Algebra of Statements and Duality

The logical equivalences of the previous section obey a tidy set of algebraic laws, closely parallel to the laws of ordinary algebra and of set theory. Together they form the algebra of statements, and they let compound statements be simplified purely by rewriting, without building a truth table each time. Throughout, $\mathbf{t}$ denotes a statement pattern that is always true (a tautology) and $\mathbf{c}$ one that is always false (a contradiction).

The laws (each holds with $\wedge$ and $\vee$ interchanged, as noted below):

1. Idempotent: $p \wedge p \equiv p$, $\quad p \vee p \equiv p$.
2. Commutative: $p \wedge q \equiv q \wedge p$, $\quad p \vee q \equiv q \vee p$.
3. Associative: $(p \wedge q) \wedge r \equiv p \wedge (q \wedge r)$, $\quad (p \vee q) \vee r \equiv p \vee (q \vee r)$.
4. Distributive: $p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)$, $\quad p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)$.
5. Identity: $p \wedge \mathbf{t} \equiv p$, $\quad p \vee \mathbf{c} \equiv p$.
6. Domination: $p \vee \mathbf{t} \equiv \mathbf{t}$, $\quad p \wedge \mathbf{c} \equiv \mathbf{c}$.
7. Complement: $p \vee \sim p \equiv \mathbf{t}$, $\quad p \wedge \sim p \equiv \mathbf{c}$.
8. Involution (double negation): $\sim(\sim p) \equiv p$.
9. De Morgan: $\sim(p \wedge q) \equiv \sim p \vee \sim q$, $\quad \sim(p \vee q) \equiv \sim p \wedge \sim q$.
10. Absorption: $p \vee (p \wedge q) \equiv p$, $\quad p \wedge (p \vee q) \equiv p$. …