Question

The negation of the Boolean expression $ \sim s \vee \left( { \sim r \wedge s} \right)$ is equivalent to:
A.r
B. $s \wedge r$
C. $s \vee r$
D. $ \sim s \wedge \sim r$

Answer 1

Hint: We can apply the distributive property of conjunction and disjunction. Then we can apply the property that disjunction of a statement and its negation is always true. Then by using the definitions of conjunction and disjunction and its values we can simplify the expression. Then we take the negation of the simplified expression to get the required statement.

Complete step-by-step answer:
We have the Boolean expression $ \sim s \vee \left( { \sim r \wedge s} \right)$.
We know that $ \wedge $ represents conjunction, $ \vee $ represents disjunction and $ \sim $ represents negation.
We know that $ \wedge $ and $ \vee $ are distributive, so we can write the expression as
 \[ \sim s \vee \left( { \sim r \wedge s} \right) \equiv \left( { \sim s \vee \sim r} \right) \wedge \left( { \sim s \vee s} \right)\]
As $s$ is a statement it can be either true or false. So, either \[s\] or \[ \sim s\] will be true. So, their disjunction will always be true.
Now the expression will become
 \[\left( { \sim s \vee \sim r} \right) \wedge T\]
As the conjunction of two statements are true if and only if both the statements are true. So, the value of the expression depends on \[\left( { \sim s \vee \sim r} \right)\]
 \[ \sim s \vee \left( { \sim r \wedge s} \right) \equiv \sim s \vee \sim r\]
Now we can take the negation,
\[ \sim \left( { \sim s \vee \sim r} \right)\]
On applying the negation, the disjunction will become conjunction and the negation of each of the statements is taken
 \[\left( { \sim \sim s \wedge \sim \sim r} \right)\]
We know that $ \sim \left( { \sim p} \right) = p$ . So, the expression will become,
 \[\left( {s \wedge r} \right)\]
The negation of the Boolean expression $ \sim s \vee \left( { \sim r \wedge s} \right)$ is equivalent to \[\left( {s \wedge r} \right)\]
Therefore, the correct answer is option B.

Note: In mathematical logics statements are sentences which are either true or false. Negation of a statement gives the value of a statement as true if it is false and vice versa. Two simple statements are combined together using connectors like conjunction and disjunction. Conjunction is similar to the word ‘and’ and is represented by $ \wedge $ . The conjunction of two statements is true if and only if the two statements are true. In all other cases the conjunction will be false. Disjunction is similar to the word ‘or’ and is represented by $ \vee $ . The conjunction of two statements is false if and only if the two statements are false. In all other cases the disjunction will be true.