# Thread: Sets and Relations Question

1. ## Sets and Relations Question

Let A be a subset of Z (integers) squared and, for (a,b),(c,d) in A, define (a,b) <= (c,d) if and only if a <= c and d <= b.

A) Show that (A, <=) is a partially ordered set.
B) Is (A, <=) totally ordered? Explain

I am pretty confused in tackling this question, help is appreciated

2. For lexigraphic ordering shouldn't it be that (a,b) <= (c,d) iff a<=c and b<= d? How does this change the result in proving a PO set? And how do I prove that this is a PO set?

3. Well take each of the three properties, check each out with the change.
reflexive, anti-symmetric, and transitive.

4. So therefore,
A is reflexive because a <= a
A is antisymmetric because if a <= c, c > a
A is transitive because a <= c and a <= d therefore a <= d

Is this correct or am I not showing it properly?

5. Originally Posted by kanoro
So therefore,
A is reflexive because a <= a
A is antisymmetric because if a <= c, c > a
A is transitive because a <= c and a <= d therefore a <= d
Is this correct or am I not showing it properly?
Those are not pairs. This question is about ordering on a set of pairs.

Are these true?
$\begin{gathered}
\left( {x,y} \right) \prec \left( {x,y} \right) \hfill \\
\left( {x,y} \right) \prec \left( {w,z} \right) \wedge \left( {w,z} \right) \prec \left( {x,y} \right)\, \Rightarrow \,\left( {x,y} \right) = \left( {w,z} \right) \hfill \\
\left( {x,y} \right) \prec \left( {w,z} \right) \wedge \left( {w,z} \right) \prec \left( {s,t} \right)\, \Rightarrow \,\left( {x,y} \right) \prec \left( {s,t} \right) \hfill \\
\end{gathered}$

6. Okay so the top one (x,y) < (x,y) . I don't think this would be true? Maybe I am just confused on the notation. The way I am going about it is that these would seem to be equal

7. Also, how do the "define (a,b) <= (c,d) iff (a<=c) and d<=b" come into play? Where do account for a being less than or equal to c and d being less than or equal to b? I am still unsure how to show P.O. Anymore help is appreciated.

8. I don’t think that you understand this relation.
I will show you the transitive property.

Suppose that $\left( {a,b} \right) \prec \left( {c,d} \right)~\&~ \left( {c,d} \right) \prec \left( {e,f} \right)$.

That means that $a\le c~\&~d\le b$ and $c\le e~\&~f\le d$.

That gives $a\le c\le e~\&~f\le d\le b$ so $\left( {a,b} \right) \prec \left( {e,f} \right)$.