The half blood prince

in harry potter and the half blood prince harry finds a book and on the bottom it states ~this is property of the half blood prince~ hence the name of the book. now what disturbs me is that if the half blood prince (snape) went to hogwarts how did he get the name the half blood prince. Snape of course is the half blood prince BUT snape is not a half blood. snape and narcrissa are related (i think) and in the order of the pheonix suris stated that all pure bloods are related and narcrissa is malfoys mum so of course that make narcrissa a pure blood. malfoy + narcrissa +snape+ = pure blood.