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**aman_cc** I picked up this book to read Set Theory. "Intro to Modern Set Theory - J Roitman"

It defines a partial order, $\displaystyle \leq$, on any set $\displaystyle S$ as follows:

For all $\displaystyle x,y,z \in S$

Axiom 1: $\displaystyle x \leq x$

Axiom 2: If $\displaystyle x \leq y$ and $\displaystyle y \leq x$, then $\displaystyle x=y$

Axiom 3: If $\displaystyle x \leq y$ and $\displaystyle y \leq z$, then $\displaystyle x \leq z$

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 '$\displaystyle \leq$' is a subset of SxS, such that:

1. $\displaystyle (x,x)$ in '$\displaystyle \leq$'

2. If $\displaystyle (x,y)$ and $\displaystyle (y,x)$ in '$\displaystyle \leq$', then $\displaystyle x=x$

3. If $\displaystyle (x,y)$ and $\displaystyle (y,z)$ in '$\displaystyle \leq$', then $\displaystyle (x,z) \in '\leq'$

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 $\displaystyle \leq$

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