarchies.
a total order is a sequence.
partial :
* a
\
\
* --*
/
/
* b
notice either a<=b or b>=a, we don't know
total :
* - * - *
Herc |
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| |-|erc |
| Posted: Fri Dec 19, 2003 1:53 am |
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that would be irreflexive though,
reflexive :
* - * - *
*
Herc |
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| Matt |
| Posted: Sat Dec 20, 2003 9:28 pm |
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Helen wrote:
Quote: Hi I am reading ttm's algebra book
it defines particial order as:
Given a set T and a relation >= on T, if the relation >= satisfies
the following conditions, then it is called partial ordering
a) t1>=t1 for all t1 \in T
b) t1 >= t2 and t2>=t3 => t1>=t3
c) t1>=t2 and t2>=t1 => t1=t2
and defines the total order by
Given a set T and a partial ordering >= on T, if the relation >= satisfies
the following conditions (d), then it is called total ordering
d) for any t1,t2 \in T we always have either t1>=t2 or t2>=t1
I don't understand the difference between those two order
thanks a lot
Tell us whether you understand this much:
Every total order is a partial order. There are partial orders that are
not total orders. |
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