Question

If $p$ : It is raining and $q$ : It is pleasant, then the symbolic form of "It is neither raining nor pleasant" is

• A. $\sim p \wedge q$
• B. $\sim p \vee q$
• C. $(\sim p) \wedge (\sim q)$
• D. $(\sim p) \vee (\sim q)$

Hint:

Always memorize this phrase translation template for logic exams: "Neither $p$ nor $q$" translates directly to $\sim p \wedge \sim q$. It helps to think of it as "not $p$ AND not $q$".

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given two simple verbal statements: $p$ representing "It is raining" and $q$ representing "It is pleasant". We need to translate the compound statement "It is neither raining nor pleasant" into its correct symbolic mathematical logic form.

Step 2: Key Formula or Approach:
The phrase "neither $A$ nor $B$" logically means "$A$ is false

AND $B$ is false". In formal logic notation, this translates directly to a conjunction of two negated propositions: $$\sim A \wedge \sim B$$

Step 3: Detailed Explanation:
Let's translate each part of the given phrase systematically:

• "It is not raining" is the direct negation of $p$, which is written as $\sim p$.

• "It is not pleasant" is the direct negation of $q$, which is written as $\sim q$.

• The word "nor" serves as a conjunction connecting these two conditions, which means both statements must happen simultaneously ("and").
Combining these individual components with the conjunction operator ($\wedge$) gives the final symbolic form: $$(\sim p) \wedge (\sim q)$$ Note that by De Morgan's laws, this is also logically equivalent to $\sim(p \vee q)$ ("it is not the case that it is raining or pleasant"), which perfectly matches the colloquial phrasing. Among the given options, option (C) fits this exact structural layout.

Step 4: Final Answer:
The symbolic representation of the sentence is $(\sim p) \wedge (\sim q)$, corresponding to option (C).