I'm having some trouble sorting out what the definitions for R-minimal and R-smallest elements mean in a general sense. The definitions I'm using are as follows: b is a smallest element of B if (∀x∈B)(bRx), and b is a minimal element of B if (¬∃x∈B)(xRb∧x=b).
Say B = {1,2,3,4,5} and R = {(y,x)∈B×B|y>x}
If "b" is a number we are plugging in for y, then by the definition of smallest element it would seem that we need to find a b greater than every x∈B, so that we can satisfy the requirement that (∀x∈B)(bRx) or (∀x∈B)((b,x)∈R). This would seem impossible though, since it's never the case that b>b. So I would conclude that B has no smallest element.
For minimal elements, I would interpret the definition as looking for an element that cannot satisfy the requirement y>x unless y=x. This would seem to reverse things from looking for a largest number to looking for a smallest number. 0 would be the smallest element here, since 0>x cannot hold for any element in B.
I know that you have a smallest element if you have one minimal element, but I can't seem to see why 0 is the smallest element when thinking in terms of the definition for smallest element I gave above. Can someone please correct my reasoning?