Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Join our platform to connect with experts ready to provide detailed answers to your questions in various areas. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
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}.
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.
