Determine which, if any, of the three statements are equivalent.
I) If there is an opera performance, then Paula will not wear jeans.
II) If there is no opera performance, then Paula will wear jeans.
III) Either there is an opera performance or Paula will wear jeans.
III is II and I combined.
so all 3 statements are equivalent?
To determine which, if any, of the three statements are equivalent, we need to assess the logical relationships among them.
Let's break down each statement and analyze them individually:
I) "If there is an opera performance, then Paula will not wear jeans."
This statement implies that the occurrence of an opera performance will result in Paula not wearing jeans. In other words, the absence of an opera performance allows for the possibility of Paula wearing jeans.
II) "If there is no opera performance, then Paula will wear jeans."
This statement suggests that the lack of an opera performance leads to Paula wearing jeans. It is essentially the opposite of statement I. Here, the absence of an opera performance directly leads to Paula wearing jeans.
III) "Either there is an opera performance or Paula will wear jeans."
This statement presents a logical disjunction, combining the two possibilities. It asserts that either an opera performance will occur or Paula will wear jeans. It does not provide specific cause-and-effect relationships between the two events.
Now, let's compare the statements to determine their equivalence:
To establish equivalence, two statements must have the same truth value under all possible scenarios. Let's consider several scenarios:
1) If there is an opera performance:
- Statement I is true because it states that Paula will not wear jeans.
- Statement II is false because it states that Paula will wear jeans.
- Statement III is true since there is an opera performance.
2) If there is no opera performance:
- Statement I is false since it is negated by the absence of an opera performance.
- Statement II is true since Paula will wear jeans.
- Statement III is true because there is no opera performance.
Based on these scenarios, we can see that only statement II is equivalent to statement III. Both statements II and III agree about Paula wearing jeans when there is no opera performance.
Therefore, the conclusion is that statements II and III are equivalent, while statement I is not equivalent to either of them.