If Grayson is at home, then he is not in Grant County.
If Grayson is not at home, then he is in Grant County.
Is the second conditional the contrapositive, converse, or inverse of the first conditional?
The second conditional is the contrapositive of the first conditional.
The second conditional, "If Grayson is not at home, then he is in Grant County," is the contrapositive of the first conditional, "If Grayson is at home, then he is not in Grant County." The contrapositive of a conditional statement switches the hypothesis and the conclusion while also negating both.
To determine whether the second conditional is the contrapositive, converse, or inverse of the first conditional, let's define these terms:
1. Contrapositive: Swapping the hypothesis and conclusion of a conditional statement, and negating both.
2. Converse: Swapping the hypothesis and conclusion of a conditional statement.
3. Inverse: Negating both the hypothesis and the conclusion of a conditional statement.
Let's break down the conditionals provided:
First Conditional: If Grayson is at home, then he is not in Grant County.
- Hypothesis: Grayson is at home.
- Conclusion: Grayson is not in Grant County.
Second Conditional: If Grayson is not at home, then he is in Grant County.
- Hypothesis: Grayson is not at home.
- Conclusion: Grayson is in Grant County.
Now, let's analyze each option:
- The contrapositive of the first conditional would be: If Grayson is not in Grant County, then he is at home. (Swapped and negated)
- The converse of the first conditional would be: If Grayson is not in Grant County, then he is at home. (Simply swapped)
- The inverse of the first conditional would be: If Grayson is not at home, then he is in Grant County. (Negated both)
Comparing the second conditional to the options, we can see that it matches the inverse of the first conditional since it negates both the hypothesis and conclusion. Therefore, the second conditional is the inverse of the first conditional.