I've come up with the following k-map truth table for "D" input (~ is NOT).

~Z Z

~X~Y 0 0

~X Y 1 1

X Y 1 0

X~Y 1 1

I used k-maps to get the following simplified expression Hi=~XY+X~Z+X~Y. For the third term I use an invert after ~XY, but that makes one expression ORd with its inverse. This is giving, as expected 1's for all possible inputs since I have one term ORd with its inverse all possible inputs give a 1. I'm wondering where I went wrong, any help is much appreciated.