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: Understanding the Question:
We need to translate the given English logical statement into its symbolic mathematical logic form using the provided propositions.

Step 2: Detailed Explanation:
Let's break down the given statement into its components:
1. "It does not rain today" corresponds to the negation of p, which is $\sim p$.
2. "I won't go to school" corresponds to the negation of q, which is $\sim q$.
3. These two are connected by "or", giving us the antecedent: $(\sim p \lor \sim q)$.
4. "I will meet my friend" is r.
5. "I will go to watch a movie" is s.
6. These two are connected by "and", giving us the consequent: $(r \land s)$.
The full statement is an "If ... then ..." conditional, which connects the antecedent and consequent with an implication arrow ($\rightarrow$):
$$(\sim p \lor \sim q) \rightarrow (r \land s)$$
By applying De Morgan's Law, we know that $(\sim p \lor \sim q)$ is logically equivalent to $\sim(p \land q)$.
Substituting this back gives the final symbolic form:
$$\sim(p \land q) \rightarrow (r \land s)$$

Step 3: Final Answer:
The symbolic form is $\sim(p \land q) \rightarrow (r \land s)$, matching option (C).