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Conjunction

Definition 2.3.5        The connector and is defined as follows: if $ p$ and $ q$ are two propositions, the $ p$    and $ q$ is true if, and only if, both components are true. This connector is called the conjunction and is denoted by the symbol $ \wedge$.

The connector and is described in the following table.
$ p$ $ q$ $ p \wedge q$
T T T
T F F
F T F
F F F
For example:

Proposition 2.3.6 (Commutative Law)   For any two propositions $ p$ and $ q$, the statement $ p \wedge q$ is logically equivalent to $ q \wedge p$.

Proof.
$ p$ $ q$ $ p \wedge q$ $ q \wedge p$
T T T T
T F F F
F T F F
F F F F
As the two last colums are identical we are done. $ \qedsymbol$

Proposition 2.3.7 (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 \wedge r$ $ p \wedge (q \wedge r)$ $ p \wedge q$ $ (p \wedge q) \wedge r$
T T T T T T T
T T F F F T F
T F T F F F F
T F F F F F F
F T T T F F F
F T F F F F F
F F T F F F F
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: Distributivity Up: Logical connectors Previous: Inclusive or   Contents
root 2002-06-10