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

A. 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].

B. 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].

C. 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].

D. 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].

E. 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].

Sagot :

GinnyAnswer
Let's break down the given logical statements step by step. The original statement is given as:

"If a number is negative, the additive inverse is positive." This statement can be symbolized using logic notation where:
- [tex]\( p \)[/tex] represents "a number is negative"
- [tex]\( q \)[/tex] represents "the additive inverse is positive"

Thus, the original statement symbolically is:
[tex]\[ p \rightarrow q \][/tex]

Now, let's analyze each of the given options:

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

2. [tex]\( \sim p \rightarrow \sim q \)[/tex]:
- This represents the inverse of the original statement.
- The inverse changes the condition to "If a number is not negative, the additive inverse is not positive."
- This is the correct inverse form.

3. [tex]\( \sim q \rightarrow \sim p \)[/tex]:
- This does not represent the proper converse form. The correct converse changes the order of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] without negation applied directly.
- So, this is not the converse.

4. [tex]\( q \rightarrow p \)[/tex]:
- This represents the converse of the original statement.
- The converse reverses the implications: "If the additive inverse is positive, then the number is negative."
- This is true as a logical converse.

Thus, analyzing each option, we can confirm:

1. [tex]\( p \rightarrow q \)[/tex] (original statement)
2. [tex]\( \sim p \rightarrow \sim q \)[/tex] (inverse)
3. [tex]\( q \rightarrow p \)[/tex] (converse)

Therefore, the selections (1, 2, 3) align correctly with the provided logical breakdown:
- Selection 1: [tex]\( p \rightarrow q \)[/tex]
- Selection 2: [tex]\( \sim p \rightarrow \sim q \)[/tex]
- Selection 3: [tex]\( q \rightarrow p \)[/tex]

Thus, the three true options are:

- [tex]\( p \rightarrow q \)[/tex] (Option 1)
- [tex]\( \sim p \rightarrow \sim q \)[/tex] (Option 2)
- [tex]\( q \rightarrow p \)[/tex] (Option 5)
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