logic - Prove satisfiability of F(v,&) by induction

This is my question.

1) we want to prove that the set F(v,&) only contains satisfiable formulas. Explain why it is difficult to show this with a straightforward induction.

2) Instead, we have to prove a stronger statement using structural induction:
Find a variable assignment ψ, which satisfies all formulas in F(V,&) and prove this property by induction.

I know that to say a formula is satisfiable at least one interpretation should be TRUE. My answer to this question is, By induction, we assume our hypothesis true and then we prove this for n+1 step. That means for all interpretations this set F(v,&) is true. As I think the difficulty is that we cannot show this for all the possible formulas in the set F(v,&). But I am not sure about my answer. Can anyone help to understand this question? And what does it mean by the stronger statement?

Answer

Ok, the claim is that any statement built up from atomic propsitions and using the operators $\land $ and $\lor$ is satiafiable. Let's try and prove this directly using structural induction:

Base: Take any atomic proposition, like $P$. Is it satisfiable? Yes, because we can set $P=T$, and of course do this for any atomic proposiion. Check!

Step: Assume that statements $\varphi$ and $\psi$ are satisfiable. Ok, so there is some truth-assignment $h_1$ that sets $\varphi$ to true and some truth-assignment $h_2$ that sets $\psi$ to true. Now, we want to show that there exists some truth-assignment $h$ that sets $\varphi \land \psi$ to true .... but now what? Obviously we would like to somehow combine $h_1$ and $h_2$ into $h$ ... But how? If, for example, $h_1(P) =True$ and $h_2(P)=False$ for some atomic $P$, then there is no way in which we can just 'merge' these truth-assignments together. In other words, we are really just stuck here!

OK, so let's try method 2. Let's consider the variable assignment $h$ that sets every atomic proposition to True. Now let's use structural induction to prove that for any statement $\varphi$ built up using $\land $ and $\lor$ we have that $h(\varphi) =True$:

Base: Take any atomic proposition $P$. By definition of $h$, $h(P) = True$. Check!

Step: Assume $h(\varphi)=True$ and $h(\psi) = True$ for arbitrary $\varphi$ and$\psi$. Then $h(\varphi \land \psi)=True$ and $h(\varphi \lor \psi)=True$, Check!

Ok, so we have proven that $h(\varphi)=True$ for any $\varphi$ built up from $\land$ and $\lor$. But that means that any such statement is satisfiable, and our proof is done: every statement built up from $\land$ and $\lor$ is satisfiable.