anonymous
  • anonymous
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
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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saifoo.khan
  • saifoo.khan
this is hard.
anonymous
  • anonymous
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 \(a\)
anonymous
  • anonymous
To be more specific the converse of a statement is not logically equivalent to the statement.

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anonymous
  • anonymous
So to be clear p->q does not imply q->p
anonymous
  • anonymous
Do you understand?
anonymous
  • anonymous
heck no!!!
saifoo.khan
  • saifoo.khan
LOL.
anonymous
  • anonymous
saifoo not funny!
anonymous
  • anonymous
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?
anonymous
  • anonymous
You could be getting wet, even when it is not raining (shower for instance).
anonymous
  • anonymous
saifoo not funny!
anonymous
  • anonymous
okay i get that part!
anonymous
  • anonymous
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?
anonymous
  • anonymous
yes i understand now!
anonymous
  • anonymous
So now apply this to the original problem. Do you see how only the first two statements about the cat are logically equivalent?
anonymous
  • anonymous
saifoo not funny!
anonymous
  • anonymous
so what is my awser
anonymous
  • anonymous
Well look at the choices. What do you think is the correct answer?
anonymous
  • anonymous
A sound reasonable!
anonymous
  • anonymous
Yes, A is the correct answer.
anonymous
  • anonymous
so how wouldi right this!
anonymous
  • anonymous
I suggest you read about contrapositive and converse statements.
anonymous
  • anonymous
Once you fully understand those two concepts, it will be immediately clear that only 1 & 2 are equivalent.
anonymous
  • anonymous
http://en.wikipedia.org/wiki/Contraposition http://en.wikipedia.org/wiki/Conversion_(logic)

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