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Partial order vs. total order


Partial order vs. total order

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ta.bu.shi.da.yu
ta.bu.shi.da.yu
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Hi folks,

I've been trying to understand transactions better, so I've started reading Transactional Information Systems by Morgan-Kaufman Press.

Unfortunately, I've gotten stuck in the second chapter, where they differentiate between total and partial orders.

It states that a relation R that is a proper subset of the cartesian product of set A is a partial order on A if it is reflexive, antisymmetric and transitive.

So far so good. Now total order is different only in the respect that the relation R is not reflexive but total (aRb ∨ bRa).

Where I'm getting stuck is to find an example of a partial order that is NOT a total order. Could anyone help?

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Tom Thomson
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ta.bu.shi.da.yu (10/25/2010)
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Where I'm getting stuck is to find an example of a partial order that is NOT a total order. Could anyone help?

An easy example is a set of places ordered by average daily temperature: if there are two with the same average daily temperature, the order is partial (and if not, it is total).

Another is a set of points in a circular area, ordered by distance from the centre - some will be further away than others, but there can be two different points the same distance from the centre so again the order is partial.

For a different sort of example take the set of all subsets of a given set, ordered by the "is a proper subset of" relationship. If has the original set has two or more members, this is a partial order. For example take the set {0,1}; the complete set of subsets is {},{0},{1},{0,1} and the proper subset relation has
{}<{0} and {}<{1} and {}<{0,1}
{0}<{0,1}
{1}<{0,1}
but neither {0}<{1} nor {1}<{0}

Tom

ta.bu.shi.da.yu
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Tom.Thomson (10/26/2010)
ta.bu.shi.da.yu (10/25/2010)
...
Where I'm getting stuck is to find an example of a partial order that is NOT a total order. Could anyone help?

An easy example is a set of places ordered by average daily temperature: if there are two with the same average daily temperature, the order is partial (and if not, it is total).

Another is a set of points in a circular area, ordered by distance from the centre - some will be further away than others, but there can be two different points the same distance from the centre so again the order is partial.

For a different sort of example take the set of all subsets of a given set, ordered by the "is a proper subset of" relationship. If has the original set has two or more members, this is a partial order. For example take the set {0,1}; the complete set of subsets is {},{0},{1},{0,1} and the proper subset relation has
{}<{0} and {}<{1} and {}<{0,1}
{0}<{0,1}
{1}<{0,1}
but neither {0}<{1} nor {1}<{0}


OH... now I get it! Thanks Tom.

So basically, their is a partial order when it comes to a set of read/write operations... if there is a read and then write operation, or a number of write operations then these need to be ordered to satisfy the ACID principle.

Hope I'm explaining myself well enough. This is quite new to me, and I've only recently learned enough set theory for this to make sense to me :-)

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