In this article, you will know the list of known logical equivalences and their corresponding truth table as a proof of them being a tautology. To understand more in-depth analysis of each of the identities, you can watch my YouTube channel.
Note that these logical identities are also found in Boolean algebra and each of the logical identity has its dual obtained by inverting the connectives and True or False if any.
Logical Equivalences with And, Or, Not
These are logical equivalence that use following connectives – conjunction, disjunction and negation.
Identity Laws
\begin{aligned} p \wedge T \equiv p\end{aligned} | T |  |
| T | T | T |
| F | T | F |
\begin{aligned}p \vee F \equiv p\end{aligned} | F |  |
| T | F | T |
| F | F | F |
Domination Laws
\begin{aligned}p \vee T \equiv T\end{aligned} | T |  |
| T | T | T |
| F | T | T |
\begin{aligned}p \wedge F \equiv F\end{aligned} | F |  |
| T | F | F |
| F | F | F |
Idempotent Laws
\begin{aligned}p \wedge p \equiv p\end{aligned} |  |
| T | T |
| F | F |
\begin{aligned} p \vee p \equiv p\end{aligned} |  |
| T | T |
| F | F |
Double Negation Law
\begin{aligned}\neg (\neg p) \equiv p\end{aligned} |  |  |
| T | F | T |
| F | T | F |
Commutative Laws
\begin{aligned}p \wedge q \equiv q \wedge p\end{aligned} |  |  |  |
| T | T | T | T |
| T | F | F | F |
| F | T | F | F |
| F | F | F | F |
\begin{aligned}p \vee q \equiv q \vee p\end{aligned} | |  |  |
| T | T | T | T |
| T | F | T | T |
| F | T | T | T |
| F | F | F | F |
Associative Laws
\begin{aligned}p \wedge (q \wedge r) \equiv (p \wedge q) \wedge r\end{aligned} | |  |  |  | |  |
| 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 |
\begin{aligned}p \vee (q \vee r) \equiv (p \vee q) \vee r\end{aligned} | | |  |  | |  |
| 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 |
Distributive Laws
\begin{aligned} p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)\end{aligned} | | | |  | |  |  |
| T | T | T | T | T | T | T | T |
| T | T | F | T | T | T | F | T |
| T | F | T | T | T | F | T | T |
| T | F | F | F | F | F | F | F |
| F | T | T | T | F | F | F | F |
| F | T | F | T | F | F | F | F |
| F | F | T | F | F | F | F | F |
| F | F | F | F | F | F | F | F |
\begin{aligned} p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)\end{aligned} | | | |  | |  |  |
| T | T | T | T | T | T | T | T |
| T | T | F | F | T | T | T | T |
| T | F | T | F | T | T | T | T |
| T | F | F | F | T | T | T | T |
| F | T | T | T | T | T | T | T |
| F | T | F | F | F | T | F | F |
| F | F | T | F | F | F | T | F |
| F | F | F | F | F | F | F | F |
Absorption Laws
\begin{aligned} p \wedge (p \vee q) \equiv p\end{aligned} | | |  |
| T | T | T | T |
| T | F | T | T |
| F | T | T | F |
| F | F | F | F |
\begin{aligned} p \vee (p \wedge q) \equiv p\end{aligned} | | |  |
| T | T | T | T |
| T | F | F | T |
| F | T | F | F |
| F | F | F | F |
Negation Laws
\begin{aligned}p \wedge \neg p \equiv F\end{aligned} | |  | F |
| T | F | F | F |
| F | T | F | F |
\begin{aligned}p \vee \neg p \equiv T\end{aligned} | |  | T |
| T | F | T | T |
| F | T | T | T |
De Morgan’s Laws
\begin{aligned}\neg(p \wedge q) \equiv \neg p \vee \neg q\end{aligned} | | |  |  |  |  |
| T | T | F | F | T | F | F |
| T | F | F | T | F | T | T |
| F | T | T | F | F | T | T |
| F | F | T | T | F | T | T |
\begin{aligned}\neg(p \vee q) \equiv \neg p \wedge \neg q\end{aligned} | | | |  |  |  |
| T | T | F | F | T | F | F |
| T | F | F | T | T | F | F |
| F | T | T | F | T | F | F |
| F | F | T | T | F | T | T |
