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keebler01
 3 years ago
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
keebler01
 3 years ago
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

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Alchemista
 3 years ago
Best ResponseYou've already chosen the best response.1The 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 \(a\)

Alchemista
 3 years ago
Best ResponseYou've already chosen the best response.1To be more specific the converse of a statement is not logically equivalent to the statement.

Alchemista
 3 years ago
Best ResponseYou've already chosen the best response.1So to be clear p>q does not imply q>p

Alchemista
 3 years ago
Best ResponseYou've already chosen the best response.1Here, 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?

Alchemista
 3 years ago
Best ResponseYou've already chosen the best response.1You could be getting wet, even when it is not raining (shower for instance).

keebler01
 3 years ago
Best ResponseYou've already chosen the best response.0okay i get that part!

Alchemista
 3 years ago
Best ResponseYou've already chosen the best response.1So 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?

keebler01
 3 years ago
Best ResponseYou've already chosen the best response.0yes i understand now!

Alchemista
 3 years ago
Best ResponseYou've already chosen the best response.1So now apply this to the original problem. Do you see how only the first two statements about the cat are logically equivalent?

Alchemista
 3 years ago
Best ResponseYou've already chosen the best response.1Well look at the choices. What do you think is the correct answer?

Alchemista
 3 years ago
Best ResponseYou've already chosen the best response.1Yes, A is the correct answer.

keebler01
 3 years ago
Best ResponseYou've already chosen the best response.0so how wouldi right this!

Alchemista
 3 years ago
Best ResponseYou've already chosen the best response.1I suggest you read about contrapositive and converse statements.

Alchemista
 3 years ago
Best ResponseYou've already chosen the best response.1Once you fully understand those two concepts, it will be immediately clear that only 1 & 2 are equivalent.

Alchemista
 3 years ago
Best ResponseYou've already chosen the best response.1http://en.wikipedia.org/wiki/Contraposition http://en.wikipedia.org/wiki/Conversion_(logic)
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