Consider the following statements :
p: It rains today
q: I go to school
r: I Shall meet any friends
s: I shall go for a movie
Then which of the following proposition represents 'If it does not rain or if I do not go to school, then I shall meet my friend and go for a movie .'?

Options

A. `~(p^^q) to (r^^s)`
B. `~(p^^~q) to (r^^s)`
C. `~(p^^q) to (rvvs)`
D. None of these

Explanation

To solve the problem, we need to translate the given statement into a logical proposition using the symbols for the statements provided.

1. **Identify the statements:**
- Let \( p \): It rains today.
- Let \( q \): I go to school.
- Let \( r \): I shall meet my friends.
- Let \( s \): I shall go for a movie.

2. **Translate the given statement:**
- The statement we need to represent is: "If it does not rain or if I do not go to school, then I shall meet my friends and go for a movie."
- "It does not rain" translates to \( \neg p \) (the negation of \( p \)).
- "I do not go to school" translates to \( \neg q \) (the negation of \( q \)).
- "I shall meet my friends" translates to \( r \).
- "I shall go for a movie" translates to \( s \).
- The phrase "or" translates to \( \lor \) (logical OR).
- The phrase "and" translates to \( \land \) (logical AND).
- The phrase "If... then..." translates to \( \rightarrow \) (logical implication).

3. **Construct the logical proposition:**
- The part "If it does not rain or if I do not go to school" can be expressed as:
\[ \neg p \lor \neg q \]
- The part "then I shall meet my friends and go for a movie" can be expressed as:
\[ r \land s \]
- Combining these, we get the full proposition:
\[ (\neg p \lor \neg q) \rightarrow (r \land s) \]

4. **Final representation:**
- Therefore, the logical proposition that represents the statement is:
\[ (\neg p \lor \neg q) \rightarrow (r \land s) \]