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Inclusive or

Definition 2.3.2   The connector or is defined as follows: if $ p$ and $ q$ are two propositions, the $ p$    or $ q$ is true if, and only if, at least one of the components is true. This connector is called the inclusive or and is denoted by the symbol $ \vee$.

The inclusive or is described in the following table.
$ p$ $ q$ $ p \vee q$
T T T
T F T
F T T
F F F
For example:

Proposition 2.3.3 (Commutative Law)   For any two propositions $ p$ and $ q$, the statement $ p \vee q$ is logically equivalent to $ q \vee p$.

Proof. We use a truth table.
$ p$ $ q$ $ p \vee q$ $ q \vee p$
T T T T
T F T T
F T T T
F F F F
$ \qedsymbol$

Proposition 2.3.4 (Associative Law)   For any three propositions $ p$, $ q$ and $ r$, the statement $ p \vee (q
\vee r)$ is logically equivalent to $ (p \vee q ) \vee r$.

Proof. We use a truth table.
$ p$ $ q$ $ r$ $ q \vee r$ $ p \vee (q
\vee r)$ $ p \vee q$ $ (p \vee q ) \vee r$
T T T T T T T
T T F T T T T
T F T T T T T
T F F F T T T
F T T T T T T
F T F T T T T
F F T T T F T
F F F F F F F
The fifth column is identical to the last one, whence the claim. $ \qedsymbol$


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