Looking for some help with this problem, Thanks ahead of time

. Given the relation defined on N XN by (a, b) "<" (c, d) iffb<d.

(a). Why is the relation well-founded?

(b). What are the minimal elements?

(Rock)

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- Apr 30th 2008, 06:08 PMcrowned1relations
Looking for some help with this problem, Thanks ahead of time

. Given the relation defined on N XN by (*a, b*) "<" (*c, d*) iff*b*<*d*.

(a). Why is the relation well-founded?

(b). What are the minimal elements?

(Rock) - May 1st 2008, 03:21 AMMoo
Hello,

This relation is well-founded if there doesn't exist an infinite sequence ($\displaystyle x_n$) such as $\displaystyle (a, x_{n+1})<(a, x_n)$

Assuming that it's not well-founded, this means that there will always be $\displaystyle x \in \mathbb{N}$ such as $\displaystyle x<x_n \ , \ \forall x_n \in \mathbb{N}$

If $\displaystyle x_n=0$, it's a nonsense since there doesn't exist such an x.

Hence, the relation is well-founded...

Minimal elements will be any elements such as $\displaystyle (a_m,b_m)<(c,d) \ , \ \forall c, \ d \in \mathbb{N}$

This means that $\displaystyle a_m$ can be any element in $\displaystyle \mathbb{N}$

$\displaystyle b_m$ has to be the element such as $\displaystyle \forall d \in \mathbb{N}$, $\displaystyle b_m<d$

So logically, $\displaystyle b_m=0$ but I think there is a problem (Wait)