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Posted by on Friday, July 15, 2011 at 11:12am.

Determine which, if any, of the three statements are equivalent. Give a reason for your conclusion.

I) If the garden needs watering, then the garden does not need weeding.
II) Either the garden needs watering or the garden does not need weeding.
III) If the garden does not need watering, then the garden does not need weeding.

a. None are equivalent
b. I, II, and III are equivalent
c. I and II are equivalent
d. I and III are equivalent
e. II and III are equivalent

  • Math - , Friday, July 15, 2011 at 11:40am

    I'll be glad to comment on your answer.

  • Math - , Friday, July 15, 2011 at 11:44am

    I think it's b. but i'm not sure what my reasoning would be....

  • Math - , Friday, July 15, 2011 at 11:48am

    I disagree with your answer.

    I. states that if the garden is dry, the weeds don't grow.
    II. states an either/or situation -- either it's dry or the weeds don't grow.
    III. states that if it's wet, then the weeds don't grow.

  • Math - , Friday, July 15, 2011 at 12:46pm

    Disregard my previous post on a similar question. I now have a better understanding.

    I suggest converting each statement into symbolic form.

    Let p = the garden needs watering.
    Let p = the garden needs weeding.

    I. p --> ~q
    II. p V ~q
    III. ~p --> q

    Looks like the answer is a; none of them are equivalent.

    Here's a side note: a statement is always logically equivalent to it's contrapositive, i.e., p --> q is equivalent to ~q --> ~p.

    If I am in Paris, then I am in France.
    If I am not in France, then I am not in Paris.
    II.

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