Find solutions to your questions with the help of IDNLearn.com's expert community. Discover prompt and accurate answers from our community of experienced professionals.
Sagot :
To solve this question, let's analyze the logical operations involved:
1. Original Statement:
- Given: "If a number is negative, the additive inverse is positive."
- Let [tex]\( p \)[/tex] = "a number is negative" and [tex]\( q \)[/tex] = "the additive inverse is positive".
- The original statement in standard logical form is: [tex]\( p \rightarrow q \)[/tex].
2. Inverse:
- The inverse of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim p \rightarrow \sim q \)[/tex].
- In our case, this translates to: "If a number is not negative, then the additive inverse is not positive."
3. Converse:
- The converse of [tex]\( p \rightarrow q \)[/tex] is [tex]\( q \rightarrow p \)[/tex].
- In our case, this translates to: "If the additive inverse is positive, then the number is negative."
4. Contrapositive:
- The contrapositive of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim q \rightarrow \sim p \)[/tex].
- In our case, this translates to: "If the additive inverse is not positive, then the number is not negative."
Let's now map these ideas to the given options:
1. Option 1:
- "If [tex]\( p = \)[/tex] a number is negative and [tex]\( q = \)[/tex] the additive inverse is positive, the original statement is [tex]\( p \rightarrow q \)[/tex]."
- This is true because it exactly follows the original statement.
2. Option 2:
- "If [tex]\( p = \)[/tex] a number is negative and [tex]\( q = \)[/tex] the additive inverse is positive, the inverse of the original statement is [tex]\( \sim p \rightarrow \sim q \)[/tex]."
- This is true. It correctly identifies the inverse of the original statement.
3. Option 3:
- "If [tex]\( p = \)[/tex] a number is negative and [tex]\( q = \)[/tex] the additive inverse is positive, the converse of the original statement is [tex]\( \sim q \rightarrow \sim p \)[/tex]."
- This is incorrect. The converse should be [tex]\( q \rightarrow p \)[/tex] not [tex]\( \sim q \rightarrow \sim p \)[/tex].
4. Option 4:
- "If [tex]\( q = \)[/tex] a number is negative and [tex]\( p = \)[/tex] the additive inverse is positive, the contrapositive of the original statement is [tex]\( \sim p \rightarrow \sim q \)[/tex]."
- This is incorrect because it improperly reverses the definitions of [tex]\( p \)[/tex] and [tex]\( q \)[/tex].
5. Option 5:
- "If [tex]\( q = \)[/tex] a number is negative and [tex]\( p = \)[/tex] the additive inverse is positive, the converse of the original statement is [tex]\( q \rightarrow p \)[/tex]."
- This is correct assuming the definitions of [tex]\( q \)[/tex] and [tex]\( p \)[/tex] are reversed from the original setup. This makes the statement itself true in terms of logical structure, creating the converse properly.
Thus, the three options that are true are:
- If [tex]\( p = \)[/tex] a number is negative and [tex]\( q = \)[/tex] the additive inverse is positive, the original statement is [tex]\( p \rightarrow q \)[/tex].
- If [tex]\( p = \)[/tex] a number is negative and [tex]\( q = \)[/tex] the additive inverse is positive, the inverse of the original statement is [tex]\( \sim p \rightarrow \sim q \)[/tex].
- If [tex]\( q = \)[/tex] a number is negative and [tex]\( p = \)[/tex] the additive inverse is positive, the converse of the original statement is [tex]\( q \rightarrow p \)[/tex].
So, the correct options are:
1, 2, and 5.
1. Original Statement:
- Given: "If a number is negative, the additive inverse is positive."
- Let [tex]\( p \)[/tex] = "a number is negative" and [tex]\( q \)[/tex] = "the additive inverse is positive".
- The original statement in standard logical form is: [tex]\( p \rightarrow q \)[/tex].
2. Inverse:
- The inverse of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim p \rightarrow \sim q \)[/tex].
- In our case, this translates to: "If a number is not negative, then the additive inverse is not positive."
3. Converse:
- The converse of [tex]\( p \rightarrow q \)[/tex] is [tex]\( q \rightarrow p \)[/tex].
- In our case, this translates to: "If the additive inverse is positive, then the number is negative."
4. Contrapositive:
- The contrapositive of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim q \rightarrow \sim p \)[/tex].
- In our case, this translates to: "If the additive inverse is not positive, then the number is not negative."
Let's now map these ideas to the given options:
1. Option 1:
- "If [tex]\( p = \)[/tex] a number is negative and [tex]\( q = \)[/tex] the additive inverse is positive, the original statement is [tex]\( p \rightarrow q \)[/tex]."
- This is true because it exactly follows the original statement.
2. Option 2:
- "If [tex]\( p = \)[/tex] a number is negative and [tex]\( q = \)[/tex] the additive inverse is positive, the inverse of the original statement is [tex]\( \sim p \rightarrow \sim q \)[/tex]."
- This is true. It correctly identifies the inverse of the original statement.
3. Option 3:
- "If [tex]\( p = \)[/tex] a number is negative and [tex]\( q = \)[/tex] the additive inverse is positive, the converse of the original statement is [tex]\( \sim q \rightarrow \sim p \)[/tex]."
- This is incorrect. The converse should be [tex]\( q \rightarrow p \)[/tex] not [tex]\( \sim q \rightarrow \sim p \)[/tex].
4. Option 4:
- "If [tex]\( q = \)[/tex] a number is negative and [tex]\( p = \)[/tex] the additive inverse is positive, the contrapositive of the original statement is [tex]\( \sim p \rightarrow \sim q \)[/tex]."
- This is incorrect because it improperly reverses the definitions of [tex]\( p \)[/tex] and [tex]\( q \)[/tex].
5. Option 5:
- "If [tex]\( q = \)[/tex] a number is negative and [tex]\( p = \)[/tex] the additive inverse is positive, the converse of the original statement is [tex]\( q \rightarrow p \)[/tex]."
- This is correct assuming the definitions of [tex]\( q \)[/tex] and [tex]\( p \)[/tex] are reversed from the original setup. This makes the statement itself true in terms of logical structure, creating the converse properly.
Thus, the three options that are true are:
- If [tex]\( p = \)[/tex] a number is negative and [tex]\( q = \)[/tex] the additive inverse is positive, the original statement is [tex]\( p \rightarrow q \)[/tex].
- If [tex]\( p = \)[/tex] a number is negative and [tex]\( q = \)[/tex] the additive inverse is positive, the inverse of the original statement is [tex]\( \sim p \rightarrow \sim q \)[/tex].
- If [tex]\( q = \)[/tex] a number is negative and [tex]\( p = \)[/tex] the additive inverse is positive, the converse of the original statement is [tex]\( q \rightarrow p \)[/tex].
So, the correct options are:
1, 2, and 5.
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Thank you for visiting IDNLearn.com. We’re here to provide clear and concise answers, so visit us again soon.
