Question

The negation of \( \sim S \vee ( \sim R \wedge S) \) \(\text{ is equivalent to}\)

• A. \( S \vee (R \vee \sim S) \)
• B. \( S \wedge \sim R \)
• C. \( S \wedge R \)
• D. \( S \wedge (R \wedge \sim S) \)

Hint:

When negating logical expressions, use De Morgan's law to convert between conjunctions and disjunctions and simplify the terms.

Answer 1

The Correct Option is C

Step 1: The given expression is \( \sim S \vee ( \sim R \wedge S) \). 

Step 2: To find the negation, apply De Morgan's law. The negation of a disjunction (\( \vee \)) becomes a conjunction (\( \wedge \)) of the negations of each term. So we need to negate each term separately: 
\[ \sim (\sim S \vee (\sim R \wedge S)) = \sim \sim S \wedge \sim (\sim R \wedge S) \] 

Step 3: Simplify \( \sim \sim S \) to \( S \). Now apply De Morgan's law to \( \sim (\sim R \wedge S) \), which becomes \( \sim \sim R \vee \sim S \), i.e., \( R \vee \sim S \). Therefore, the expression becomes: 
\[ S \wedge (R \vee \sim S) \] 

Step 4: However, the original expression has \( \sim S \) outside the parentheses, so this simplifies to: 
\[ S \wedge R \] Thus, the negation is \( S \wedge R \), which is option (c).