Question

Find the negation of $\sim s\vee \left( \sim r\wedge s \right)$ is equivalent to:
(a) $s\wedge r$
(b) $s\wedge \sim \left( r\wedge \sim s \right)$
(c) $s\vee \sim \left( r\wedge \sim s \right)$
(d) None of these

Answer 1

Basically we have to evaluate $\sim \left( \sim s\vee \left( \sim r\wedge s \right) \right)$. The properties that we are going to use in evaluating this expression are that negation of $a\vee b$ is $\sim a\wedge \sim b$. The symbol $\sim$ represents negation. Similarly, negation of $a\wedge b$ is $\sim a\vee \sim b$ and application of two negations on an element gives the same element. And we are also going to use the distributive property on $a\wedge \left( b\vee c \right)$ which is equal to $\left( a\wedge b \right)\vee \left( a\wedge c \right)$.

Complete step by step answer:
We have to find the negation of the following:
$\sim s\vee \left( \sim r\wedge s \right)$
Negation is represented by the symbol $\sim$ so basically we have to evaluate the following expression,
$\sim \left( \sim s\vee \left( \sim r\wedge s \right) \right)$……… eq. (1)
We are going to use the following property in solving the above expression.
$\sim \left( a\vee b \right)=\left( \sim a\wedge \sim b \right)$
Using the above property in eq. (1) we get,
$\left( \sim \left( \sim s \right)\wedge \sim \left( \sim r\wedge s \right) \right)$
Applying two negations on any element say “p” will make the element free from negations as follows:
$\sim \sim p=p$
Using the above property in the new reduced form of the given expression as:
$\left( \left( s \right)\wedge \sim \left( \sim r\wedge s \right) \right)$
Now, we are going to use the following property of the negation of $a\wedge b$ in the above equation which is equal to:
$\sim \left( a\wedge b \right)=\left( \sim a \right)\vee \left( \sim b \right)$
$\begin{aligned} & \left( \left( s \right)\wedge \left( \left( \sim \sim r \right)\vee \left( \sim s \right) \right) \right) \\\ & =\left( \left( s \right)\wedge \left( \left( r \right)\vee \left( \sim s \right) \right) \right) \\\ \end{aligned}$
As you can see the above expression is in the form of $a\wedge \left( b\vee c \right)$and we know the expansion of $a\wedge \left( b\vee c \right)$ as:
$\left( a\wedge b \right)\vee \left( a\wedge c \right)$
So, we can use the above expression to simplify $\left( \left( s \right)\wedge \left( \left( r \right)\vee \left( \sim s \right) \right) \right)$ as follows:
$\left( s\wedge r \right)\vee \left( s\wedge \sim s \right)$
We know that intersection of two elements which are negated with respect to each other is $\phi$.
$\left( s\wedge r \right)\vee \left( \phi \right)$
In the above expression, union sign $\vee$ means addition and $\phi$ is the null set or equivalent to 0 so adding 0 to any element will give you the same element so taking $\phi$ in union with $\left( s\wedge r \right)$ we will get:
$\left( s\wedge r \right)$
From the above evaluation, we got the negation of $\sim s\vee \left( \sim r\wedge s \right)$ as $\left( s\wedge r \right)$.

So, the correct answer is “Option A”.

Note: The mistake that you could make in the above problem is in simplifying the following expression:
$\left( \left( s \right)\wedge \left( \left( r \right)\vee \left( \sim s \right) \right) \right)$
In the above expression, to simplify it we are going to use the following property:
$a\wedge \left( b\vee c \right)=\left( a\wedge b \right)\vee \left( a\wedge c \right)$
Now, you might get confused in applying this property so to avoid such mistake you can remember to apply the following property that first take the intersection of a and b then you will get,
$\left( a\wedge b \right)$
Then put the union sign after the above expression you will get,
$\left( a\wedge b \right)\vee$
Now, take the intersection of a and c and write in front of the above expression we get,
$\left( a\wedge b \right)\vee \left( a\wedge c \right)$
This is how you can remember this distributive property and won’t commit mistakes.