Math

posted by .

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.

  • Math -

    And your answer is?

  • Math -

    I think I and III are equivalent??

  • Math -

    Right! :-)

  • Math -

    great!! thank you :)

  • Math -

    You're welcome.

  • Math -

    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)

  • Math -

    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.

  • Math -

    Okay.... so I and III are not equivalent??

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. CRT 205

    Crt 205 Vacuum Sales Digital Story Text Axia College Material Text of the Vacuum Sales Digital Story This text provides the script of the Vacuum Sales digital story. Use this text as an alternative or supplement to viewing the digital …
  2. college mathematics

    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 …
  3. Math

    Carpet plus installs carpet for $100 plus $8 per square yard of carpet. Carpet World charges $75 for installation and $10 per square yard of carpet. Find the number of square yards of carpet for which the cost including carpet and …
  4. Pre-Algebra

    Carpet Masters charges $9.50 per square yard to clean a carpet. If you have two rooms with an area of six square yards each, how much will it cost to have the carpets cleaned?
  5. geometry

    you plan to carpet 468ft of space. carpet costs $26.75 per square yard and the tax is 7%. the carpet is sold in 12 ft rolls. what is the cost of the carpet?
  6. algebra

    CLEAN CARPET COMPANYS CLEAN 3 ROOMS OF CARPETING FOR 144 DOLLARS. WHICH OF THE FOLLOWING IS AN EQUIVALENT RATE ?
  7. Math

    If the cost is 3.00 doars per sq ft to clean a carpet, how much will it cost to clean a carpet 12.5 ft x 23ft?
  8. English

    What are you going to do for your home this weekend?
  9. Math

    A contractor buys 14 yd of nylon carpet and 20 yd of wool carpet for $1738. A second purchase, at the same prices, includes 16 yd of nylon carpet and 25 yd of wool carpet for $2102. Find the cost per yard of the wool carpet.
  10. math

    Sheila needs to hire a babysitter for the evening. The babysitter charges $10 for each hour and also charges for the time she spends driving to and from the house. She thinks it will take 1.5 hours of total driving time. Sheila is …

More Similar Questions