Question

If 
\( p \): It is raining today, 
\( q \): I go to school, 
\( r \): I shall meet my friends, 
and \( s \): I shall go for a movie, then which of the following 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?"

• A. \( \neg (p \wedge q) \Rightarrow (r \wedge s) \)
• B. \( \neg (p \wedge \neg q) \Rightarrow (r \wedge s) \)
• C. \( \neg (p \wedge q) \Rightarrow (r \vee s) \)
• D. None of these

Hint:

Use logical operators systematically to convert English statements into symbolic form.

Answer 1

The Correct Option is A

The given statement translates as: {"If it does not rain or I do not go to school, then I shall meet my friend and go for a movie."} The phrase "does not rain or do not go to school" can be written as: \[ \neg (p \wedge q) \] The phrase "I shall meet my friend and go for a movie" translates to: \[ r \wedge s \] Thus, the logical expression becomes: \[ \neg (p \wedge q) \Rightarrow (r \wedge s) \]

Answer 2

Step 1: Understanding the logical statements
We are given the following propositions: - \( p \): It is raining today, - \( q \): I go to school, - \( r \): I shall meet my friends, - \( s \): I shall go for a movie.

Step 2: Interpreting the statement
The given statement 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." This can be broken down as follows: - "If it does not rain" is \( \neg p \), - "If I do not go to school" is \( \neg q \), - "Then I shall meet my friends" is \( r \), - "Then I shall go for a movie" is \( s \).

The logical structure of this statement is: \[ \neg p \vee \neg q \Rightarrow (r \wedge s) \] This means that if either it does not rain or I do not go to school, then both \( r \) (meeting my friends) and \( s \) (going for a movie) will happen.

Step 3: Simplifying the expression
We can simplify the left-hand side of the implication: \[ \neg p \vee \neg q = \neg (p \wedge q) \] Thus, the statement becomes: \[ \neg (p \wedge q) \Rightarrow (r \wedge s) \] Step 4: Final Answer
The correct representation is:

\( \neg (p \wedge q) \Rightarrow (r \wedge s) \)