Answer :
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.
