# 2 Boolean Algebra expressions

• Jan 3rd 2009, 03:53 AM
rushhour
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:

http://e.imagehost.org/0078/boolean_expr.jpg
Just to clarify, I need help simplifying both expressions.
Thank you in advance if you can help.
• Jan 3rd 2009, 04:16 AM
Isomorphism
Quote:

Originally Posted by rushhour
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.
• Jan 3rd 2009, 04:30 AM
rushhour
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:

http://e.imagehost.org/0536/boolean_expr2.jpg

I hope you can help me format it in this type of way.
• Jan 3rd 2009, 05:35 AM
Plato
You must supply the reasons according to your textbook/notes.
$\begin{gathered}
\left( {P \Rightarrow \neg Q} \right) \vee \left( {P \wedge Q} \right) \hfill \\
\left( {\neg P \vee \neg Q} \right) \vee \left( {P \wedge Q} \right) \hfill \\
\neg P \vee \left[ {\neg Q \vee \left( {P \wedge Q} \right)} \right] \hfill \\
\neg P \vee \left( {\neg Q \vee P} \right) \wedge \underbrace {\left( {\neg Q \vee Q} \right)}_{TRUE} \hfill \\
\end{gathered}$

$\begin{gathered}
\neg P \vee \left( {\neg Q \vee P} \right) \hfill \\
\neg Q \vee \underbrace {\left( {\neg P \vee P} \right)}_{TRUE} \hfill \\
TRUE \hfill \\
\end{gathered}$
• Jan 3rd 2009, 06:41 AM
rushhour
May I ask if this is the solution to both or just the bottom one?
• Jan 3rd 2009, 08:02 AM
rushhour
Just wish to clarify that there are 2 problems I need help solving, the first is this:

http://e.imagehost.org/0720/boolean_expr3.jpg

the second one is this:

http://e.imagehost.org/0084/boolean_expr4.jpg
Thanks if you can help me simplify them.
• Jan 3rd 2009, 08:26 AM
Plato
$\begin{gathered}
\left( {P \wedge Q} \right) \vee \left( {\neg P \wedge Q} \right) \hfill \\
\underbrace {\left( {P \vee \neg P} \right)}_{TRUE} \wedge Q \hfill \\
Q \hfill \\
\end{gathered}$
• Jan 3rd 2009, 10:10 AM
rushhour
Quote:

Originally Posted by Plato
$\begin{gathered}
\left( {P \wedge Q} \right) \vee \left( {\neg P \wedge Q} \right) \hfill \\
\underbrace {\left( {P \vee \neg P} \right)}_{TRUE} \wedge Q \hfill \\
Q \hfill \\
\end{gathered}$

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
$\begin{gathered}
\left( {P \Rightarrow \neg Q} \right) \vee \left( {P \wedge Q} \right) \hfill \\
\left( {\neg P \vee \neg Q} \right) \vee \left( {P \wedge Q} \right) \hfill \\
\neg P \vee \left[ {\neg Q \vee \left( {P \wedge Q} \right)} \right] \hfill \\
\neg P \vee \left( {\neg Q \vee P} \right) \wedge \underbrace {\left( {\neg Q \vee Q} \right)}_{TRUE} \hfill \\
\end{gathered}$

$\begin{gathered}
\neg P \vee \left( {\neg Q \vee P} \right) \hfill \\
\neg Q \vee \underbrace {\left( {\neg P \vee P} \right)}_{TRUE} \hfill \\
TRUE \hfill \\
\end{gathered}$

Could you please state the rules you used for this if possible. Thanks.