If p: It rains today, q: I go to the school, r: I shall meet my friends and s: I shall go for a movie, then which of the following is the proportion? It does not rain or if I do not go to school, then I shall meet my friend and go for a movie.
(a) \[\left( \sim p\wedge \sim q \right)\Rightarrow \left( r\wedge s \right)\],
(b) $ \sim\left( p\wedge q \right)\Rightarrow \left( r\wedge s \right) $ ,
(c) $ \sim\left( p\vee q \right)\Rightarrow \left( r\vee s \right) $ ,
(d) None of the above.

Hint: We first recall all the symbols and notations used before solving the problem. We first find the mathematical logic for the first part of the statement “It does not rain or if I do not go to school”. We then find the mathematical logic for the second part of the statement “I shall meet my friend and go for a movie”. We can see that both the statements are if-then statement to find the final answer.

Complete step-by-step answer:
Given that we have four statements given as p: It rains today, q: I go to the school, r: I shall meet my friends and s: I shall go for a movie. We need to find the mathematical logic for the given statement: It does not rain or if I do not go to school, then I shall meet my friend and go for a movie.
We know that logical symbol for or is $ \vee $ , for and is $ \wedge $ , for if-then statement is $ \Rightarrow $ .
Let us find the mathematical logic for the statement: It does not rain or if I do not go to school.
We know that negation of any statement is denoted by $ \sim $ . We use this now.
So, we get mathematical logic for the statement: It does not rain or if I do not go to school as $ \sim p\vee \sim q $ . (As both statements are negative statements and are connected by logic).
Now, we find the mathematical logic for the statement: I shall meet my friend and go for a movie.
Since both statements are positive statements as given in the problem and are connected by logic. We get mathematical logic for the statement: I shall meet my friend and go for a movie as $ r\wedge s $ .
Since both logical statements $ \sim p\vee \sim q $ and $ r\wedge s $ are connected by if-then statement, we get the mathematical logic as $ \left( \sim p\vee \sim q \right)\Rightarrow \left( r\wedge s \right) $ ---(1).
We know that $ \left( \sim p\vee \sim q \right)=\sim\left( p\wedge q \right) $ . We use this statement in equation (1).
So, we get mathematical logic as $ \sim\left( p\wedge q \right)\Rightarrow \left( r\wedge s \right) $ .
∴ The statements get proportioned with $ \sim\left( p\wedge q \right)\Rightarrow \left( r\wedge s \right) $ .
So, the correct answer is “Option B”.

Note: Whenever we get problems containing two or more statements to find the logic, we divide them into small statements and find the logic for those. In the end we find logic for all together of the statements. We should not confuse $ \left( \sim p\vee \sim q \right)=\sim\left( p\wedge q \right) $ with $ \left( \sim p\vee \sim q \right)=\sim\left( p\vee q \right) $ .