Let's analyze the statements you've provided.
Statement B
- Original Statement: If this month is not September, then next month does not end in the letter “r.”
- Converse: If next month ends in the letter “r,” then this month is September.
Analysis:
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When is the original statement true?
- It is true for all months except for August (which is not September, but next month is September, which ends in “r”).
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When is the converse true?
- The converse is true because if next month ends in the letter “r,” it must be September, making this month's condition correct as well.
Statement C
- Original Statement: If next month ends in the letter “r,” then this month is September.
- Converse: If this month is not September, then next month does not end in the letter “r.”
Analysis:
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When is the original statement true?
- The original statement is true because if next month ends in "r" (i.e., September, December, or such), this month must be August or November, respectively, making the month not September.
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When is the converse false?
- A counterexample is given: December, where this month is not September, but the next month (January) ends in "r."
Statement D
- Original Statement: If this month is not September, then next month does not end in the letter “r.”
- Converse: If next month ends in the letter “r,” then this month is September.
Analysis:
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When is the original statement true?
- Like B, it is not true if this month is August, because next month will be September which does end in "r."
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When is the converse false?
- The counterexample provided is again December, because even though this month is not September, January ends in "r."
Summary:
- Statement B is true and its converse is true.
- Statement C is logically consistent in that the original is true, and the converse is indeed false (the December counterexample stands).
- Statement D contains an original statement that is the same as B (true) and provides a converse that is the same as C (false).
Thus, statements B and D are effectively discussing the same premise regarding September and the months that follow, while C explores the implications of next months ending in "r" correctly.