Htykuut

I'm trying to prove partial order for a set, but I can't seem to be able to fully understand what I'm doing and weather I'm doing it right at all. I don't think I understand how to work with all the conditions between the two partial oders when proving antisymmetry and transitivity because of all the conditions I have to work with to get there.

The exercise is as follows:

Let X be a partial order with the sign $$≤_X$$ and Y be a partial order with the sign $$≤_Y$$ Prove that the following is a partial order for $$Xtimes Y$$ given that: $$(x_1,y_1 ) le (x_2,y_2 ) iff (x_1 le_X x_2 text{ and } x_2 not le_X x_1)text{ or }(x_1 =_X x_2 text { and } y_1 le_Y y_2)$$

I preassume that $=_X$ denotes the equality relation on $X$.

Then relation $leq$ on $Xtimes Y$ is defined by:$$(x_1,y_1)leq(x_2,y_2)iff [x_1<_X x_2text{ or }(x_1=x_2text{ and }y_1leq_Y y_2)]$$where $x_1<_X x_2$ abbreviates $x_1leq_X x_2text{ and }x_1neq x_2$.

Based on $x_1=x_1text{ and }y_1=y_1$ we then find that $(x_1,y_1)leq(x_1,y_1)$, proving reflexivity.

In order to prove transitivity let $(x_1,y_1)leq(x_2,y_2)$ and $(x_2,y_2)leq(x_3,y_3)$.

Now discern the following cases:

• 
  $x_1<_X x_2$ and $x_2<_Xx_3$ leading to $x_1<_X x_3$


• 
  $x_1<_X x_2$ and $[x_2=x_3text{ and }y_2leq_Y y_3]$ leading to $x_1<_X x_3$


• 
  $[x_1=x_2text{ and }y_1leq_Y y_2]$ and $x_2<_X x_3$ leading to $x_1<_X x_3$


• 
  $[x_1=x_2text{ and }y_1leq_Y y_2]$ and $[x_2=x_3text{ and }y_2leq_Y y_3]$ leading to $[x_1=x_3text{ and }y_1leq_Y y_3]$
  

In each case the conclusion $(x_1,y_1)leq(x_3,y_3)$ is justified, so this proves transitivity.

Now it remains to prove antisymmetry, and for that let $(x_1,y_1)leq(x_2,y_2)$ and $(x_2,y_2)leq(x_1,y_1)$.

On base of this it must be shown that $(x_1,y_1)=(x_3,y_3)$.

As above discern $4$ cases and give it a try yourself.

Let me know if you run into troubles.