1. ## Tautology

Is this a Tautology...

(A => (B v C)) V (A => (¬B ^ ¬C))

*

2. Hello, *skywalker*!

Is this a Tautology?

. . $[A \to (B \vee C)] \;\vee \;[A \to (\sim B \:\wedge \sim C)]$

The two "sides" are logically equivalent (by DeMorgan's Law)
. . but they are not neccesarily always true.

If we set up the standard truth table, we have:

. . $\begin{array}{ccccc}A & B & C & [A\to(B \vee C)] & [A \to (\sim B \wedge \sim C)]\\ \hline
T & T & T & T & T\\ T & T & F & T & T\\ T & F & T & T & T\\ T & F & F & F& F\\
F & T & T & T & T\\ F & T & F & T & T\\ F & F & T & T & T\\ F & F & F & T & T\end{array}$

The left side and right side form this truth table:

. . $\begin{array}{ccc}\text{Left} & \vee & \text{Right} \\ \hline
T & T & T \\ T & T & T \\ T & T & T \\ F & {\bf{\color{red}F}} & F \\
T & T & T \\ T & T & T \\ T & T & T \\ T & T & T \end{array}$

It is not a tautology.

3. It is a tautology. There is a mistake in the truth table.

$\begin{array}{ccccc}A & B & C & [A\to(B \vee C)] & [A \to (\sim B \wedge \sim C)]\\ \hline
T & T & T & T & {\bf{\color{red}F}}\\ T & T & F & T & {\bf{\color{red}F}}\\ T & F & T & T & {\bf{\color{red}F}}\\ T & F & F & F& {\bf{\color{red}T}}\\
F & T & T & T & T\\ F & T & F & T & T\\ F & F & T & T & T\\ F & F & F & T & T\end{array}$

$\begin{array}{l}
\left[ {\neg A \vee (B \vee C)} \right] \vee \left[ {\neg A \vee \left( {\neg B \wedge \neg C} \right)} \right] \\
\neg A \vee \left[ {(B \vee C) \vee \left( {\neg B \wedge \neg C} \right)} \right] \\
\neg A \vee \left[ {(B \vee C) \vee \neg \left( {B \vee C} \right)} \right] \\
\end{array}$