Determine which if any of the three statements are equivalent, give a reason for your conclusion. Show complete work.
I). If the dog wags its tail, then the dog is not calm
II). Either the dog does not wag its tail or the dog is not calm.
III). If the dog is not calm, then the dog wags its tail
To determine if any of the three statements are equivalent, we need to evaluate their logical relationships.
Statement I: If the dog wags its tail, then the dog is not calm.
This is a conditional statement (if-then form). It states that whenever the dog wags its tail, the dog is not calm.
Statement II: Either the dog does not wag its tail or the dog is not calm.
This statement is a disjunction (either-or form). It claims that one of two conditions must be true: either the dog does not wag its tail, or the dog is not calm.
Statement III: If the dog is not calm, then the dog wags its tail.
Similar to Statement I, this is a conditional statement. It asserts that whenever the dog is not calm, the dog wags its tail.
To determine if the statements are equivalent, we need to compare their truth values under various conditions. We can use truth tables to evaluate this.
Let's assign the following variables:
P: The dog wags its tail.
Q: The dog is not calm.
Now, let's create a truth table for each statement:
For Statement I:
| P | Q | If P, then Q |
|----|----|-------------|
| T | T | F |
| T | F | F |
| F | T | T |
| F | F | T |
For Statement II:
| P | Q | Either not ot Q |
|----|----|----------------------|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
For Statement III:
| P | Q | If not Q, then P |
|----|----|-----------------|
| T | T | T |
| T | F | T |
| F | T | F |
| F | F | T |
Analyzing the truth tables, we can conclude that statements I and III are equivalent because they have identical truth values. In both cases, the dog wags its tail whenever it is not calm.
Statement II, on the other hand, is not equivalent to statements I or III. It is also a true statement, but it allows for the possibility that the dog is not calm without wagging its tail.
Therefore, the conclusion is that statements I and III are equivalent because both express the same logical relationship between the dog wagging its tail and being calm.