# Partial Orders and Cardinality

• Nov 6th 2007, 02:32 PM
kevi555
Partial Orders and Cardinality
Just a question I was having some trouble with, any thoughts would be appreciated.

- Suppose S is a (finite) set containing at least two elements for A,B that are elements of P(S), we define A is a poset on B to mean |A| =< |B|. Is the relation a partial order on P(S) and explain.

Thanks!
• Nov 6th 2007, 02:40 PM
Jhevon
Quote:

Originally Posted by kevi555
Just a question I was having some trouble with, any thoughts would be appreciated.

- Suppose S is a (finite) set containing at least two elements for A,B that are elements of P(S), we define A is a poset on B to mean |A| =< |B|. Is the relation a partial order on P(S) and explain.

Thanks!

to determine if we have a partial ordering, we must verify that the set has three properties: reflexivity, anti-symmetry and transitivity. that is, we must show that:

1. $\displaystyle A \le A$ for all $\displaystyle A \in P(S)$

2. For all $\displaystyle A,B \in P(S)$, If $\displaystyle A \le B$ and $\displaystyle B \le A$ then $\displaystyle A = B$

3. For all $\displaystyle A,B,C \in P(S)$, If $\displaystyle A \le B$ and $\displaystyle B \le C$ then $\displaystyle A \le C$

and it seems that you have defined $\displaystyle A \le B$ to mean $\displaystyle |A| \le |B|$

by the way, what does "poset" mean?
• Nov 6th 2007, 02:50 PM
kevi555
poset meaning Partial ordered set...its symbol is the curly =< sign.
• Nov 6th 2007, 02:58 PM
Jhevon
Quote:

Originally Posted by kevi555
poset meaning Partial ordered set...its symbol is the curly =< sign.

i don't get what you are saying. i'm not familiar with the symbol you are talking about. you mean the less than or equal sign with a wavy line under it? in any case, i think that's just giving us a criterion by which to compare any two elements in the set in terms of the $\displaystyle \le$ relation.

i stick by my answer. check if the properties i've mentioned hold.
• Nov 6th 2007, 03:08 PM
Plato
Quote:

Originally Posted by kevi555
Suppose S is a (finite) set containing at least two elements for A,B that are elements of P(S), we define A is a poset on B to mean |A| =< |B|. Is the relation a partial order on P(S) and explain.

It appears that you are confused about the terminology of ‘posets’.
I think what the problem surely must be is the following
Suppose S is a (finite) set containing at least two elements, define a relation on P(S) by this ARB if and only if |A|<|B|. Now the question is if (P(S), R) is a partial ordering or a poset?
Or, is R a partial order on P(S)?
Now clearly ordinary comparison of counting numbers is both reflexive and transitive.