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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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    $\begingroup$ There is no part of $(p \land q)$ that is outside of $p$ and all of $(p \land q)$ is in $p$, so $p$ on its own suffices. $\endgroup$ – turkeyhundt Jan 24 '15 at 4:03
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    $\begingroup$ If somebody says "I like both pizza and hamburgers, or I like pizza," you'll wonder why they didn't just say "I like pizza," which is the same thing, but put more simply. $\endgroup$ – user208259 Jan 24 '15 at 4:45
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From a Venn Diagram standpoint, all of $A$ plus any subset of $A$ will still just be $A$ enter image description here

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  • $\begingroup$ I get it @turkeyhundt :) . And Thanks also . But can there be another more intuitive explanation , which maybe doe snot make use of Truth Tables OR Venn Diagrams and the like . Just , wondering ... $\endgroup$ – pranav Jan 24 '15 at 4:09
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    $\begingroup$ Hmm, I can't come up with a great intuitive way right now. Just examples like, What is the set that is the union of the alphabet and the vowels. Or what is the union of the numbers from 1 to 10 and the numbers from 7 to 10. If you know one of the sets is completely in the larger set, the union of them will be the larger set. $\endgroup$ – turkeyhundt Jan 24 '15 at 4:12
  • $\begingroup$ Well , I get it @turkeyhundt . Thanks :) $\endgroup$ – pranav Jan 24 '15 at 4:13
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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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