Prove the logical equivalence of two formulas which use $\bigwedge$

Question

I have to argue whether the following statement holds. $X_i, Y_i \in A$ where $A$ is the set of atoms both formulas are defined over:

$$\bigwedge_{i=1}^n (X_i \vee Y_i) \models \bigwedge_{i=1}^n X_i \vee \bigwedge_{i=1}^n Y_i$$

I thought I could use induction as it is easy enough to anchor with $i=1$. But I am unsure how to continue the proof. Maybe there is a better way to solve this not using induction. I am open for any suggestions.

Answer 1

A couple of approaches for proving the statement false. Let all $X_i$ be $0$ or $1$. Let $Y_i$ be $\lnot X_i$. Then the left side is true, the right side is false.

If $n=2$ ad distribution law yields

$(X_1\lor Y_1)\land (X_2 \lor Y_2)=? (X_1 \land X_2)\lor(X_1 \land Y_2)\lor (Y_1 \land X_2) \lor (Y_1 \land Y_2)$

Can you take it from there?

Answer 2

This is false. Hint: Consider the case $n=2$.