### Proposition Theorems

Propositions can be simplified to make them simpler to understand.

• Idempotence: $p\vee p \equiv p$ and $p\wedge p \equiv p$.
• Excluded middle: $p\vee p \equiv T$ and $p\wedge p \equiv F$
• Identity: $p\vee F \equiv p$ and $p\wedge T \equiv p$.
• Strictness: $p\vee T \equiv T$ and $p\vee F \equiv F$.
• Double negation $\lnot \lnot p \equiv p$

This is easily provable: construct truth tables with rows for $p=F$ and $p=T$.

Associativity
$(p\vee q) \vee r \equiv p \vee (q \vee r)$
$(p\wedge q) \wedge r \equiv p \wedge (q \wedge r)$

Communtativity
$p \vee q \equiv q \vee p$
$p \wedge q \equiv q \wedge p$

Distributivity
$p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)$
$p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)$

de Morgan
$\lnot (p \vee q) \equiv (\lnot p) \wedge (\lnot q)$
$\lnot (p \wedge q) \equiv (\lnot p) \vee (\lnot q)$

There is a convention of priority among connectives, together with the associativity of $\vee$ and $\wedge$, reduces the need for brackets. The list below is order from the highest priority to lowest:

1. $\lnot$ not
2. $\wedge$ and
3. $\vee$ or