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:
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?
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
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?