Let (A,<) be a strictly ordered set and b not in A. Define a relation in B=A {b} as follows:
x y if and only if (x,y in A and x<y) or (x in A and y=b).
Show that is a strict ordering of B and =<. (Intuitively, keeps A ordered in the same way as < and makes b greaters than every element of A.)
Can anyone help with this? Thanks!