Final answer:
The correct answer is option d, which declares p to be true, aligning with the original true statement p ∧ q. Other options introduce undefined elements or do not guarantee both p and q are true.
Step-by-step explanation:
If p ∧ q is true, then p is true and q is also true. The only compound statement that has the same truth value under these conditions would be one where p is true, and its combination with q or a statement derived from q maintains a true value.
Let's examine each option:
- a. p - 9 is false and q + p is true. - This option does not provide enough information about the truth values of p and q separately, it involves additional elements (such as 9) which are not defined in the context of the logical statement.
- b. p - q is false and q - p is false. - This expression suggests contradictions and does not guarantee that p and q are both true.
- c. p - q is true and q - p is false. - Again, this includes undefined operations and does not reliably indicate that both p and q are true.
- d. p - q is true and p is true. - This option indicates that p is true. Even though we have an undefined operation (p - q), the statement explicitly states that p is true, which aligns with our original statement (p ∧ q) being true.
Thus, the correct answer is d. Statements a, b, and c either introduce undefined elements or suggest that both p and q might not be true simultaneously.