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}