Question

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

Answer 1

this is hard.

Answer 2

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 \(a\)

Answer 3

To be more specific the converse of a statement is not logically equivalent to the statement.

Answer 4

So to be clear p->q does not imply q->p

Answer 5

Do you understand?

Answer 6

heck no!!!

Answer 7

LOL.

Answer 8

saifoo not funny!

Answer 9

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?

Answer 10

You could be getting wet, even when it is not raining (shower for instance).

Answer 11

saifoo not funny!

Answer 12

okay i get that part!

Answer 13

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?

Answer 14

yes i understand now!

Answer 15

So now apply this to the original problem. Do you see how only the first two statements about the cat are logically equivalent?

Answer 16

saifoo not funny!

Answer 17

so what is my awser

Answer 18

Well look at the choices. What do you think is the correct answer?

Answer 19

A sound reasonable!

Answer 20

Yes, A is the correct answer.

Answer 21

so how wouldi right this!

Answer 22

I suggest you read about contrapositive and converse statements.

Answer 23

Once you fully understand those two concepts, it will be immediately clear that only 1 & 2 are equivalent.

Answer 24

http://en.wikipedia.org/wiki/Contraposition
http://en.wikipedia.org/wiki/Conversion_(logic)