To solve the problem where [tex]\( p \)[/tex] is true and [tex]\( q \)[/tex] is false, and we need to evaluate the logical implication [tex]\( p \rightarrow q \)[/tex], we proceed as follows:
1. Recall the definition of logical implication [tex]\( p \rightarrow q \)[/tex]. The statement [tex]\( p \rightarrow q \)[/tex] is false only when [tex]\( p \)[/tex] is true and [tex]\( q \)[/tex] is false. In all other cases, [tex]\( p \rightarrow q \)[/tex] is true.
2. In our situation, [tex]\( p \)[/tex] is given as true, and [tex]\( q \)[/tex] is given as false.
3. Substitute the values:
- Since [tex]\( p \)[/tex] is true and [tex]\( q \)[/tex] is false, the case where [tex]\( p \)[/tex] is true and [tex]\( q \)[/tex] is false satisfies the condition that makes the implication [tex]\( p \rightarrow q \)[/tex] false.
4. Therefore, the statement [tex]\( p \rightarrow q \)[/tex] is false.
The result is:
[tex]\[ p \rightarrow q \][/tex] is [tex]\(\boxed{false}\)[/tex].
