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)
