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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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3 Answers

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From a Venn Diagram standpoint, all of $A$ plus any subset of $A$ will still just be $A$

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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.

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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)

so

p ∨ (p ∧ q) ≡ (p ∧ T) ∨ (p ∧ q)

(p ∧ T) ∨ (p ∧ q) ≡ p ∧ (T ∨ q) (by inverting distribution law)

(T ∨ q) ≡ T (by absorbent element)

so

p ∧ (T ∨ q) ≡ p ∧ T

p ∧ T ≡ p (by null element)

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