Rule of Distribution

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Theorem

Conjunction Distributes over Disjunction

Conjunction is Left Distributive over Disjunction

Formulation 1

$p \land \paren {q \lor r} \dashv \vdash \paren {p \land q} \lor \paren {p \land r}$


Formulation 2

$\vdash \paren {p \land \paren {q \lor r} } \iff \paren {\paren {p \land q} \lor \paren {p \land r} }$


Conjunction is Right Distributive over Disjunction

Formulation 1

$\paren {q \lor r} \land p \dashv \vdash \paren {q \land p} \lor \paren {r \land p}$


Formulation 2

$\vdash \paren {\paren {q \lor r} \land p} \iff \paren {\paren {q \land p} \lor \paren {r \land p} }$

Disjunction Distributes over Conjunction

Disjunction is Left Distributive over Conjunction

Formulation 1

$p \lor \paren {q \land r} \dashv \vdash \paren {p \lor q} \land \paren {p \lor r}$

Formulation 2

$\vdash \paren {p \lor \paren {q \land r} } \iff \paren {\paren {p \lor q} \land \paren {p \lor r} }$


Disjunction is Right Distributive over Conjunction

Formulation 1

$\paren {q \land r} \lor p \dashv \vdash \paren {q \lor p} \land \paren {r \lor p}$

Formulation 2

$\vdash \paren {\paren {q \land r} \lor p} \iff \paren {\paren {q \lor p} \land \paren {r \lor p} }$

Their abbreviation in a tableau proof is $\text{Dist}$.


Sources