The statement [tex]p \rightarrow q[/tex] represents "If a number is doubled, the result is even."
Which represents the inverse?
A. [tex]\sim p \rightarrow \sim q[/tex] where [tex]p[/tex] = a number is doubled and [tex]q[/tex] = the result is even B. [tex]q \rightarrow p[/tex] where [tex]p[/tex] = a number is doubled and [tex]q[/tex] = the result is even C. [tex]\sim p \rightarrow \sim q[/tex] where [tex]p[/tex] = the result is even and [tex]q[/tex] = a number is doubled D. [tex]q \rightarrow p[/tex] where [tex]p[/tex] = the result is even and [tex]q[/tex] = a number is doubled
To solve this problem, let's break down the logical implications and their inverses.
Given: - [tex]\( p \)[/tex]: A number is doubled. - [tex]\( q \)[/tex]: The result is even. - The original statement is [tex]\( p \rightarrow q \)[/tex], which reads "If a number is doubled, the result is even."
The inverse of a logical implication [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim q \rightarrow \sim p \)[/tex].
We want to identify the inverse in the context of this specific problem. Let's look at each of the available answer choices in detail:
1. [tex]\(\sim p \rightarrow \sim q\)[/tex] where [tex]\( p \)[/tex] = a number is doubled and [tex]\( q \)[/tex] = the result is even. - This translates to: "If a number is not doubled, the result is not even." - This description does not accurately capture the meaning of the inverse of the original statement [tex]\( p \rightarrow q \)[/tex], because it does not follow the correct form [tex]\( \sim q \rightarrow \sim p \)[/tex].
2. [tex]\( q \rightarrow p \)[/tex] where [tex]\( p \)[/tex] = a number is doubled and [tex]\( q \)[/tex] = the result is even. - This translates to: "If the result is even, then the number is doubled." - This also does not correspond to the inverse of the original statement [tex]\( p \rightarrow q \)[/tex].
3. [tex]\(\sim p \rightarrow \sim q\)[/tex] where [tex]\( p \)[/tex] = the result is even and [tex]\( q \)[/tex] = a number is doubled. - This translates to: "If the result is not even, the number is not doubled." - This option switches the definitions of [tex]\( p \)[/tex] and [tex]\( q \)[/tex], which is incorrect.
4. [tex]\( q \rightarrow p \)[/tex] where [tex]\( p \)[/tex] = the result is even and [tex]\( q \)[/tex] = a number is doubled. - This translates to: "If the result is even, a number is doubled." - This follows the correct form of the inverse: [tex]\( \sim q \rightarrow \sim p \)[/tex] when correctly interpreted in the context of the problem statement.
Therefore, the correct representation of the inverse is: [tex]\[ q \rightarrow p \text{ where } p = \text{the result is even and } q = \text{a number is doubled}. \][/tex]
The correct answer is: [tex]\[ q \rightarrow p \text{ where } p = \text{the result is even and } q = \text{a number is doubled}.\][/tex]
