Question

for exercises 15–18, write the converse, inverse, and contrapositive of each true conditional statement. determine whether each related conditional is true or false. if a statement is false, then find a counterexample. 15. ulma is waiting to board an airplane. over the speakers she hears a flight attendant say “if you are seated in rows 10 through 20, you may now board.” 16. if you have five dollars, then you can buy five raffle tickets. 17. if two angles are complementary angles, then the angles are acute. 18. a medicine bottle says “if you will be driving, then you should not take this medicine.” 19. you are evaluating a conditional statement in which the hypothesis is true, but the conclusion is false. is the inverse of the statement true or false? explain your reasoning.

Answer:

15

Let \(p\): You are seated in rows 10 through 20. Let \(q\): You may now board.

• Converse: If you may now board, then you are seated in rows 10 through 20. (\(q

ightarrow p\))

• Truth - value: False. Counter - example: You could be in a priority - boarding group (e.g., first - class passengers) and be allowed to board even if you are not in rows 10 through 20.
• Inverse: If you are not seated in rows 10 through 20, then you may not now board. (\(

eg p
ightarrow
eg q\))

• Truth - value: False. Counter - example: You could be in a priority - boarding group (e.g., first - class passengers) and be allowed to board even if you are not in rows 10 through 20.
• Contrapositive: If you may not now board, then you are not seated in rows 10 through 20. (\(

eg q
ightarrow
eg p\))

• Truth - value: True. If you are not allowed to board, it means you don't meet the criteria of being in rows 10 through 20.

16

Let \(p\): You have five dollars. Let \(q\): You can buy five raffle tickets.

• Converse: If you can buy five raffle tickets, then you have five dollars. (\(q

ightarrow p\))

• Truth - value: True. Assuming the price of each raffle ticket is one dollar, if you can buy five raffle tickets, you must have five dollars.
• Inverse: If you do not have five dollars, then you cannot buy five raffle tickets. (\(

eg p
ightarrow
eg q\))

• Truth - value: True. Assuming the price of each raffle ticket is one dollar, if you don't have five dollars, you can't buy five raffle tickets.
• Contrapositive: If you cannot buy five raffle tickets, then you do not have five dollars. (\(

eg q
ightarrow
eg p\))

• Truth - value: True. Assuming the price of each raffle ticket is one dollar, if you can't buy five raffle tickets, you don't have five dollars.

17

Let \(p\): Two angles are complementary angles. Let \(q\): The angles are acute.

• Converse: If the angles are acute, then the two angles are complementary angles. (\(q

ightarrow p\))

• Truth - value: False. Counter - example: Two angles of \(30^{\circ}\) and \(40^{\circ}\) are acute, but \(30 + 40=70

eq90\), so they are not complementary.

• Inverse: If two angles are not complementary angles, then the angles are not acute. (\(

eg p
ightarrow
eg q\))

• Truth - value: False. Counter - example: Two angles of \(100^{\circ}\) and \(20^{\circ}\) are not complementary, but the \(20^{\circ}\) angle is acute.
• Contrapositive: If the angles are not acute, then the two angles are not complementary angles. (\(

eg q
ightarrow
eg p\))

• Truth - value: True. Complementary angles add up to \(90^{\circ}\), and non - acute angles (obtuse or right) cannot add up to \(90^{\circ}\) with another angle.

18

Let \(p\): You will be driving. Let \(q\): You should not take this medicine.

• Converse: If you should not take this medicine, then you will be driving. (\(q

ightarrow p\))

• Truth - value: False. Counter - example: You might not take the medicine because it has other side - effects not related to driving, like it makes you extremely drowsy and you plan to sleep.
• Inverse: If you will not be driving, then you should take this medicine. (\(

eg p
ightarrow
eg q\))

• Truth - value: False. Counter - example: You might still not take the medicine because it has other side - effects not related to driving, like it causes an allergic reaction.
• Contrapositive: If you should take this medicine, then you will not be driving. (\(

eg q
ightarrow
eg p\))

• Truth - value: True. If it is okay to take the medicine, it means the restriction of driving while taking it does not apply.

19

Let the conditional statement be \(p
ightarrow q\). Given that \(p\) is true and \(q\) is false, so \(p
ightarrow q\) is false.
The inverse of \(p
ightarrow q\) is \(
eg p
ightarrow
eg q\).
Since \(p\) is true, \(
eg p\) is false. A conditional statement \(
eg p
ightarrow
eg q\) with a false hypothesis is always true (by the truth - table of conditional statements: when the hypothesis is false, the entire conditional statement is true regardless of the truth - value of the conclusion).

• Answer: The inverse of the statement is true. Reasoning: A conditional statement with a false hypothesis is always true, and since the hypothesis of the inverse is the negation of a true statement (so it is false), the inverse is true.