Sure, let's analyze the given conditionals step-by-step.
### Step 1: Define the Propositions
We start by defining the propositions of the original conditional statement:
- Let [tex]\( P \)[/tex]: "Roy does a cartwheel."
- Let [tex]\( Q \)[/tex]: "Lee does not stand on his head."
### Step 2: Original Conditional Statement
The original statement can be written formally as:
[tex]\[ P \rightarrow Q \][/tex]
(If Roy does a cartwheel, then Lee does not stand on his head.)
### Step 3: Forms of Related Conditionals
We need to examine the second conditional and decide whether it is the contrapositive, converse, or inverse of the original.
#### Inverse
The inverse of the original statement [tex]\( P \rightarrow Q \)[/tex] is:
[tex]\[ \neg P \rightarrow \neg Q \][/tex]
(If Roy does not do a cartwheel, then Lee stands on his head.)
#### Converse
The converse of the original statement [tex]\( P \rightarrow Q \)[/tex] is:
[tex]\[ Q \rightarrow P \][/tex]
(If Lee does not stand on his head, then Roy does a cartwheel.)
#### Contrapositive
The contrapositive of the original statement [tex]\( P \rightarrow Q \)[/tex] is:
[tex]\[ \neg Q \rightarrow \neg P \][/tex]
(If Lee stands on his head, then Roy does not do a cartwheel.)
### Step 4: Analyze the Second Conditional
The second conditional statement is:
"If Lee stands on his head, then Roy does not do a cartwheel."
This can be rewritten in propositional form as:
[tex]\[ \neg Q \rightarrow \neg P \][/tex]
### Step 5: Compare and Identify
Now, we compare the second conditional with the forms we identified previously:
- The second conditional ([tex]\( \neg Q \rightarrow \neg P \)[/tex]) matches the contrapositive of the original conditional.
### Conclusion
Therefore, the second conditional is the contrapositive of the first conditional.
The answer is:
[tex]\[ \text{contrapositive} \][/tex]
Thus, the correct choice is:
[tex]\[ \boxed{contrapositive} \][/tex]
