Question

p : It rains today
q : I am going to school
r : I will meet my friend
s : I will go to watch a movie.
Then symbolic form of the statement "If it does not rain today or I won't go to school, then I will meet my friend and I will go to watch a movie" is

• A. $\sim(p \lor q) \rightarrow (r \lor s)$
• B. $(p \land q) \rightarrow (r \lor s)$
• C. $\sim(p \land q) \rightarrow (r \land s)$
• D. $(\sim p \land q) \rightarrow (r \land s)$

Hint:

De Morgan's Laws are crucial in mathematical logic: $\sim(p \land q) \equiv \sim p \lor \sim q$ and $\sim(p \lor q) \equiv \sim p \land \sim q$. Always check if an option uses an equivalent expression.

Answer 1

The Correct Option is C

Step 1: List the simple statements.
$p$: it rains today. $q$: I go to school. $r$: I meet my friend. $s$: I watch a movie.

Step 2: Translate the first part.
"It does not rain" is $\sim p$. "I won't go to school" is $\sim q$. They are joined by "or", giving $(\sim p \lor \sim q)$.

Step 3: Translate the second part.
"I will meet my friend and I will watch a movie" is $(r \land s)$.

Step 4: Join with "if then".
The whole sentence is an implication.
\[ (\sim p \lor \sim q) \rightarrow (r \land s) \]
Step 5: Use De Morgan's law.
$\sim p \lor \sim q$ is the same as $\sim(p \land q)$.
\[ \sim(p \land q) \rightarrow (r \land s) \]
Step 6: Conclusion.
This matches the option written with the De Morgan form. \[ \boxed{\sim(p \land q) \rightarrow (r \land s) \text{ (Option 3)}} \]