Example

Truth table for ${\displaystyle p\to (q\to p)}$

${\displaystyle p\,\!}$	${\displaystyle q\,\!}$	${\displaystyle q\to p}$	${\displaystyle p\to (q\to p)}$
T	T	T	T
T	F	T	T
F	T	F	T
F	F	T	T

Definition

A formula ${\displaystyle \theta \,\!}$ of propositional logic is a tautology if only T's occur in the ${\displaystyle \theta \,\!}$ column of the truth table.

Examples

Truth table for ${\displaystyle (p\to \lnot p)\to \lnot p=\theta }$

${\displaystyle p\,\!}$	${\displaystyle \lnot p}$	${\displaystyle p\to \lnot p}$	${\displaystyle \theta \,\!}$
T	F	F	T
F	T	T	T

Truth table for ${\displaystyle (p\to (q\leftrightarrow \lnot q))\to \lnot p=\theta }$

${\displaystyle p\,\!}$	${\displaystyle q\,\!}$	${\displaystyle \lnot q}$	${\displaystyle q\leftrightarrow \lnot q}$	${\displaystyle p\to (q\leftrightarrow \lnot q)}$	${\displaystyle \lnot p}$	${\displaystyle \theta \,\!}$
T	T	F	F	F	F	T
T	F	T	F	F	F	T
F	T	F	F	T	T	T
F	F	T	F	T	T	T

Truth table for ${\displaystyle (\lnot p\vee q)\to (p\to q)=\theta }$

${\displaystyle p\,\!}$	${\displaystyle q\,\!}$	${\displaystyle \lnot p}$	${\displaystyle \lnot p\vee q}$	${\displaystyle p\to q}$	${\displaystyle \theta \,\!}$
T	T	F	T	T	T
T	F	F	F	F	T
F	T	T	T	T	T
F	F	T	T	T	T

Definition

The proposition formulas ${\displaystyle \varphi \,\!}$ and ${\displaystyle \theta \,\!}$ are tautologically equivalent if ${\displaystyle \varphi \leftrightarrow \theta }$ is a tautology.

Examples

Contraposition: ${\displaystyle p\to q=\theta }$ is tautologically equivalent to ${\displaystyle \lnot q\to \lnot p=\varphi }$.

${\displaystyle p\,\!}$	${\displaystyle q\,\!}$	${\displaystyle \lnot q}$	${\displaystyle \lnot p}$	${\displaystyle \theta \,\!}$	${\displaystyle \varphi \,\!}$	${\displaystyle \theta \leftrightarrow \varphi }$
T	T	F	F	T	T	T
T	F	T	F	F	F	T
F	T	F	T	T	T	T
F	F	T	T	T	T	T

de Morgan's Law I: ${\displaystyle \lnot (p\lor q)=\theta }$ is tautologically equivalent to ${\displaystyle \lnot p\land \lnot q=\varphi }$.

${\displaystyle p\,\!}$	${\displaystyle q\,\!}$	${\displaystyle \lnot p}$	${\displaystyle \lnot q}$	${\displaystyle p\lor q}$	${\displaystyle \theta \,\!}$	${\displaystyle \varphi \,\!}$	${\displaystyle \theta \leftrightarrow \varphi }$
T	T	F	F	T	F	F	T
T	F	F	T	T	F	F	T
F	T	T	F	T	F	F	T
F	F	T	T	F	T	T	T

de Morgan's Law II: ${\displaystyle \lnot (p\land q)=\theta }$ is tautologically equivalent to ${\displaystyle \lnot p\lor \lnot q=\varphi }$. Truth table for Assignment #1