To determine which of the compound statements involving [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are true, let's evaluate each one step by step, given that [tex]\( p \)[/tex] is true and [tex]\( q \)[/tex] is false.
1. [tex]\( p \wedge q \)[/tex] (p AND q):
- For [tex]\( p \wedge q \)[/tex] to be true, both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] need to be true.
- Since [tex]\( p \)[/tex] is true but [tex]\( q \)[/tex] is false, [tex]\( p \wedge q \)[/tex] is false.
2. [tex]\( p \vee q \)[/tex] (p OR q):
- For [tex]\( p \vee q \)[/tex] to be true, at least one of [tex]\( p \)[/tex] or [tex]\( q \)[/tex] must be true.
- Since [tex]\( p \)[/tex] is true, [tex]\( p \vee q \)[/tex] is true regardless of [tex]\( q \)[/tex].
3. [tex]\( p \rightarrow q \)[/tex] (p implies q):
- [tex]\( p \rightarrow q \)[/tex] is true unless [tex]\( p \)[/tex] is true and [tex]\( q \)[/tex] is false.
- Since [tex]\( p \)[/tex] is true and [tex]\( q \)[/tex] is false in our case, [tex]\( p \rightarrow q \)[/tex] is false.
4. [tex]\( q \rightarrow p \)[/tex] (q implies p):
- [tex]\( q \rightarrow p \)[/tex] is true unless [tex]\( q \)[/tex] is true and [tex]\( p \)[/tex] is false.
- Since [tex]\( q \)[/tex] is false, [tex]\( q \rightarrow p \)[/tex] is true regardless of [tex]\( p \)[/tex].
Summarizing the evaluations:
- [tex]\( p \wedge q \)[/tex] is false.
- [tex]\( p \vee q \)[/tex] is true.
- [tex]\( p \rightarrow q \)[/tex] is false.
- [tex]\( q \rightarrow p \)[/tex] is true.
Thus, given [tex]\( p \)[/tex] is true and [tex]\( q \)[/tex] is false, the compound statements that are true are:
- [tex]\( p \vee q \)[/tex]
- [tex]\( q \rightarrow p \)[/tex]
