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keebler01
Determine which, if any, of the three statements are equivalent. Give a reason for your conclusion. I) If the cat does not have claws, then the cat cannot scratch the furniture. II) If the cat can scratch the furniture, then the cat has claws. III) If the cat has claws, then the cat can scratch the furniture. a. I and II are equivalent b. I and III are equivalent c. II and III are equivalent d. I, II, and III are equivalent e. None are equivalent
The first thing you need to do is transform these statements into statements using propositional logic: \[p = \text{cat has claws}\]\[q = \text{cat can scratch furniture}\] 1) \(\neg p \implies \neg q\) 2) \(q \implies p\) 3) \(p \implies q\) Since 1 & 2 are contrapositions of each other, therefore they are logically equivalent. However 3 is not equivalent to 1 or 2. Therefore I would say the answer is \(a\)
To be more specific the converse of a statement is not logically equivalent to the statement.
So to be clear p->q does not imply q->p
Here, I will explain. Lets look at these two statements: \[p=\text{"it is raining"}\]\[q=\text{"i am getting wet"}\] Suppose \(p \implies q\) If it is raining, then I am getting wet. Is that the same as \(q \implies p\)? If I am getting wet then it is raining. The answer is clearly no. Do you understand so far?
You could be getting wet, even when it is not raining (shower for instance).
okay i get that part!
So we've established that \(p \implies q\) does not necessarily mean \(q \implies p\). However, as I have stated before. I have made the claim that if \(p \implies q\) then \(\neg q \implies \neg p\). So \(p \implies q\) is If it is raining then I am getting wet. Is that the same as \(\neg q \implies \neg p\) If I am not getting wet, then it is not raining. Well we have stated that if it is raining then I am getting wet. So if I am not getting wet then it is not raining, or the first implication would not hold. Do you see how these two are logically equivalent?
yes i understand now!
So now apply this to the original problem. Do you see how only the first two statements about the cat are logically equivalent?
Well look at the choices. What do you think is the correct answer?
Yes, A is the correct answer.
so how wouldi right this!
I suggest you read about contrapositive and converse statements.
Once you fully understand those two concepts, it will be immediately clear that only 1 & 2 are equivalent.
http://en.wikipedia.org/wiki/Contraposition http://en.wikipedia.org/wiki/Conversion_(logic)