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Sagot :
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]
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]
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