To determine whether the logical implication [tex]\( p \rightarrow q \)[/tex] is true or false, we need to understand what [tex]\( p \rightarrow q \)[/tex] represents. The logical implication [tex]\( p \rightarrow q \)[/tex] is read as "if [tex]\( p \)[/tex], then [tex]\( q \)[/tex]" and it is equivalent to the expression "not [tex]\( p \)[/tex] or [tex]\( q \)[/tex]" ([tex]\( \neg p \)[/tex] ∨ [tex]\( q \)[/tex]) in propositional logic.
Given:
- [tex]\( p \)[/tex] is true.
- [tex]\( q \)[/tex] is false.
Let's evaluate [tex]\( p \rightarrow q \)[/tex]:
1. Write the logical expression for the implication [tex]\( p \rightarrow q \)[/tex]:
[tex]\[ p \rightarrow q \equiv \neg p \vee q \][/tex]
2. Substitute the given values:
- Since [tex]\( p \)[/tex] is true, [tex]\( \neg p \)[/tex] (not [tex]\( p \)[/tex]) is false.
- Since [tex]\( q \)[/tex] is false, [tex]\( q \)[/tex] remains false.
3. Evaluate the disjunction ([tex]\( \neg p \vee q \)[/tex]):
- [tex]\( \neg p \)[/tex] is false.
- [tex]\( q \)[/tex] is false.
So we have:
[tex]\[ \neg p \vee q = \text{false} \vee \text{false} = \text{false} \][/tex]
Therefore, the statement [tex]\( p \rightarrow q \)[/tex] evaluates to false.
Thus, if [tex]\( p \)[/tex] is true and [tex]\( q \)[/tex] is false, the implication [tex]\( p \rightarrow q \)[/tex] is [tex]\(\boxed{\text{false}}\)[/tex].
