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Distributivity

Proposition 2.3.8   For any three propositions $ p$, $ q$ and $ r$, the following hold:
  1. The statement $ p \vee (q \wedge r)$ is logically equivalent to $ (p \vee q ) \wedge (p \vee r)$.
  2. The statement $ p \wedge (q \vee r)$ is logically equivalent to $ (p \wedge q ) \vee (p \wedge r)$.

Proof. We use truth tables.
  1. $ p$ $ q$ $ r$ $ q \wedge r$ $ p \vee (q \wedge r)$ $ p \vee q$ $ p \vee r$ $ (p \vee q ) \wedge (p \vee r)$
    T T T T T T T T
    T T F F T T T T
    T F T F T T T T
    T F F F T T T T
    F T T T T T T T
    F T F F F T F F
    F F T F F F T F
    F F F F F F F F
  2. $ p$ $ q$ $ r$ $ q \vee r$ $ p \wedge (q \vee r)$ $ p \wedge q$ $ p \wedge r$ $ (p \wedge q ) \vee (p \wedge r)$
    T T T T T T T T
    T T F T T T F T
    T F T T T F T T
    T F F F F F F F
    F T T T F F F F
    F T F T F F F F
    F F T T F F F F
    F F F F F F F F
In both tables the fifth column and the eigth column are identical, whence the claims. $ \qedsymbol$


next up previous contents
Next: De Morgan Laws Up: Logical connectors Previous: Conjunction   Contents
root 2002-06-10