Intuition for Absorption and Distribution Rules (of Replacement in Propositional Logic)

Question

$P ∧ (Q ∨ R) \equiv (P ∧ Q) ∨ (P ∧ R) \tag{1. Distributivity of ∧ over ∨}$
$P ∨ (Q ∧ R) \equiv (P ∨ Q) ∧ (P ∨ R) \tag{2. Distributivity of ∨ over ∧}$
$P ∨ (P ∧ Q) \equiv P \tag{3. ∨ Absorbs ∧}$
$P ∧ (P ∨ Q) \equiv P \tag{4. ∧ Absorbs ∨}$

What are the intuitions for the four Rules of Replacement above? I pursue only intuition;
so please do not rely on or use Truth Tables, Venn Diagrams, or any formal proofs.
This question also enquires about Rule 2 above.

_{Sources: p 49, Mathematical Proofs, 2nd ed. by Chartrand et al, $\qquad$ p 21, How to Prove It by Velleman.
I apprehended the names of the Absorption Laws herefrom and the Distribution Laws herefrom.}

Answer 1

I abbreviate foods as follows in my explanation, where $P :=$ I have a burger, $Q :=$ I have fish, and $R :=$ I have chips. I'll just explain the $\Rightarrow$ direction of each equivalence for now. In each case, keep in mind that "or" in math means inclusive or, so there is no need to ever say the phrase "or both."

$\huge{1.}$ If I have a burger with (ketchup or mustard) then either I have a burger with ketchup, or I have a burger with mustard. My burger must have at least one of the two condiments on it.

$\huge{2.}$ If I have a burger or (fish and chips) then statements
(a) "I have a burger or fish" $\qquad \qquad $ and $\qquad \qquad$ (b) "I have a burger or chips"
must be true. To see this, let's break $(2)$ down by cases.
Case 1: I have a burger. Then (a) and (b) are both true because of the burger, regardless of the fish and chips.
Case 2: I have fish and chips. Again (a) and (b) are both true because of the fish and the chips respectively, regardless of the burger.

Alternatively, note that from "I have a burger or (fish and chips)" we can conclude "I have a burger or fish"; this just weakens the second disjunct (fish and chips) by forgetting the about the chips. Formally, $P \vee (Q \wedge R) \implies P \vee Q$. Likewise we can conclude "I have a burger or chips", while forgetting about the fish. So we can conclude the conjunction: "I have a burger and fish, or I have a burger and chips."

$\huge{3.}$ If I have a burger or (a burger with cheese) then in any case I must have a burger. (I can't say for certain whether it has cheese on it, though.)

$\huge{4.}$ If I have a burger and (a burger or fish) then I must have a burger. (The fish is just a red herring.)

This is not a complete explanation, so if something still doesn't make sense, please ask.