Is $p \lor (q \land r)\equiv p \land (q \lor r)$? Before writing, I'm not good at English. As I developed the law of distribution, I felt strange.
$$\begin{align}
p \land (q \lor r)& \equiv (p \land q) \lor (p \land r)\\
&\equiv X \lor (p\land r)\tag{$X=p \land q$}\\
&\equiv (X \lor p) \land (X \lor r)\\
&\equiv [(p \land q) \lor p] \land [(p \land q) \lor r]\\
&\equiv (p \lor p) \land (p \lor q) \land (p \lor r) \land (q \lor r)\\
&\equiv p \land (q \lor r) \land (p \lor q) \land (p \lor r)\\
&\equiv [p \land (q \lor r)] \land p \lor (q \land r)\end{align}
$$
When called $p \land (q \lor r)=A,\ p \lor (q \land r)=B.$
I have never heard of $A≡A \land B$. (When they are not the same)
So I developed $p \lor (q \land r)$
$$\begin{align}
p \lor (q \land r) &\equiv (p \lor q) \land (p \lor r)\\
&\equiv Y \land (p \lor r)\ \ \ \ (let\ Y=p \lor q)\\
&\equiv (Y \land p) \lor (Y \land r)\\
&\equiv [(p \lor q) \land p] \lor [(p \lor q) \land r]\\
&\equiv (p \land p) \lor (p \land q) \lor (p \land r) \lor (q \land r)\\
&\equiv [p \lor (q \land r)] \lor [p \land (q \lor r)]\end{align}
$$
$A\equiv A \land B,\ B\equiv A \lor B$
I don't know if they are equal or if I made a mistake.

Consider when $p$ is true but both $q$ and $r$ are false.
Here is a link to a useful online tool.
