Question

Consider a propositional language where:

p means “Joan is happy”
q means “Joan paints a picture”
r means “Kupa is happy”.
Which of the options below is the correct one for the statement “Joan is happy only if she paints a picture”?
Question 5 options:

p ∧ q → ¬r



p → q



¬(p ∧ ¬q)



Both b and c are correct

Answer 1

The statement "Joan is happy only if she paints a picture" can be analyzed logically. The phrase "only if" introduces a conditional relationship where the first part (Joan is happy, represented by \( p \)) can occur only under the condition that the second part (Joan paints a picture, represented by \( q \)) is true.

This can be expressed as:
- If Joan is happy (p), then she paints a picture (q), which is logically represented as:

\[ p \rightarrow q \]

Now, let’s evaluate the provided options regarding the statement:

1. **p ∧ q → ¬r**: This states that if Joan is happy and paints a picture, then Kupa is not happy. This does not represent the stated condition and is incorrect.

2. **p → q**: This is correct as it directly translates the statement “Joan is happy only if she paints a picture.”

3. **¬(p ∧ ¬q)**: This states that it is not the case that Joan is happy and does not paint a picture, which is logically equivalent to saying "if Joan is happy, then she paints a picture." This is also correct.

4. **Both b and c are correct**: Since both \( p \rightarrow q \) (option b) and \( ¬(p ∧ ¬q) \) (option c) are valid representations of the statement, this option is correct.

Thus, the correct answer is:

**Both b and c are correct**.