# Thread: 2 Boolean Algebra expressions

1. ## 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.

2. 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:

$\displaystyle \overline{P}$ for NOT P.

A+B for OR of A and B.

A.B for AND of A and B.

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

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

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

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

Hence $\displaystyle (P \implies \overline{Q}) + P.Q$ is a tautology.

3. 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.

4. You must supply the reasons according to your textbook/notes.
$\displaystyle \begin{gathered} \left( {P \Rightarrow \neg Q} \right) \vee \left( {P \wedge Q} \right) \hfill \\ \left( {\neg P \vee \neg Q} \right) \vee \left( {P \wedge Q} \right) \hfill \\ \neg P \vee \left[ {\neg Q \vee \left( {P \wedge Q} \right)} \right] \hfill \\ \neg P \vee \left( {\neg Q \vee P} \right) \wedge \underbrace {\left( {\neg Q \vee Q} \right)}_{TRUE} \hfill \\ \end{gathered}$
$\displaystyle \begin{gathered} \neg P \vee \left( {\neg Q \vee P} \right) \hfill \\ \neg Q \vee \underbrace {\left( {\neg P \vee P} \right)}_{TRUE} \hfill \\ TRUE \hfill \\ \end{gathered}$

5. May I ask if this is the solution to both or just the bottom one?

6. 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.

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

8. Originally Posted by Plato
$\displaystyle \begin{gathered} \left( {P \wedge Q} \right) \vee \left( {\neg P \wedge Q} \right) \hfill \\ \underbrace {\left( {P \vee \neg P} \right)}_{TRUE} \wedge Q \hfill \\ Q \hfill \\ \end{gathered}$
Thanks this helped alot, and I included the rules that you used, I am just having trouble understanding what you wrote for this:

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