Rule of Distribution

Definition
Conjunction distributes over disjunction:


 * $$p \and \left({q \or r}\right) \dashv \vdash \left({p \and q}\right) \or \left({p \and r}\right)$$
 * $$\left({q \or r}\right) \and p \dashv \vdash \left({q \and p}\right) \or \left({r \and p}\right)$$

Disjunction distributes over conjunction:


 * $$p \or \left({q \and r}\right) \dashv \vdash \left({p \or q}\right) \and \left({p \or r}\right)$$
 * $$\left({q \and r}\right) \or p \dashv \vdash \left({q \or p}\right) \and \left({r \or p}\right)$$

Its abbreviation in a tableau proof is $$\textrm{Dist}$$.

Proof by Natural Deduction
This is proved by the Tableau method.

We can use the Rule of Commutation to show:

$$\left({q \or r}\right) \and p \vdash \left({q \and p}\right) \or \left({r \and p}\right)$$

$$\left({q \and p}\right) \or \left({r \and p}\right) \vdash \left({q \or r}\right) \and p$$

Proof by Truth Table
Let $$v: \left\{{p}\right\} \to \left\{{T, F}\right\}$$ be an interpretation for a boolean variable $$p$$.

$$\left({q \or r}\right) \and p \dashv \vdash \left({q \and p}\right) \or \left({r \and p}\right)$$ is demonstrated similarly.

Proof by Natural Deduction
This is proved by the Tableau method.

$$p \or \left({q \and r}\right) \vdash \left({p \or q}\right) \and \left({p \or r}\right)$$

$$\left({p \or q}\right) \and \left({p \or r}\right) \vdash p \or \left({q \and r}\right) $$

We can use the Rule of Commutation to show:

$$\left({q \and r}\right) \or p \vdash \left({q \or p}\right) \and \left({r \or p}\right)$$

$$\left({q \or p}\right) \and \left({r \or p}\right) \vdash \left({q \and r}\right) \or p$$

Proof by Truth Table
Let $$v: \left\{{p}\right\} \to \left\{{T, F}\right\}$$ be an interpretation for a boolean variable $$p$$.

$$\left({q \and r}\right) \or p \dashv \vdash \left({q \or p}\right) \and \left({r \or p}\right)$$ is demonstrated similarly.