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Math Help - 2 Boolean Algebra expressions

  1. #1
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    2 Boolean Algebra expressions

    Hi

    I need help solving these two boolean algebra expressions. I need to simplify them using the rules of boolean algebra. The two expressions in question are:




    Just to clarify, I need help simplifying both expressions.
    Thank you in advance if you can help.
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  2. #2
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    Quote Originally Posted by rushhour View Post
    Hi

    I need help solving these two boolean algebra expressions. I need to simplify them using the rules of boolean algebra. The two expressions in question are:



    Just to clarify, I need help simplifying both expressions.
    Thank you in advance if you can help.
    Well I will follow different notation and I apologize for that...

    NOTATIONS I shall use:

    \overline{P} for NOT P.

    A+B for OR of A and B.

    A.B for AND of A and B.


    (P.Q) + (\overline{P} . Q) = (P + \overline{P}) . Q = 1.Q = Q

    P \implies Q is equivalent to \overline{P} + Q

    So (P \implies \overline{Q}) + P.Q = \overline{P} + \overline{Q} + P.Q = \overline{P} + \overline{Q} + P = 1

    Here I have used the Boolean relation X + \overline{X}.Y = X + Y

    Hence (P \implies \overline{Q}) + P.Q is a tautology.
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  3. #3
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    Thanks, but could you possibly convert that into my version, simply because I really have no idea how to follow what you have done! The type of way I do it is the following:




    I hope you can help me format it in this type of way.
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  4. #4
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    You must supply the reasons according to your textbook/notes.
    \begin{gathered}<br />
  \left( {P \Rightarrow \neg Q} \right) \vee \left( {P \wedge Q} \right) \hfill \\<br />
  \left( {\neg P \vee \neg Q} \right) \vee \left( {P \wedge Q} \right) \hfill \\<br />
  \neg P \vee \left[ {\neg Q \vee \left( {P \wedge Q} \right)} \right] \hfill \\<br />
  \neg P \vee \left( {\neg Q \vee P} \right) \wedge \underbrace {\left( {\neg Q \vee Q} \right)}_{TRUE} \hfill \\ <br />
\end{gathered}
    \begin{gathered}<br />
  \neg P \vee \left( {\neg Q \vee P} \right) \hfill \\<br />
  \neg Q \vee \underbrace {\left( {\neg P \vee P} \right)}_{TRUE} \hfill \\<br />
  TRUE \hfill \\ <br />
\end{gathered}
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  5. #5
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    May I ask if this is the solution to both or just the bottom one?
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  6. #6
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    Just wish to clarify that there are 2 problems I need help solving, the first is this:



    the second one is this:


    Thanks if you can help me simplify them.
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  7. #7
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    \begin{gathered}<br />
  \left( {P \wedge Q} \right) \vee \left( {\neg P \wedge Q} \right) \hfill \\<br />
  \underbrace {\left( {P \vee \neg P} \right)}_{TRUE} \wedge Q \hfill \\<br />
  Q \hfill \\ <br />
\end{gathered}
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  8. #8
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    Quote Originally Posted by Plato View Post
    \begin{gathered}<br />
  \left( {P \wedge Q} \right) \vee \left( {\neg P \wedge Q} \right) \hfill \\<br />
  \underbrace {\left( {P \vee \neg P} \right)}_{TRUE} \wedge Q \hfill \\<br />
  Q \hfill \\ <br />
\end{gathered}
    Thanks this helped alot, and I included the rules that you used, I am just having trouble understanding what you wrote for this:

    Quote Originally Posted by Plato View Post
    \begin{gathered}<br />
  \left( {P \Rightarrow \neg Q} \right) \vee \left( {P \wedge Q} \right) \hfill \\<br />
  \left( {\neg P \vee \neg Q} \right) \vee \left( {P \wedge Q} \right) \hfill \\<br />
  \neg P \vee \left[ {\neg Q \vee \left( {P \wedge Q} \right)} \right] \hfill \\<br />
  \neg P \vee \left( {\neg Q \vee P} \right) \wedge \underbrace {\left( {\neg Q \vee Q} \right)}_{TRUE} \hfill \\ <br />
\end{gathered}
    \begin{gathered}<br />
  \neg P \vee \left( {\neg Q \vee P} \right) \hfill \\<br />
  \neg Q \vee \underbrace {\left( {\neg P \vee P} \right)}_{TRUE} \hfill \\<br />
  TRUE \hfill \\ <br />
\end{gathered}
    Could you please state the rules you used for this if possible. Thanks.
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