Totally ordered and Partially ordered Sets

[dpa]

Hi Everyone,

What are the difference between totally and partially ordered sets?
Any examples would help except the fact that one holds reflexivity and another totality. Clarification of this would also be fine.

Thank You

 

[Robert1986]

Consider the set: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. As you can see, there is a total order on this set defined by: a < b if a-b < 0. (This is just the regular ordering of integers.)

On the other hand, there is a partial order: a < b if and only if a divides b. So that, in this partial order, 5 < 10, as in the total order, but 5 is not less than 7 in this partial order.

 

[HallsofIvy]

Any "order relation" obeys the "transitive law": if a<b and b<c then a< c. A "total order" also obey the "trichotomy law": Given any a, b in the set, one and only one must be true- a< b, b< a, a= b. We can say that all member so the set are "comparable"- given any two distinct members, we can "compare" them, one is < the other.

The important thing for Robert1986's second example is not just that "5 is not less than 7" but also that "7 is not less than 5" while of course 5 is not equal to 7 so trichotomy does not hold.

A very important example of a partial order is "set inclusion". We define "A< B" if and only if A is a subset of B. Certainly, if A is a subset of B and B is a subset of C, then A is a subset of C. If, for example, our 'universal set' is the set of positive integers, A= {1, 2, 3} and B= {2, 3, 4}, A and B are certainly not equal but also neither is a subset of the other. None of "A= B", "A is a subset of B", or "B is a subset of A".