Let $X$ be a projective, noetherian $k$-scheme for an algebraically closed field $k$ of characteristic zero. Let $Y\_1,...,Y\_r$ be locally closed subschemes (open subschemes of closed subschemes) of $X$. Does the scheme structure on $X$ necessarily induce a scheme structure on $Y\_1 \cup ... \cup Y\_r$? 



Is finite union of locally closed subscheme, a scheme
You just discovered *constructible sets*! 


It is really easy to give counter-examples to your suggestion (as *Ja ok* already has), but here is a general idea:
Take your favorite locally closed but neither open nor closed subscheme of your favorite irreducible scheme. Then prove (as a homework) that its complement is a union of locally closed subschemes (hint: an open and a closed), but itself is not locally closed (and hence not a subscheme). 


