Math
posted by Jen .
Determine which, if any, of the three statements are equivalent.
I) If the carpet is not clean, then Sheila will run the vacuum.
II) Either the carpet is not clean or Sheila will not run the vacuum.
III) If the carpet is clean, Sheila will not run the vacuum.

And your answer is?

I think I and III are equivalent??

Right! :)

great!! thank you :)

You're welcome.

I don't agree with this answer. I think you are studying on how to solve such problems using truth tables, so you should write down the three truth tables for the three statements and see which ones match.
E.g., put:
A = carpet is not clean.
B = Sheila will run the vacuum.
For statement 1 (S1) we have:
A=False, B=True, S1 = True
A=False, B=False, S1 = True
A=True, B=True, S1 = True
A=True, B=False, S1 = False
If you do this for the two others you see that the table for statement 2 and 3 are the same.
You can also see this from the fact that
A implies B is equivalent to
Not(A) OR B
A implies B means that if A is true, then B has to be true. Terefore, only if A is true, there is a requirement for B to be true. The only way to violate this condition is thus when B is false and A is True. So, for the statement to be true requires that either A is false or that B is true.
Now A being false making A implies B true can be counterintuitive, and this makes it dangerous to rely on intuition to analyze such problems.
Statement 3 is equivalent to:
Not(A) implies Not(B) =
Not(Not A) Or Not(B) =
A Or Not(B)
which is exactly what statement 2 says.
Another thing to remember is how to write the statement:
A implies B
in terms of Not(A) and Not(B), as you see statments 1 and 3 are not equivalent. We have:
A implies B = Not(A) OR B =
B OR Not(A) =
Not(Not(B)) OR Not(A)
If you put Not(B) = X and Not(A) = Y, you see that the last line is:
Not(X) OR Y
and that is the same as:
X implies Y
So, we find that:
A implies B =
Not(B) implies Not(A) 
Simply put, p > q is not equivalent to ~p > ~q.
Example:
If I am in Paris, then I am in France.
Therefore, if I am not in Paris, then I am not in France.
The conclusion is invalid.
Sure, I can be in some other country such as England, but I can be in another city in France. 
Okay.... so I and III are not equivalent??
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