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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 -

    I'll be glad to comment on your answer.

  • Math -

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

  • Math -

    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 -

    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.

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