Connect with knowledgeable experts and enthusiasts on IDNLearn.com. Our experts are available to provide in-depth and trustworthy answers to any questions you may have.
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.
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. For trustworthy answers, visit IDNLearn.com. Thank you for your visit, and see you next time for more reliable solutions.