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Sagot :
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.
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.
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