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:
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.
