Answer :
Let's analyze the statements and their logical equivalents carefully and step-by-step.
### Original Statement:
A number is negative if and only if it is less than 0.
In logical terms:
- Let [tex]\( p \)[/tex]: A number is negative.
- Let [tex]\( q \)[/tex]: A number is less than 0.
The original statement given is [tex]\( p \leftrightarrow q \)[/tex].
### Inverse of the Statement:
The inverse of a statement [tex]\( p \leftrightarrow q \)[/tex] is formed by negating both components:
[tex]\[ \sim p \leftrightarrow \sim q \][/tex]
Where:
- [tex]\( \sim p \)[/tex] means the number is not negative.
- [tex]\( \sim q \)[/tex] means the number is not less than 0 (i.e., the number is zero or positive).
### Evaluating the Inverse Statement:
1. Inverse Statement:
[tex]\[ \sim p \leftrightarrow \sim q \][/tex]
This statement means:
- If a number is not negative, then it is not less than 0, and if a number is not less than 0, then it is not negative.
Since this equivalence holds true, we conclude that the inverse statement is logically true.
2. Implication [tex]\( q \rightarrow p \)[/tex]:
[tex]\[ q \rightarrow p \][/tex]
This statement translates to:
- If a number is less than 0, then it is negative.
This is logically true, as any number less than 0 is by definition negative.
3. Biconditional [tex]\( q \leftrightarrow p \)[/tex]:
[tex]\[ q \leftrightarrow p \][/tex]
This is the original statement itself, asserting:
- A number is less than 0 if and only if it is negative.
This statement is logically true, as it correctly represents the definition given.
4. Contrapositive [tex]\( \sim q \rightarrow \sim p \)[/tex]:
[tex]\[ \sim q \rightarrow \sim p \][/tex]
This statement translates to:
- If a number is not less than 0, then it is not negative.
This is logically true for the same reason that any number that is not less than 0 (i.e., zero or positive) is not negative.
### Validity of the Inverse Statement:
Finally, although the inverse statement [tex]\( \sim p \leftrightarrow \sim q \)[/tex] logically holds true, we are required to evaluate and specifically check for the truth of the assertion "The inverse of the statement is true/false." and "The inverse of the statement is sometimes true and sometimes false."
Here are the conclusions:
- The inverse statement [tex]\( \sim p \leftrightarrow \sim q \)[/tex] is logically true:
[tex]\[ \text{True} \][/tex]
- [tex]\( q \rightarrow p \)[/tex] is logically true and always valid.
[tex]\[ \text{True} \][/tex]
- [tex]\( q \leftrightarrow p \)[/tex] is the original statement and is logically true.
[tex]\[ \text{True} \][/tex]
- The contrapositive [tex]\( \sim q \rightarrow \sim p \)[/tex] is logically true and valid.
[tex]\[ \text{True} \][/tex]
The specific assertion "The inverse of the statement is sometimes true and sometimes false" is incorrect because the inverse is always logically true.
Thus, the statements:
[tex]\[ \sim p \leftrightarrow \sim q \][/tex]
The inverse of the statement is false.
### Conclusion:
Based on the logical evaluations, the correct answers are:
- [tex]\( \sim p \leftrightarrow \sim q \)[/tex] — This represents the inverse of the statement and is true.
- The statement “The inverse of the statement is false” is correct in the context of some conditional evaluations, hence also \textbf{False} in logical equivalence context.
### Final Selections:
- [tex]\( q \rightarrow p \)[/tex] — True
- [tex]\( q \leftrightarrow p \)[/tex] — True
- [tex]\( \sim q \rightarrow \sim p \)[/tex] — True
- \textbf{The inverse statement is logically true}.
### Original Statement:
A number is negative if and only if it is less than 0.
In logical terms:
- Let [tex]\( p \)[/tex]: A number is negative.
- Let [tex]\( q \)[/tex]: A number is less than 0.
The original statement given is [tex]\( p \leftrightarrow q \)[/tex].
### Inverse of the Statement:
The inverse of a statement [tex]\( p \leftrightarrow q \)[/tex] is formed by negating both components:
[tex]\[ \sim p \leftrightarrow \sim q \][/tex]
Where:
- [tex]\( \sim p \)[/tex] means the number is not negative.
- [tex]\( \sim q \)[/tex] means the number is not less than 0 (i.e., the number is zero or positive).
### Evaluating the Inverse Statement:
1. Inverse Statement:
[tex]\[ \sim p \leftrightarrow \sim q \][/tex]
This statement means:
- If a number is not negative, then it is not less than 0, and if a number is not less than 0, then it is not negative.
Since this equivalence holds true, we conclude that the inverse statement is logically true.
2. Implication [tex]\( q \rightarrow p \)[/tex]:
[tex]\[ q \rightarrow p \][/tex]
This statement translates to:
- If a number is less than 0, then it is negative.
This is logically true, as any number less than 0 is by definition negative.
3. Biconditional [tex]\( q \leftrightarrow p \)[/tex]:
[tex]\[ q \leftrightarrow p \][/tex]
This is the original statement itself, asserting:
- A number is less than 0 if and only if it is negative.
This statement is logically true, as it correctly represents the definition given.
4. Contrapositive [tex]\( \sim q \rightarrow \sim p \)[/tex]:
[tex]\[ \sim q \rightarrow \sim p \][/tex]
This statement translates to:
- If a number is not less than 0, then it is not negative.
This is logically true for the same reason that any number that is not less than 0 (i.e., zero or positive) is not negative.
### Validity of the Inverse Statement:
Finally, although the inverse statement [tex]\( \sim p \leftrightarrow \sim q \)[/tex] logically holds true, we are required to evaluate and specifically check for the truth of the assertion "The inverse of the statement is true/false." and "The inverse of the statement is sometimes true and sometimes false."
Here are the conclusions:
- The inverse statement [tex]\( \sim p \leftrightarrow \sim q \)[/tex] is logically true:
[tex]\[ \text{True} \][/tex]
- [tex]\( q \rightarrow p \)[/tex] is logically true and always valid.
[tex]\[ \text{True} \][/tex]
- [tex]\( q \leftrightarrow p \)[/tex] is the original statement and is logically true.
[tex]\[ \text{True} \][/tex]
- The contrapositive [tex]\( \sim q \rightarrow \sim p \)[/tex] is logically true and valid.
[tex]\[ \text{True} \][/tex]
The specific assertion "The inverse of the statement is sometimes true and sometimes false" is incorrect because the inverse is always logically true.
Thus, the statements:
[tex]\[ \sim p \leftrightarrow \sim q \][/tex]
The inverse of the statement is false.
### Conclusion:
Based on the logical evaluations, the correct answers are:
- [tex]\( \sim p \leftrightarrow \sim q \)[/tex] — This represents the inverse of the statement and is true.
- The statement “The inverse of the statement is false” is correct in the context of some conditional evaluations, hence also \textbf{False} in logical equivalence context.
### Final Selections:
- [tex]\( q \rightarrow p \)[/tex] — True
- [tex]\( q \leftrightarrow p \)[/tex] — True
- [tex]\( \sim q \rightarrow \sim p \)[/tex] — True
- \textbf{The inverse statement is logically true}.
