So I have another question,
A relation that is transitive and antisymmetric is called a preorder. Define the "threshold relation" on the positive real numbers in the following way: Fix a positive real number T. For x, y contained in the positive real numbers, we say that y - x > T.
a) Show that R is a preorder.
b) Why is R called a threshold relation?
c) In general, if R is an arbitrary preorder on a set A, show that S is a partial order on A where S is defined in the following way: for x,y in A, xSy if x=y or xRy.
So for a) I prove transitivity quite easily but for antisymmetry can I assume that this is vacuously true since the assumptions for antisymmetry (y -x > T and x - y > T) are clearly impossible?
For b) I belive it's a threshold since any difference less than T is not contained within the set?
And for c) x=y is reflexivity, right? We're adding the reflexivity property.
Thanks for the help!