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
I'll be glad to comment on your answer.
I think it's b. but i'm not sure what my reasoning would be....
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.
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.
To determine which statements, if any, are equivalent, we can compare them and look for similarities in their logical structure.
Statement I) If the garden needs watering, then the garden does not need weeding.
Statement III) If the garden does not need watering, then the garden does not need weeding.
Both statements I and III have a similar structure: "If A, then not B." In both cases, A represents the garden's watering needs, and B represents the garden's weeding needs.
Statement II) Either the garden needs watering or the garden does not need weeding.
Statement II has a different structure compared to statements I and III, using the logical operator "or" instead of "if...then." In this case, the statement is saying that either the garden needs watering or the garden does not need weeding.
Now, let's compare the statements:
If we consider the conditions in each statement, we can see that statement I implies statement III. If the garden needs watering (A is true), it automatically means that the garden does not need weeding (so B is false). Thus, statement I being true implies statement III being true.
On the other hand, statement III does not imply statement I, as it is possible for the garden not to need watering (A is false), and still require weeding (B is true). So statement III does not guarantee the truth value of statement I.
Considering statement II, it is different from both I and III in terms of structure and meaning. While it is related to the concept of the garden's needs, it does not have the same logical structure as statement I and III. It presents a choice between two options: watering or not needing weeding.
Based on the analysis, we can conclude that statement I and statement III are equivalent, but statement II is not equivalent to either statement I or III.
Therefore, the correct answer is option d) I and III are equivalent.