Questions: Suppose that the given statements are true. Find the other true statements. (a) Given: If Jina knows how to dance, then the tires are old. Jina knows how to dance. Which statement must also be true? (b) Given: Which statement must also be true? (c) Given: The tires are not old. Which statement must also be true?

Solution Steps

• The first statement is: If Jina knows how to dance, then the tires are old. This can be written as \( \text{Jina knows how to dance} \rightarrow \text{The tires are old} \).
• The second statement is: Jina knows how to dance. This is \( \text{Jina knows how to dance} \).

• Since \( \text{Jina knows how to dance} \) is true, and \( \text{Jina knows how to dance} \rightarrow \text{The tires are old} \) is also true, we can conclude that \( \text{The tires are old} \) must be true.

• The first statement is: The tires are old. This is \( \text{The tires are old} \).
• The second statement is: Jina does not know how to dance. This is \( \neg \text{Jina knows how to dance} \).

• From the given statements, we cannot directly infer any additional true statements because the relationship between Jina knowing how to dance and the tires being old is only one-directional (from Jina knowing how to dance to the tires being old).

• The given statement is: The tires are not old. This is \( \neg \text{The tires are old} \).

• From the original implication \( \text{Jina knows how to dance} \rightarrow \text{The tires are old} \), if \( \text{The tires are not old} \) is true, then \( \text{Jina knows how to dance} \) must be false. This is because if Jina knew how to dance, the tires would have to be old, which contradicts the given statement. Therefore, \( \neg \text{Jina knows how to dance} \) must be true.

The remaining parts of the question are not addressed as per the instructions.

Final Answer