I picked up this book to read Set Theory. "Intro to Modern Set Theory - J Roitman"
It defines a partial order,
, on any set
as follows:
For all
Axiom 1:
Axiom 2: If
and
, then
Axiom 3: If
and
, then
Now I have a few questions:
Q1. I'm not comfortable with the definition above. What is a 'partial order'? It has to some object which we should have defined earlier? (I mean, for e.g. a function from A to B is defines as a subset of the product set, AxB)
So I tried to define it like that:
Partial Order '
' is a subset of SxS, such that:
1.
in '
'
2. If
and
in '
', then
3. If
and
in '
', then
Is the above good?
Excellent. This is exactly what is meant by order: it is a relation and thus a subset of the corresponding cartesian product.
Q2. I think there is another requirement
If x and y are not equal, then exactly one of the (x,y) OR (y,x) should belong to
No, this is not required unless the partial order is complete.
Where is this covered? Is it a theorem - if yes, can someone help me to prove it?
Q3. The book I am referring isn't too convincing (for me). Is there a better text which anyone can recommend me?
You may want to try the classic texts by Church, or Halmos, or Monk. You can also search in google "set theory": sometimes you can find surprisingly nice and well-written course notes from teacher.
Tonio
Thanks