Write the converse, inverse, and contrapositive of the statement below. Determine the truth value of each statement. If a statement is false, give a counterexample.
If i am in Florida, then i am in the south
Converse: If I am in the south, then I am in Florida.
Truth value: False. Counterexample: If I am in Texas, I am in the south but not in Florida.
Inverse: If I am not in Florida, then I am not in the south.
Truth value: False. Counterexample: If I am in Georgia, I am not in Florida but I am still in the south.
Contrapositive: If I am not in the south, then I am not in Florida.
Truth value: True. Since the contrapositive is logically equivalent to the original statement, it is true.
Question content area bottom
Part 1
Write the converse of the conditional. Choose the correct answer below.
A.
If I am not in the , then I am not in .
B.
If I am not in , then I am not in the .
C.
If I am in the , then I am in .
D.
If I am in the , then I am not in .
Part 2
Determine the truth value of the converse. Choose the correct answer below.
A.
The converse is false. I can be in and not be in the .
B.
The converse is false. I can be in and still be in the .
C.
The converse is true.
Part 3
Write the inverse of the conditional. Choose the correct answer below.
A.
If I am not in the , then I am not in .
B.
If I am in , then I am not in the .
C.
If I am in the , then I am in .
D.
If I am not in , then I am not in the .
Part 4
Determine the truth value of the inverse. Choose the correct answer below.
A.
The inverse is false. I can be in and not be in the .
B.
The inverse is true.
C.
The inverse is false. I can be in and still be in the .
Part 5
Write the contrapositive of the conditional. Choose the correct answer below.
A.
If I am not in the , then I am in .
B.
If I am not in the , then I am not in .
C.
If I am not in , then I am not in the .
D.
If I am in the , then I am in .
Part 6
Determine the truth value of the contrapositive. Choose the correct answer below.
A.
The contrapositive is false. I can be in and not be in the .
B.
The contrapositive is false. I can be in and still be in the .
C.
The contrapositive is true.
Part 1: The correct answer is D. If I am in the south, then I am not in Florida.
Part 2: The correct answer is B. The converse is false. I can be in the south and still not be in Florida.
Part 3: The correct answer is A. If I am not in Florida, then I am not in the south.
Part 4: The correct answer is B. The inverse is true.
Part 5: The correct answer is B. If I am not in the south, then I am not in Florida.
Part 6: The correct answer is C. The contrapositive is true.
To find the converse, inverse, and contrapositive of a statement, we need to understand a few definitions:
1. Converse: Switching the hypothesis and conclusion of a conditional statement.
2. Inverse: Negating both the hypothesis and conclusion of a conditional statement.
3. Contrapositive: Switching and negating both the hypothesis and conclusion of a conditional statement.
Given the statement: "If I am in Florida, then I am in the south," we can determine its converse, inverse, and contrapositive.
1. Converse: If I am in the south, then I am in Florida.
- This statement switches the hypothesis ("I am in Florida") and conclusion ("I am in the south").
2. Inverse: If I am not in Florida, then I am not in the south.
- This statement negates both the hypothesis ("I am in Florida") and conclusion ("I am in the south").
3. Contrapositive: If I am not in the south, then I am not in Florida.
- This statement switches and negates both the hypothesis ("I am in Florida") and conclusion ("I am in the south").
Now let's evaluate the truth value of each statement:
- The original statement: If I am in Florida, then I am in the south.
- This is generally true because Florida is located in the south.
- The converse: If I am in the south, then I am in Florida.
- This is false because someone can be in the south without being in Florida. For example, someone can be in Texas.
- The inverse: If I am not in Florida, then I am not in the south.
- This is also false because one can be outside Florida and still be in the south. For example, someone can be in Georgia.
- The contrapositive: If I am not in the south, then I am not in Florida.
- This statement is true because if someone is not in the south, it means they are not in Florida.
Thus, the original statement and contrapositive are true, while the converse and inverse are false.