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Given the original statement "If a number is negative, the additive inverse is positive," which are true? Select three options.

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].
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].
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]\( q \rightarrow p \)[/tex].
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]\( \sim q \rightarrow \sim p \)[/tex].
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].



Answer :

GinnyAnswer
To determine which statements are true given the original statement "If a number is negative, the additive inverse is positive," we will analyze the original statement and its logical equivalents.

Firstly, let's establish the variables:
- Let [tex]\( p \)[/tex]: "a number is negative"
- Let [tex]\( q \)[/tex]: "the additive inverse is positive".

The original statement given can be written in logical form as:
- Original statement: [tex]\( p \rightarrow q \)[/tex] (If [tex]\( p \)[/tex], then [tex]\( q \)[/tex])

Next, we define and evaluate the logical equivalents:

1. Inverse of the original statement: This involves negating both the hypothesis and conclusion of the original statement.
- Inverse: [tex]\( \sim p \rightarrow \sim q \)[/tex]
- Meaning: "If a number is not negative, then the additive inverse is not positive."

2. Converse of the original statement: This involves switching the hypothesis and conclusion.
- Converse: [tex]\( q \rightarrow p \)[/tex]
- Meaning: "If the additive inverse is positive, then the number is negative."

3. Contrapositive of the original statement: This involves negating and switching the hypothesis and conclusion.
- Contrapositive: [tex]\( \sim q \rightarrow \sim p \)[/tex]
- Meaning: "If the additive inverse is not positive, then the number is not negative."

Based on our definitions and the relationships between these logical statements, the true statements involve the original statement, inverse, and contrapositive since these represent valid logical transformations:

1. The original statement [tex]\( p \rightarrow q \)[/tex]:
- "a number is negative \rightarrow the additive inverse is positive"

2. The inverse of the original statement [tex]\( \sim p \rightarrow \sim q \)[/tex]:
- "not(a number is negative) \rightarrow not(the additive inverse is positive)"

3. The contrapositive of the original statement [tex]\( \sim q \rightarrow \sim p \)[/tex]:
- "not(the additive inverse is positive) \rightarrow not(a number is negative)"

The true statements are thus:
1. [tex]\( p \rightarrow q \)[/tex]: "a number is negative \rightarrow the additive inverse is positive"
2. [tex]\( \sim p \rightarrow \sim q \)[/tex]: "not(a number is negative) \rightarrow not(the additive inverse is positive)"
3. [tex]\( \sim q \rightarrow \sim p \)[/tex]: "not(the additive inverse is positive) \rightarrow not(a number is negative)"

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