To determine the inverse of the conditional statement [tex]\( p \rightarrow q \)[/tex], we need to understand the logical structure of such statements.
First, let's examine the given conditional statement:
[tex]\[ p \rightarrow q \][/tex]
where:
- [tex]\( p \)[/tex] is "a number is doubled"
- [tex]\( q \)[/tex] is "the result is even"
The inverse of the conditional statement [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim p \rightarrow \sim q \)[/tex]. This means that if [tex]\( p \)[/tex] is false, then [tex]\( q \)[/tex] should also be false.
Now, let's break it down step by step:
1. Identify [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
- [tex]\( p \)[/tex]: a number is doubled
- [tex]\( q \)[/tex]: the result is even
2. Determine the negations ([tex]\(\sim p\)[/tex] and [tex]\(\sim q\)[/tex]):
- [tex]\(\sim p\)[/tex]: a number is not doubled
- [tex]\(\sim q\)[/tex]: the result is not even
3. Form the inverse statement ([tex]\(\sim p \rightarrow \sim q\)[/tex]):
- If a number is not doubled, then the result is not even.
Given this breakdown, let's compare the provided options:
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 correctly represents the inverse of the original statement.
2. [tex]\( q \rightarrow p\)[/tex] where [tex]\( p = \)[/tex] a number is doubled and [tex]\( q = \)[/tex] the result is even:
- This represents the converse, not the inverse.
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 reverses the roles of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] and 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 again represents the converse but with [tex]\( p \)[/tex] and [tex]\( q \)[/tex] reversed, and is incorrect.
The correct option is:
[tex]\[ \sim p \rightarrow \sim q \text{ where } p = \text{a number is doubled and } q = \text{the result is even} \][/tex]
Thus, the inverse of the given statement is represented by the first option. The correct answer is:
[tex]\[ 1 \][/tex]
