The negation of \( \sim S \vee ( \sim R \wedge S) \) \(\text{ is equivalent to}\)
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).
A remote island has a unique social structure. Individuals are either "Truth-tellers" (who always speak the truth) or "Tricksters" (who always lie). You encounter three inhabitants: X, Y, and Z.
X says: "Y is a Trickster"
Y says: "Exactly one of us is a Truth-teller."
What can you definitively conclude about Z?
Consider the following statements followed by two conclusions.
Statements: 1. Some men are great. 2. Some men are wise.
Conclusions: 1. Men are either great or wise. 2. Some men are neither great nor wise. Choose the correct option: