8. Determine which, if any, of the three statements are equivalent.
I) If I am hungry, then I will not be able to concentrate at the meeting.
II) Either I am not hungry or I will be able to concentrate at the meeting.
III) If I am able to concentrate at the meeting, then I am not hungry.
1 I, II, and III are equivalent
2 I and II are equivalent
3 II and III are equivalent
4 I and III are equivalent
5 None are equivalent
To determine which, if any, of the three statements are equivalent, we can compare each statement to the other two and check for logical equivalence.
Let's analyze each statement:
I) "If I am hungry, then I will not be able to concentrate at the meeting."
II) "Either I am not hungry or I will be able to concentrate at the meeting."
III) "If I am able to concentrate at the meeting, then I am not hungry."
To determine if two statements are equivalent, we need to check if they have the same truth value under all possible scenarios.
Now, let's compare the statements:
Comparing I and II:
Statement I states that if you are hungry, you cannot concentrate at the meeting. Statement II states that either you are not hungry or you can concentrate at the meeting. These statements are not equivalent because statement II allows for the possibility of being not hungry while still being unable to concentrate at the meeting. Therefore, I and II are not equivalent.
Comparing I and III:
Statement I states that if you are hungry, you cannot concentrate at the meeting. Statement III states that if you can concentrate at the meeting, you are not hungry. These statements are not equivalent because statement III allows for the possibility of not being able to concentrate at the meeting even if you are not hungry. Therefore, I and III are not equivalent.
Comparing II and III:
Statement II states that either you are not hungry or you can concentrate at the meeting. Statement III states that if you can concentrate at the meeting, you are not hungry. These statements are equivalent because they both describe the same scenarios. If you are not hungry, you can concentrate at the meeting, and if you can concentrate at the meeting, you are not hungry. Therefore, II and III are equivalent.
Based on our analysis, the answer is:
3) II and III are equivalent.