8. 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.
a.)I,II, and III are equivalent
b.)I and II are equivalent
c.)II and III are equivalent
d.)I and III are equivalent
e.)None are equivalent
To determine whether the three statements are equivalent, we need to examine their logical relationships.
Statement I: "If the dog wags its tail, then the dog is not calm."
Statement II: "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."
To check for equivalence, we can compare the two statements by converting them into logical form using "p" for "the dog wags its tail" and "q" for "the dog is not calm."
Statement I: p → q
Statement II: ¬p ∨ ¬q (negation of egation of q)
Statement III: ¬q → p
We can determine equivalence by comparing the truth tables of the statements. Here are the truth tables for each statement:
Statement I:
| p | q | p → q |
|---|---|-------|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Statement II:
| p | q | ¬p ∨ ¬q |
|---|---|---------|
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | T |
Statement III:
| p | q | ¬q → p |
|---|---|--------|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
By comparing the truth tables, we can see that Statement I and Statement III have the same truth values for all combinations of p and q. Thus, Statement I and III are equivalent statements.
Based on this analysis, the correct answer is:
d) I and III are equivalent