Propositional calculus-Valuation of a group of propositions

Question

I've encountered these two statements below, and I fail to understand why one of them is true and the other isn't. Let $X_1,X_2$ be groups of propositions,and $Z_1,Z_2$ be their satisfying groups of valuation then: $Z_1 \cap Z_2$ satisfies $X_1 \cap X_2$ (is false according to the text book),the other statement is $Z_1 \setminus Z_2$ satisfies $X_1 \setminus X_2$ (which is true according to the text book). Every counter argument I tried to come up with also counters the "true" statement so I'm lost and would gladly use some help.