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 |