Rule of Distribution
Jump to navigation
Jump to search
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 $\text{Dist}$.