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Can the absorption law, $p \vee (p \wedge q) \equiv p$, be proved using other laws in propositional logic?

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closed as off-topic by B. Goddard, Namaste, user91500, kingW3, Shaun Sep 8 '17 at 17:51

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Let $p,q \in S$.

Then, as desired. :

$\begin{array}{l:l}\quad p\lor(p\land q) \\=(p\land \top)\lor(p\land q) & \top\text{ is the identity for }\land \\=p\land(\top\lor q)&\land \text{ distributes over }\lor \\ =p\land\top&\top\text{ is the annihilator for }\lor&\text{Identities of Boolean Operators are also Zeroes of their Dual} \\ =p&\top\text{ is the identity for }\land \end{array}$

The result: $p=p\lor (p\land q)$

By Dual Symmetry : $p=p\land (p\lor q)$

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