Alright, let's break down the provided logical statements to find which one is equivalent to [tex]\(\sim q \rightarrow \sim p\)[/tex].
Given:
- [tex]\( p \)[/tex]: [tex]\( n \)[/tex] is a multiple of two.
- [tex]\( q \)[/tex]: [tex]\( n \)[/tex] is an even number.
We need to determine the statement equivalent to [tex]\(\sim q \rightarrow \sim p\)[/tex].
### Understanding the Given Statement:
1. [tex]\(\sim q\)[/tex]: [tex]\( n \)[/tex] is not an even number.
2. [tex]\(\sim p\)[/tex]: [tex]\( n \)[/tex] is not a multiple of two.
Thus, [tex]\(\sim q \rightarrow \sim p\)[/tex] is read as:
"If [tex]\( n \)[/tex] is not an even number, then [tex]\( n \)[/tex] is not a multiple of two."
### Finding the Equivalent Statement:
To find the equivalent statement, we can use the rule of contrapositive. The contrapositive of a statement "A implies B" is "not B implies not A", and both are logically equivalent.
Here's how we apply this:
- The statement we have is [tex]\(\sim q \rightarrow \sim p\)[/tex].
- The contrapositive of [tex]\(\sim q \rightarrow \sim p\)[/tex] is [tex]\( p \rightarrow q\)[/tex].
Here’s why [tex]\( p \rightarrow q\)[/tex] is equivalent:
- [tex]\( p\)[/tex]: [tex]\( n \)[/tex] is a multiple of two.
- [tex]\( q\)[/tex]: [tex]\( n \)[/tex] is an even number.
So, [tex]\( p \rightarrow q \)[/tex] means:
"If [tex]\( n \)[/tex] is a multiple of two, then [tex]\( n \)[/tex] is an even number."
### Conclusion:
After the analysis, we find that [tex]\( p \rightarrow q \)[/tex] is equivalent to [tex]\(\sim q \rightarrow \sim p\)[/tex].
Therefore, the correct option is:
[tex]\[ p \rightarrow q \][/tex]
So, the statement equivalent to [tex]\(\sim q \rightarrow \sim p\)[/tex] is [tex]\( p \rightarrow q \)[/tex]. The correct answer is:
[tex]\[ \text{3) } p \rightarrow q \][/tex]
