Let's break down the given conditional statement step-by-step:
The given statement is: "If 3 is an even number, then [tex]\(5 + 3 = 8\)[/tex]."
1. Identify the Hypothesis and Conclusion:
- Hypothesis ([tex]\( p \)[/tex]): "3 is an even number."
- Conclusion ([tex]\( q \)[/tex]): "[tex]\(5 + 3 = 8\)[/tex]."
2. Determine the truth value of the Hypothesis:
- The hypothesis "3 is an even number" is false because 3 is not an even number; it is an odd number.
3. Determine the truth value of the Conclusion:
- The conclusion "[tex]\(5 + 3 = 8\)[/tex]" is true because, indeed, [tex]\(5 + 3\)[/tex] equals 8.
4. Analyze the Conditional Statement:
- A conditional statement "If [tex]\( p \)[/tex], then [tex]\( q \)[/tex]" ([tex]\( p \rightarrow q \)[/tex]) is evaluated based on the truth values of [tex]\( p \)[/tex] (the hypothesis) and [tex]\( q \)[/tex] (the conclusion).
- A conditional statement is true in all cases except when [tex]\( p \)[/tex] is true and [tex]\( q \)[/tex] is false.
The possible scenarios are:
- [tex]\( p \)[/tex] is true and [tex]\( q \)[/tex] is true → Conditional statement is true.
- [tex]\( p \)[/tex] is true and [tex]\( q \)[/tex] is false → Conditional statement is false.
- [tex]\( p \)[/tex] is false and [tex]\( q \)[/tex] is true → Conditional statement is true.
- [tex]\( p \)[/tex] is false and [tex]\( q \)[/tex] is false → Conditional statement is true.
In our case:
- [tex]\( p \)[/tex] (the hypothesis) is false.
- [tex]\( q \)[/tex] (the conclusion) is true.
- Since [tex]\( p \)[/tex] is false and [tex]\( q \)[/tex] is true, the conditional statement "If [tex]\( p \)[/tex], then [tex]\( q \)[/tex]" is true.
So, the truth value of the given conditional statement "If 3 is an even number, then [tex]\( 5 + 3 = 8 \)[/tex]" is true.
