Answer :
Let's analyze the logical statement and the given options step by step.
We are given the logical statement:
[tex]\[ (\sim p \rightarrow q) \vee r \][/tex]
where:
- \( p \) represents "a number is divisible by 2",
- \( q \) represents "a number is odd",
- \( r \) represents "a number is even".
### Step-by-Step Solution:
1. Understanding \(\sim p\):
- \(\sim p\) is the negation of \( p \).
- If \( p \) means "a number is divisible by 2", then \(\sim p\) means "a number is not divisible by 2".
2. Understanding \(\sim p \rightarrow q\):
- The implication \(\sim p \rightarrow q\) means "If a number is not divisible by 2, then it is odd".
3. Understanding \((\sim p \rightarrow q) \vee r\):
- The logical OR (\(\vee\)) operation means that the expression will be true if at least one of the components (\(\sim p \rightarrow q\) or \( r \)) is true.
- \( r \) is "the number is even".
Thus, the statement \((\sim p \rightarrow q) \vee r\) means:
- Either "If a number is not divisible by 2 (\(\sim p\)), then it is odd (\( q \))" is true,
- Or "the number is even (\( r \))".
### Translating to plain language:
This implies:
- If a number is not divisible by 2, then it must be odd.
- If a number is not odd, then it must be even.
This matches the option where we state:
- "If a number isn't divisible by 2, then it's odd. Otherwise, it's even."
Thus, the correct answer is:
[tex]\[ \boxed{C} \][/tex]
We are given the logical statement:
[tex]\[ (\sim p \rightarrow q) \vee r \][/tex]
where:
- \( p \) represents "a number is divisible by 2",
- \( q \) represents "a number is odd",
- \( r \) represents "a number is even".
### Step-by-Step Solution:
1. Understanding \(\sim p\):
- \(\sim p\) is the negation of \( p \).
- If \( p \) means "a number is divisible by 2", then \(\sim p\) means "a number is not divisible by 2".
2. Understanding \(\sim p \rightarrow q\):
- The implication \(\sim p \rightarrow q\) means "If a number is not divisible by 2, then it is odd".
3. Understanding \((\sim p \rightarrow q) \vee r\):
- The logical OR (\(\vee\)) operation means that the expression will be true if at least one of the components (\(\sim p \rightarrow q\) or \( r \)) is true.
- \( r \) is "the number is even".
Thus, the statement \((\sim p \rightarrow q) \vee r\) means:
- Either "If a number is not divisible by 2 (\(\sim p\)), then it is odd (\( q \))" is true,
- Or "the number is even (\( r \))".
### Translating to plain language:
This implies:
- If a number is not divisible by 2, then it must be odd.
- If a number is not odd, then it must be even.
This matches the option where we state:
- "If a number isn't divisible by 2, then it's odd. Otherwise, it's even."
Thus, the correct answer is:
[tex]\[ \boxed{C} \][/tex]
