Question

what is the converse of the following conditional statement?

\if this month is september, then next month ends in the letter r.\

determine if the converse is true or false and give a counterexample if the converse is false.

• if next month ends in the letter r, then this month is september. the converse is true.
• if this month is not september, then next month does not end in the letter r. the converse is true.
• if next month ends in the letter r, then this month is september. the converse is false; a counterexample is december.
• if this month is not september, then next month does not end in the letter r. the converse is false; a counterexample is december.

Explanation:

Identify the hypothesis and conclusion

The original conditional statement is: "If this month is September, then next month ends in the letter r."

• Hypothesis \(p\): "this month is September"
• Conclusion \(q\): "next month ends in the letter r"

Formulate the converse statement

The converse of a conditional statement "If \(p\), then \(q\)" is "If \(q\), then \(p\)".

• Converse: "If next month ends in the letter r, then this month is September."

Evaluate the truth value of the converse

We must determine if the converse is true or false.

• Let's find if there is any case where the hypothesis of the converse is true, but the conclusion is false.
• The hypothesis of the converse is: "next month ends in the letter r".
• Months ending in 'r' are: September, October, November, December.
• If next month is December (which ends in 'r'), then this month is November.
• In this case, "this month is September" is false.
• Therefore, the converse is false, and a valid counterexample is when next month is December (meaning this month is November).

Match with the given options

Let's check the options:

• Option 1: Incorrect converse statement and claims it is true.
• Option 2: This is the inverse statement, not the converse.
• Option 3: Correct converse statement, correctly states it is false, and provides "December" as a counterexample for next month.
• Option 4: This is the inverse statement, not the converse.

Answer:

• (A) If next month ends in the letter r, then this month is September. The converse is true.
• (B) If this month is not September, then next month does not end in the letter r. The converse is true.
• (C) If next month ends in the letter r, then this month is September. The converse is false; a counterexample is December. (Correct answer)
• (D) If this month is not September, then next month does not end in the letter r. The converse is false; a counterexample is December.