Why is the Absorption Law of Propositional Logic so ?
p $\lor (p \land q) \equiv$ p
Would appreciate an intuitive explanation and not one using a Truth Table
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From a Venn Diagram standpoint, all of $A$ plus any subset of $A$ will still just be $A$
When we have an or statement either the left hand side or the right needs to be true for it to be true. In this case we have p on the left and $ (p \land q)$ on the right. But $ (p \land q)$ can only be true if p and q are both true. But if p is true we can immediately deduce that the or statement is true without knowing the value of q (since the LHS will be true). We can also immediately tell that if p were false then both the left and right would be false; therefore, the statement is false. So basically all the absorbtion law is saying is that the truth vaule of this or statement is only dependent on p, we do not need q to determine whether it is true or not.
As you can see in this link (Prove the absorption law in propositional logic)
p ∨ (p ∧ q) ≡ p ∧ (p ∨ q) ≡ p
explained by derivation with Morgan's Laws
T stands for True
p ≡ (p ∧ T) (by inverting null element)
p ∨ (p ∧ q) ≡ (p ∧ T) ∨ (p ∧ q)
(p ∧ T) ∨ (p ∧ q) ≡ p ∧ (T ∨ q) (by inverting distribution law)
(T ∨ q) ≡ T (by absorbent element)
p ∧ (T ∨ q) ≡ p ∧ T
p ∧ T ≡ p (by null element)