To analyze the given logical statement [tex]\((\sim p \rightarrow q) \vee r\)[/tex], let's break it down step-by-step with respect to the definitions of [tex]\( p \)[/tex], [tex]\( q \)[/tex], and [tex]\( r \)[/tex]:
1. Definitions:
- [tex]\( p \)[/tex]: "a number is divisible by 2"
- [tex]\( q \)[/tex]: "a number is odd"
- [tex]\( r \)[/tex]: "a number is even"
2. Negation of [tex]\( p \)[/tex]:
- [tex]\( \sim p \)[/tex]: "a number is not divisible by 2"
3. Implication:
- [tex]\( \sim p \rightarrow q \)[/tex]: "If a number is not divisible by 2, then it is odd"
- This is logically true because a number that is not divisible by 2 is indeed an odd number.
4. Disjunction with [tex]\( r \)[/tex]:
- Now, consider [tex]\((\sim p \rightarrow q) \vee r\)[/tex]:
- [tex]\((\sim p \rightarrow q)\)[/tex] is saying "if the number is not divisible by 2, then it's odd".
- [tex]\( r \)[/tex], as "the number is even", directly states the number is even.
5. Combining the Statements:
- Hence, [tex]\((\sim p \rightarrow q) \vee r\)[/tex] can be interpreted as: "If a number is not divisible by 2, then it is odd, or the number is even".
6. Choosing the Correct Option:
- We need to match this implication to the options given:
[tex]\[
\begin{aligned}
&\text{A. If a number is divisible by 2, then it's even. Otherwise, it's odd.} \\
&\text{B. If a number is divisible by 2, then it isn't even. Otherwise, it's even.} \\
&\text{C. If a number isn't divisible by 2, then it's odd. Otherwise, it's even.} \\
&\text{D. If a number is divisible by 2, then it isn't odd. Otherwise, it's odd.}
\end{aligned}
\][/tex]
- Going through each option:
- A: Doesn't match because we need to start with "a number isn't divisible by 2".
- B: Incorrect because it contradicts [tex]\( r \)[/tex] and the given conditions.
- D: Presents a rearranged implication and doesn't match the statement we're analyzing.
Correct Option:
[tex]\[
\text{C. If a number isn't divisible by 2, then it's odd. Otherwise, it's even.}
\][/tex]
Therefore, the correct answer is C.
