## keebler01 Group Title 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 3 years ago 3 years ago

1. saifoo.khan Group Title

this is hard.

2. Alchemista Group Title

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$$

3. Alchemista Group Title

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

4. Alchemista Group Title

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

5. Alchemista Group Title

Do you understand?

6. keebler01 Group Title

heck no!!!

7. saifoo.khan Group Title

LOL.

8. keebler01 Group Title

saifoo not funny!

9. Alchemista Group Title

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?

10. Alchemista Group Title

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

11. keebler01 Group Title

saifoo not funny!

12. keebler01 Group Title

okay i get that part!

13. Alchemista Group Title

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?

14. keebler01 Group Title

yes i understand now!

15. Alchemista Group Title

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

16. keebler01 Group Title

saifoo not funny!

17. keebler01 Group Title

so what is my awser

18. Alchemista Group Title

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

19. keebler01 Group Title

A sound reasonable!

20. Alchemista Group Title

Yes, A is the correct answer.

21. keebler01 Group Title

so how wouldi right this!

22. Alchemista Group Title