# Math Help - Arguments and Proofs

1. ## Arguments and Proofs

Can someone help me with those two argument problems? No matter how many times I try I reach a dead end at some point. These problems require the proof laws to show if the arguments are valid or not.

1- If the rat ate the cheese or the rate is not in the trap, then it is under the bed.
The Rat is neither under the bed nor under the table. You will see the Rat footprint unless It is under the table or it is not in the trap. The Rat did drink the water if you see its footprint. The rate ate the cheese or it did not eat the newspaper. Therefore if the rate neither under the table nor under the bed, then It did not eat the newspaper but it drank the water.

2-

And this one's seemed easy so I managed to figure out it's valid, but I just wanna check with you guys. I came up with the answer because it's not possible to make the premises all true while setting the conclusion to false, well you can do it with a truth table as well and get the answer.

Any kind of help is appreciated.

2. ## Re: Arguments and Proofs

Originally Posted by Ryuna
1- If the rat ate the cheese or the rate is not in the trap, then it is under the bed.
The Rat is neither under the bed nor under the table. You will see the Rat footprint unless It is under the table or it is not in the trap. The Rat did drink the water if you see its footprint. The rate ate the cheese or it did not eat the newspaper. Therefore if the rate neither under the table nor under the bed, then It did not eat the newspaper but it drank the water.
Well, that's quite a tangle. I assume rat, Rat and rate are the same thing. Start by listing all elementary (indivisible) propositions occurring in these statements. Give them good mnemonic one- or two-letter names. For example, "the rat drank the water" can be denoted by w. Don't denote it by x. Then carefully go through each statement and write it symbolically using the introduced variables. Post the result here for checking.

Originally Posted by Ryuna
2-
If $s\lor p\land m$ means $s\lor (p\land m)$ (as it should), then the assignment $p=q=r=m=F$, $s=T$ makes the hypotheses true and the conclusion false. If $s\lor p\land m$ means $(s\lor p)\land m$, then it alone implies $m$, independent of $s$.

Originally Posted by Ryuna
And this one's seemed easy so I managed to figure out it's valid, but I just wanna check with you guys. I came up with the answer because it's not possible to make the premises all true while setting the conclusion to false, well you can do it with a truth table as well and get the answer.
Yes, the two hypotheses are negations of each other, so together they are contradictory. Therefore, they imply everything.

3. ## Re: Arguments and Proofs

Thank you for the quick reply. Here's the symbolic for the first one. I tried doing it and again got stuck. Also for clarification, when we use neither no, is it true that we add the negation to the beginning of the sentence and use and between the two symbols like I did?

Also I have to use the proof laws (modus tollens etc) to determine their validty. Having those also help me personally because theyre a step by step breakdown.

4. ## Re: Arguments and Proofs

Hi,
Yes you translate "neither p nor q" into "not p and not q". Also the word "unless" in English can always be translated to "or".
Here's a symbolic proof for your first question:

5. ## Re: Arguments and Proofs

So neither or has a negation before each symbol. Are you sure about it? Now I'm getting confused cause my tutor must've told me the negation is only before the first.

6. ## Re: Arguments and Proofs

PSLV Discrete Mathematics: Logic

I checked this website, and it is indeed true. I can't believe I've been doing that for neither nor the whole semester. /facepalm

Thank you both for your help!

7. ## Re: Arguments and Proofs

@johng

Can you please redo the 2nd question using the same method? It helped me a lot with the 1st one.

8. ## Re: Arguments and Proofs

Hello again, I solved the whole thing, I used johng example and solved the second question, I feel like I should use the laws for this one instead of justifying it with logic in words. I also assigned proper truth values for question 3 and if anyone's interested here are my the final answers.

again thank you both for helping out.