Logical Equivalence of Formulas

# Logical Equivalence of Formulas

Definition: Let $\varphi$ and $\psi$ be formulas that are composed of the same component statements. Then $\varphi$ and $\psi$ are said to be Logically Equivalent denoted $\varphi \Leftrightarrow \psi$ if every truth assignment to the component statements cause $\varphi$ and $\psi$ to have the same truth value. |

Let $P$ and $Q$ be statements and consider the formulas $P \leftrightarrow Q$ and $(P \rightarrow Q) \wedge (Q \rightarrow P)$. We construct the truth tables for both of these formulas:

$P$ | $Q$ | $P \leftrightarrow Q$ |
---|---|---|

T | T | T |

T | F | F |

F | T | F |

F | F | T |

$P$ | $Q$ | $P \rightarrow Q$ | $Q \rightarrow P$ | $(P \rightarrow Q) \wedge (Q \rightarrow P)$ |
---|---|---|---|---|

T | T | T | T | T |

T | F | F | T | F |

F | T | T | F | F |

F | F | T | T | T |

We compare the righthand columns of both of the truth tables and see they are the same. Therefore $P \leftrightarrow Q$ is logically equivalent to $(P \rightarrow Q) \wedge (Q \rightarrow P)$, that is:

(1)\begin{align} P \leftrightarrow Q \Leftrightarrow (P \rightarrow Q) \wedge (Q \rightarrow P) \end{align}