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Math Help - Tautology

  1. #1
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    Post Tautology

    Is this a Tautology...

    (A => (B v C)) V (A => (B ^ C))


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  2. #2
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    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<br />
T & T & T & T & T\\ T & T & F & T & T\\ T & F & T & T & T\\ T & F & F & F& F\\<br />
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<br />
T & T & T \\ T & T & T \\ T & T & T \\ F & {\bf{\color{red}F}} & F \\<br />
T & T & T \\ T & T & T \\ T & T & T \\ T & T & T \end{array}

    It is not a tautology.

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  3. #3
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    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<br />
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}}\\<br />
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}<br />
 \left[ {\neg A \vee (B \vee C)} \right] \vee \left[ {\neg A \vee \left( {\neg B \wedge \neg C} \right)} \right] \\ <br />
 \neg A \vee \left[ {(B \vee C) \vee \left( {\neg B \wedge \neg C} \right)} \right] \\ <br />
 \neg A \vee \left[ {(B \vee C) \vee \neg \left( {B \vee C} \right)} \right] \\ <br />
 \end{array}
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