Preserving a propositional formula

Question

I know I must be getting stuck on notation. However, I'm having trouble following the logic in Example 1.2 in https://arxiv.org/pdf/cs/0611018.pdf. They define what preserving a propositional formula means.

So, I'm trying to understand a basic example. Take X = A or (Not B) or (Not C) as an example Horn Clause. Define the assignments f1 and f2 by f1(A) = f1(B) = f1(C) = true and f2(A) = f2(B) = f2(C) = false. Both f1 and f2 satisfy X.

What does it mean to say AND preserves X with respect f1 and f2?

Answer 1

In your example the assignment $f_1\wedge f_2$ is $(f_1\wedge f_2)(A)=f_1(A)\wedge f_2(A)=\text{True} \land \text{False} = \text{False}$. similarly: $(f_1\wedge f_2)(B)=(f_1\wedge f_2)(C)=\text{False}$ so actually $f_1\wedge f_2 \equiv f_2$.

As they define, AND is preserving $X$ if for every two assignments $f_1, f_2$ that satisfy $X$, the assignment $f_1 \wedge f_2$ (defined by $(f_1\wedge f_2)(v)=f_1(v)\wedge f_2(v)$) also satisfies $X$.