15. Consider the statements: (1) If it rains, then the picnic will be canceled. (2) If the picnic is canceled, then Frank will be sad. Which conclusion must be true?
Answer: A
If it rains, Frank will be sad.
Based on the statements provided, if it rains, the picnic will indeed be canceled, leading to Frank's sadness. Therefore, the conclusion that if it rains, Frank will be sad is logically valid.
A) If it rains, Frank will be sad.
This option is correct because it follows logically from the two given statements. If it rains, the first statement confirms that the picnic will be canceled, and the second states that a canceled picnic results in Frank being sad. Thus, rain directly leads to Frank’s sadness.
B) If Frank is sad, then it rained.
This option is incorrect as it implies a direct cause-and-effect relationship that is not supported by the statements. Frank being sad could result from other factors besides rain; hence, we cannot definitively conclude that rain is the sole reason for Frank's sadness.
C) If it does not rain, the picnic will not be canceled.
This option is incorrect because the statements do not provide information about what happens if it does not rain. The picnic could still be canceled for reasons unrelated to rain, making this conclusion invalid.
D) If the picnic is not canceled, then it did not rain.
This option is also incorrect. While the first statement indicates that rain leads to cancellation, it does not provide a reciprocal relationship. The picnic could be not canceled for other reasons, and rain does not exclusively determine the cancellation.
Conclusion
The correct conclusion, "If it rains, Frank will be sad," logically follows from the premises provided. All other options fail because they either misinterpret the logical relationships established by the statements or suggest conditions that are not guaranteed by the information given. Thus, A is the only conclusion that must be true based on the statements.