To evaluate the truth value of the conditional statement [tex]\((9 < 5) \rightarrow F\)[/tex], we need to carefully analyze the logical components of the statement.
1. Assess the Left Side of the Conditional Statement:
- The left side of the conditional statement is [tex]\(9 < 5\)[/tex].
- We need to determine whether [tex]\(9 < 5\)[/tex] is true or false.
- Since 9 is not less than 5, the statement [tex]\(9 < 5\)[/tex] is false.
- Therefore, we have [tex]\(P = \text{false}\)[/tex] where [tex]\(P\)[/tex] represents [tex]\(9 < 5\)[/tex].
2. Assess the Right Side of the Conditional Statement:
- The right side of the conditional statement is [tex]\(F\)[/tex].
- [tex]\(F\)[/tex] is explicitly stated to represent a false statement.
- Therefore, we have [tex]\(Q = \text{false}\)[/tex] where [tex]\(Q\)[/tex] represents [tex]\(F\)[/tex].
3. Evaluate the Conditional Statement [tex]\(P \rightarrow Q\)[/tex]:
- In logic, a conditional statement [tex]\(P \rightarrow Q\)[/tex] is read as "If [tex]\(P\)[/tex], then [tex]\(Q\)[/tex]".
- A conditional statement [tex]\(P \rightarrow Q\)[/tex] is true in all cases except when [tex]\(P\)[/tex] is true and [tex]\(Q\)[/tex] is false. The truth table for a conditional statement is shown below:
[tex]\[
\begin{array}{c|c|c}
P & Q & P \rightarrow Q \\
\hline
\text{true} & \text{true} & \text{true} \\
\text{true} & \text{false} & \text{false} \\
\text{false} & \text{true} & \text{true} \\
\text{false} & \text{false} & \text{true} \\
\end{array}
\][/tex]
- Given that [tex]\(P\)[/tex] is false ([tex]\(P = \text{false}\)[/tex]) and [tex]\(Q\)[/tex] is also false ([tex]\(Q = \text{false}\)[/tex]), we look at the row corresponding to [tex]\(P = \text{false}\)[/tex] and [tex]\(Q = \text{false}\)[/tex] in the truth table.
- According to the truth table, when [tex]\(P\)[/tex] is false and [tex]\(Q\)[/tex] is false, the conditional statement [tex]\(P \rightarrow Q\)[/tex] is true.
Therefore, the conditional statement [tex]\((9 < 5) \rightarrow F\)[/tex] is [tex]\(\boxed{\text{true}}\)[/tex].
