Answer :
To determine the relationship between the two conditionals given, we need to understand some basic concepts regarding logical statements.
The original statement is:
1. If Harold will put Gorgonzola cheese on a sandwich, then he is not making it for lunch.
- This can be written in logical terms as: [tex]\( P \rightarrow \neg Q \)[/tex]
- Where [tex]\( P \)[/tex] is "Harold will put Gorgonzola cheese on a sandwich" and [tex]\( \neg Q \)[/tex] is "he is not making it for lunch".
The second statement is:
2. If Harold is making a sandwich for lunch, then he will not put Gorgonzola cheese on it.
- This can be written in logical terms as: [tex]\( Q \rightarrow \neg P \)[/tex]
- Where [tex]\( Q \)[/tex] is "Harold is making a sandwich for lunch" and [tex]\( \neg P \)[/tex] is "he will not put Gorgonzola cheese on it".
Now, let's explore the definitions:
Contrapositive:
The contrapositive of [tex]\( P \rightarrow \neg Q \)[/tex] is [tex]\( \neg Q \rightarrow \neg P \)[/tex]. This flips and negates both the hypothesis and conclusion.
- Original: [tex]\( P \rightarrow \neg Q \)[/tex]
- Contrapositive: [tex]\( \neg Q \rightarrow \neg P \)[/tex]
- This is not the same as our second statement.
Converse:
The converse of [tex]\( P \rightarrow \neg Q \)[/tex] is [tex]\( \neg Q \rightarrow P \)[/tex]. This just flips the hypothesis and conclusion.
- Original: [tex]\( P \rightarrow \neg Q \)[/tex]
- Converse: [tex]\( \neg Q \rightarrow P \)[/tex]
- This is not the same as our second statement.
Inverse:
The inverse of [tex]\( P \rightarrow \neg Q \)[/tex] is [tex]\( \neg P \rightarrow Q \)[/tex]. This negates both the hypothesis and conclusion.
- Original: [tex]\( P \rightarrow \neg Q \)[/tex]
- Inverse: [tex]\( \neg P \rightarrow Q \)[/tex]
- This is not the same as our second statement.
Let's reassess the second statement in logical terms:
- Given: [tex]\( Q \rightarrow \neg P \)[/tex]
- This aligns with the format of the contrapositive.
Therefore, the second conditional statement: "If Harold is making a sandwich for lunch, then he will not put Gorgonzola cheese on it," is the contrapositive of the first conditional statement.
Hence, the correct answer is: contrapositive.
The original statement is:
1. If Harold will put Gorgonzola cheese on a sandwich, then he is not making it for lunch.
- This can be written in logical terms as: [tex]\( P \rightarrow \neg Q \)[/tex]
- Where [tex]\( P \)[/tex] is "Harold will put Gorgonzola cheese on a sandwich" and [tex]\( \neg Q \)[/tex] is "he is not making it for lunch".
The second statement is:
2. If Harold is making a sandwich for lunch, then he will not put Gorgonzola cheese on it.
- This can be written in logical terms as: [tex]\( Q \rightarrow \neg P \)[/tex]
- Where [tex]\( Q \)[/tex] is "Harold is making a sandwich for lunch" and [tex]\( \neg P \)[/tex] is "he will not put Gorgonzola cheese on it".
Now, let's explore the definitions:
Contrapositive:
The contrapositive of [tex]\( P \rightarrow \neg Q \)[/tex] is [tex]\( \neg Q \rightarrow \neg P \)[/tex]. This flips and negates both the hypothesis and conclusion.
- Original: [tex]\( P \rightarrow \neg Q \)[/tex]
- Contrapositive: [tex]\( \neg Q \rightarrow \neg P \)[/tex]
- This is not the same as our second statement.
Converse:
The converse of [tex]\( P \rightarrow \neg Q \)[/tex] is [tex]\( \neg Q \rightarrow P \)[/tex]. This just flips the hypothesis and conclusion.
- Original: [tex]\( P \rightarrow \neg Q \)[/tex]
- Converse: [tex]\( \neg Q \rightarrow P \)[/tex]
- This is not the same as our second statement.
Inverse:
The inverse of [tex]\( P \rightarrow \neg Q \)[/tex] is [tex]\( \neg P \rightarrow Q \)[/tex]. This negates both the hypothesis and conclusion.
- Original: [tex]\( P \rightarrow \neg Q \)[/tex]
- Inverse: [tex]\( \neg P \rightarrow Q \)[/tex]
- This is not the same as our second statement.
Let's reassess the second statement in logical terms:
- Given: [tex]\( Q \rightarrow \neg P \)[/tex]
- This aligns with the format of the contrapositive.
Therefore, the second conditional statement: "If Harold is making a sandwich for lunch, then he will not put Gorgonzola cheese on it," is the contrapositive of the first conditional statement.
Hence, the correct answer is: contrapositive.
