To determine which two statements must logically be true given that [tex]\( p \)[/tex] is true (i.e., it is raining), let's analyze each of the logical expressions step by step:
### 1. [tex]\( p \vee q \)[/tex] ( [tex]\( p \)[/tex] OR [tex]\( q \)[/tex] )
The disjunction (OR) of two propositions [tex]\( p \)[/tex] and [tex]\( q \)[/tex] is true if at least one of the propositions is true.
- Given: [tex]\( p \)[/tex] is true.
- Therefore, regardless of whether [tex]\( q \)[/tex] is true or false, [tex]\( p \vee q \)[/tex] will be true because [tex]\( p \)[/tex] is true.
### 2. [tex]\( p \wedge q \)[/tex] ( [tex]\( p \)[/tex] AND [tex]\( q \)[/tex] )
The conjunction (AND) of two propositions [tex]\( p \)[/tex] and [tex]\( q \)[/tex] is true if both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are true.
- Given: [tex]\( p \)[/tex] is true.
- [tex]\( q \)[/tex] is not given, so it can be either true or false.
- Hence, [tex]\( p \wedge q \)[/tex] will be true if [tex]\( q \)[/tex] is true. But if [tex]\( q \)[/tex] is false, [tex]\( p \wedge q \)[/tex] will be false. We cannot definitively say that [tex]\( p \wedge q \)[/tex] is true without knowing the value of [tex]\( q \)[/tex].
### 3. [tex]\( q \rightarrow p \)[/tex] ( [tex]\( q \)[/tex] IMPLIES [tex]\( p \)[/tex] )
The implication (if-then statement) [tex]\( q \rightarrow p \)[/tex] is true in all cases except when [tex]\( q \)[/tex] is true and [tex]\( p \)[/tex] is false.
- Given: [tex]\( p \)[/tex] is true.
- [tex]\( q \)[/tex] can be either true or false.
- Therefore, if [tex]\( q \)[/tex] is true, [tex]\( p \)[/tex] is true, making the implication true.
- If [tex]\( q \)[/tex] is false, then the implication is also true because a false proposition implies any proposition.
- Thus, [tex]\( q \rightarrow p \)[/tex] must be true in both scenarios.
### 4. [tex]\( p - q \)[/tex]
This notation does not represent a standard logical operation in classical logic. It is typically not used in standard logical expressions, so it is unclear what specific operation is intended.
### 5. [tex]\( q - p \)[/tex]
Similar to [tex]\( p - q \)[/tex], this notation does not represent a standard logical operation and is not recognized in classical logic.
### Conclusion
Given the logical analysis above, the two statements that must logically be true if [tex]\( p \)[/tex] is true are:
1. [tex]\( p \vee q \)[/tex] ( [tex]\( p \)[/tex] OR [tex]\( q \)[/tex] )
2. [tex]\( q \rightarrow p \)[/tex] ( [tex]\( q \)[/tex] IMPLIES [tex]\( p \)[/tex] )
These two logical expressions will always be true if [tex]\( p \)[/tex] (it is raining) is true.
