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:

• The sentence ``Copernic was an astronom and Kikegaard was a philosoph'' is true because both components are true.
• The sentence ``Copernic was an astronom and Kant run a grocery shop'' is false because the second component is false.
• we cannot decide whether the sentence ``Copernic was an astronom and lived close to a grocery shop'' is true or not, because we have no possibility to check the truth value of the second component.

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. $\qedsymbol$