Question

Determine the equivalent form of the Boolean expression \[\left( p\wedge q \right)\vee \left( p\vee \sim q \right)\wedge \left( \sim p\wedge \sim q \right)\].

(a) \[p\wedge \left( \sim q \right)\]
(b) \[p\vee \left( \sim q \right)\]
(c) \[\left( \sim p \right)\wedge \left( \sim q \right)\]
(d) \[p\wedge q\]

Answer 1

Hint: In this question, we have to determine the equivalent form of the Boolean expression \[\left( p\wedge q \right)\vee \left( p\vee \sim q \right)\wedge \left( \sim p\wedge \sim q \right)\]. Now we will use the following properties of Boolean expression of statements namely \[p\] , \[q\] and \[r\]. We have the distributive properties \[p\wedge \left( q\vee r \right)=\left( p\wedge q \right)\vee \left( p\wedge r \right)\] and \[p\vee \left( q\wedge r \right)=\left( p\vee q \right)\wedge \left( p\vee r \right)\] .
We will also use the De-Morgan’s law which states that \[\sim \left( p\vee q \right)=\left( \sim p \right)\wedge \left( \sim q \right)\] and vice versa in order to get the desired expression from \[\left( p\wedge q \right)\vee \left( p\vee \sim q \right)\wedge \left( \sim p\wedge \sim q \right)\].

Complete step-by-step answer:
We are given with a Boolean expression \[\left( p\wedge q \right)\vee \left( p\vee \sim q \right)\wedge \left( \sim p\wedge \sim q \right)...........(1)\].
Since we know the following properties of Boolean expression of statements namely \[p\] , \[q\] and \[r\]. We have the distributive properties \[p\wedge \left( q\vee r \right)=\left( p\wedge q \right)\vee \left( p\wedge r \right)\] and \[p\vee \left( q\wedge r \right)=\left( p\vee q \right)\wedge \left( p\vee r \right)\] .
We will first evaluate the expression \[\left( p\wedge q \right)\vee \left( p\vee \sim q \right)\] by using the above mentioned distributive laws.
We now have
\[\left( p\wedge q \right)\vee \left( p\vee \sim q \right)=\left[ \left( p\vee \left( p\vee \sim q \right) \right)\wedge \left( q\vee \left( p\vee \sim q \right) \right) \right]...........(2)\]
On evaluating the expression \[\left( p\vee \left( p\vee \sim q \right) \right)\] in the above expression, we get
\[\left( p\vee \left( p\vee \sim q \right) \right)=\left( p\vee p \right)\vee \left( p\vee \left( \sim q \right) \right)\]
Now since we know that \[\left( p\vee p \right)=p\], therefore the above expression becomes
\[\begin{align}
  & \left( p\vee \left( p\vee \sim q \right) \right)=\left( p\vee p \right)\vee \left( p\vee \left( \sim q \right) \right) \\
 & =p\vee \left( p\vee \left( \sim q \right) \right)..............(3)
\end{align}\]
Now on evaluating the expression \[\left( q\vee \left( p\vee \sim q \right) \right)\] in the above expression, we get
\[\left( q\vee \left( p\vee \sim q \right) \right)=\left( q\vee p \right)\vee \left( q\vee \left( \sim q \right) \right)\]
Since we know that \[\left( q\vee \left( \sim q \right) \right)=1\], therefore the above expression becomes
\[\begin{align}
  & \left( q\vee \left( p\vee \sim q \right) \right)=\left( q\vee p \right)\vee \left( q\vee \left( \sim q \right) \right) \\
 & =\left( q\vee p \right)..............(4)
\end{align}\]

We will now substitute the values in equation (3) and equation (4) in expression (2) to get
\[\left( p\wedge q \right)\vee \left( p\vee \sim q \right)=\left[ \left( p\vee \left( p\vee \sim q \right) \right)\wedge \left( q\vee p \right) \right]..............(5)\]
Now by substituting the value obtained in equation (5) expression (1) , we will get
\[\left( p\wedge q \right)\vee \left( p\vee \sim q \right)\wedge \left( \sim p\wedge \sim q \right)=\left[ \left( p\vee \left( p\vee \sim q \right) \right)\wedge \left( q\vee p \right) \right]\left( \sim p\wedge \sim q \right)\]
Since we know that by De-Morgan’s law which states that \[\sim \left( p\vee q \right)=\left( \sim p \right)\wedge \left( \sim q \right)\].
Therefore the above expression becomes
\[\left( p\wedge q \right)\vee \left( p\vee \sim q \right)\wedge \left( \sim p\wedge \sim q \right)=\left[ \left( p\vee \left( p\vee \sim q \right) \right)\wedge \left( q\vee p \right) \right]\sim \left( p\vee q \right)\]
Evaluating the above expression using distributive law, we get
\[\begin{align}
  & \left( p\wedge q \right)\vee \left( p\vee \sim q \right)\wedge \left( \sim p\wedge \sim q \right)=\left[ \left( p\vee \left( p\vee \sim q \right) \right)\wedge \left( q\vee p \right) \right]\sim \left( p\vee q \right) \\
 & =\left( p\vee \left( p\vee \sim q \right) \right)\sim \left( p\vee q \right)\wedge \left[ \left( q\vee p \right)\sim \left( p\vee q \right) \right]
\end{align}\]now since \[\left[ \left( q\vee p \right)\sim \left( p\vee q \right) \right]\] is empty.
Therefore we have
\[\begin{align}
  & \left( p\wedge q \right)\vee \left( p\vee \sim q \right)\wedge \left( \sim p\wedge \sim q \right)=\left( p\vee \left( p\vee \sim q \right) \right)\sim \left( p\vee q \right)\wedge \left[ \left( q\vee p \right)\sim \left( p\vee q \right) \right] \\
 & =\left( p\vee \left( p\vee \sim q \right) \right)\sim \left( p\vee q \right) \\
 & =\left( p\vee \sim q \right)\wedge \sim \left( p\vee q \right) \\
 & =\sim \left( p\vee q \right) \\
 & =\left( \sim p \right)\wedge \left( \sim q \right)
\end{align}\]
Therefore we finally get that the Boolean expression \[\left( p\wedge q \right)\vee \left( p\vee \sim q \right)\wedge \left( \sim p\wedge \sim q \right)\] is equals to \[\left( \sim p \right)\wedge \left( \sim q \right)\].

So, the correct answer is “Option (c)”.

Note: In this problem, in order determine the equivalent form of the Boolean expression \[\left( p\wedge q \right)\vee \left( p\vee \sim q \right)\wedge \left( \sim p\wedge \sim q \right)\], we are extensively using the distributive properties and the De-Morgan’s law for the given statement. It might get a bit confusing to keep using these properties on complicated statements to be used for it.