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??

To determine if any of the statements are equivalent, we need to check if their logical meanings are the same.

Let's analyze each statement one by one:

I) "If the carpet is not clean, then Sheila will run the vacuum."

This statement is a conditional statement, written in the form "If A, then B." It states that if the carpet is not clean (A), then Sheila will run the vacuum (B).

II) "Either the carpet is not clean or Sheila will not run the vacuum."

This statement is a disjunction, meaning it is a statement formed by combining two simpler statements using the word "or." It states that 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."

Similar to statement I, this statement is a conditional statement. It states that if the carpet is clean, Sheila will not run the vacuum.

Now, let's compare the statements:

Statement I says that if the carpet is not clean, Sheila will run the vacuum.
Statement II says that either the carpet is not clean or Sheila will not run the vacuum.

These two statements are equivalent because they convey the same meaning. In both cases, if the carpet is not clean (or if the carpet is clean, in the negation of statement II), Sheila will not run the vacuum.

Statement III, on the other hand, says that if the carpet is clean, Sheila will not run the vacuum. This statement is not equivalent to the other two because it states the opposite condition. If the carpet is not clean, statement III doesn't provide any information about whether Sheila will run the vacuum or not.

Therefore, the equivalent statements are I) "If the carpet is not clean, then Sheila will run the vacuum" and II) "Either the carpet is not clean or Sheila will not run the vacuum."