Answer :
Certainly! Let's analyze the logical statements given the assumption that [tex]\( p \)[/tex] (It is raining) is true. We need to determine which two statements would logically be true.
### Logical Statements Breakdown:
1. [tex]\( p \vee q \)[/tex] (p OR q)
- This statement is true if at least one of [tex]\( p \)[/tex] or [tex]\( q \)[/tex] is true.
- Since [tex]\( p \)[/tex] is true, [tex]\( p \vee q \)[/tex] will always be true regardless of the value of [tex]\( q \)[/tex].
2. [tex]\( p \wedge q \)[/tex] (p AND q)
- This statement is true only if both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are true.
- We know [tex]\( p \)[/tex] is true, but we have no information about [tex]\( q \)[/tex]. Therefore, we cannot determine the truth value of [tex]\( p \wedge q \)[/tex].
3. [tex]\( q \rightarrow p \)[/tex] (q implies p)
- This statement is true if whenever [tex]\( q \)[/tex] is true, [tex]\( p \)[/tex] must also be true. Another way to express this is that [tex]\( q \rightarrow p \)[/tex] is false only if [tex]\( q \)[/tex] is true and [tex]\( p \)[/tex] is false.
- Since [tex]\( p \)[/tex] is true, [tex]\( q \rightarrow p \)[/tex] will always be true irrespective of the value of [tex]\( q \)[/tex].
4. [tex]\( p - q \)[/tex]
- This is not a standard logical notation. Without a proper definition, we cannot evaluate this statement.
5. [tex]\( q - p \)[/tex]
- Similarly, this is not a standard logical notation. Without a clear definition, we cannot evaluate this statement.
### Conclusion:
The two statements that must logically be true given that [tex]\( p \)[/tex] is true are:
1. [tex]\( p \vee q \)[/tex] (p OR q)
3. [tex]\( q \rightarrow p \)[/tex] (q implies p)
Therefore, the two logically true statements are:
[tex]\[ \boxed{p \vee q \text{ (p OR q) and } q \rightarrow p \text{ (q implies p)}} \][/tex]
### Logical Statements Breakdown:
1. [tex]\( p \vee q \)[/tex] (p OR q)
- This statement is true if at least one of [tex]\( p \)[/tex] or [tex]\( q \)[/tex] is true.
- Since [tex]\( p \)[/tex] is true, [tex]\( p \vee q \)[/tex] will always be true regardless of the value of [tex]\( q \)[/tex].
2. [tex]\( p \wedge q \)[/tex] (p AND q)
- This statement is true only if both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are true.
- We know [tex]\( p \)[/tex] is true, but we have no information about [tex]\( q \)[/tex]. Therefore, we cannot determine the truth value of [tex]\( p \wedge q \)[/tex].
3. [tex]\( q \rightarrow p \)[/tex] (q implies p)
- This statement is true if whenever [tex]\( q \)[/tex] is true, [tex]\( p \)[/tex] must also be true. Another way to express this is that [tex]\( q \rightarrow p \)[/tex] is false only if [tex]\( q \)[/tex] is true and [tex]\( p \)[/tex] is false.
- Since [tex]\( p \)[/tex] is true, [tex]\( q \rightarrow p \)[/tex] will always be true irrespective of the value of [tex]\( q \)[/tex].
4. [tex]\( p - q \)[/tex]
- This is not a standard logical notation. Without a proper definition, we cannot evaluate this statement.
5. [tex]\( q - p \)[/tex]
- Similarly, this is not a standard logical notation. Without a clear definition, we cannot evaluate this statement.
### Conclusion:
The two statements that must logically be true given that [tex]\( p \)[/tex] is true are:
1. [tex]\( p \vee q \)[/tex] (p OR q)
3. [tex]\( q \rightarrow p \)[/tex] (q implies p)
Therefore, the two logically true statements are:
[tex]\[ \boxed{p \vee q \text{ (p OR q) and } q \rightarrow p \text{ (q implies p)}} \][/tex]
