SOLUTION: 8. Determine which, if any, of the three statements are equivalent. Give a reason for your conclusion. Show complete work. I) If the dog licks her nose, then the dog is not happ

Question 262147: 8. Determine which, if any, of the three statements are equivalent. Give a reason for your conclusion. Show complete work.
I) If the dog licks her nose, then the dog is not happy.
II) If the dog licks her nose, then the dog is happy.
III) If the dog is not happy, then the dog does not lick her nose.
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
How do I show work on this? I don't get it?
Found 2 solutions by jim_thompson5910, richwmiller:
Answer by jim_thompson5910(21667) (Show Source):
You can put this solution on YOUR website!
Here's the basic outline to solving this problem

Step 1) Assign p: the dog licks her nose and q: the dog is happy

Step 2) Translate each statement into symbolic form.

Step 3) Create a table for each symbolic statement. Any tables which have equivalent final columns will be equivalent expressions.

For example, translate "If the dog licks her nose, then the dog is not happy. " into p -> ~q and set up the following table

p q ~q p -> ~q
T T F F
T F T T
F T F T
F F F T

Do the same for the 2 other statements and compare the last columns in the table. Let me know if this helps.

Answer by richwmiller(7655) (Show Source):
You can put this solution on YOUR website!
I and II are contradictory and so not true.
II and III are called contrapositives and are therefore true by definition and for the same reason I and III are not true.