To determine the truth value of [tex]\( p \rightarrow q \)[/tex] given that [tex]\( p \)[/tex] is true and [tex]\( q \)[/tex] is false, we need to recall the definition of a logical implication. The implication [tex]\( p \rightarrow q \)[/tex] can be expressed as "if [tex]\( p \)[/tex], then [tex]\( q \)[/tex]".
The truth table for [tex]\( p \rightarrow q \)[/tex] is:
[tex]\[
\begin{array}{|c|c|c|}
\hline
p & q & p \rightarrow q \\
\hline
\text{True} & \text{True} & \text{True} \\
\text{True} & \text{False} & \text{False} \\
\text{False} & \text{True} & \text{True} \\
\text{False} & \text{False} & \text{True} \\
\hline
\end{array}
\][/tex]
From the truth table, we observe that the implication [tex]\( p \rightarrow q \)[/tex] is only false when [tex]\( p \)[/tex] is true and [tex]\( q \)[/tex] is false. In all other cases, [tex]\( p \rightarrow q \)[/tex] is true.
Given the conditions:
- [tex]\( p \)[/tex] is true
- [tex]\( q \)[/tex] is false
We locate these conditions in the truth table and find that [tex]\( p \rightarrow q \)[/tex] is false in this case.
Therefore, if [tex]\( p \)[/tex] is true and [tex]\( q \)[/tex] is false, then [tex]\( p \rightarrow q \)[/tex] is false.
