## No-data Group Title How can I simplify this: $\left(P\wedge\neg R\right)\vee\left(P\wedge\neg Q\right)\vee\left(Q\wedge\neg R\right)$ one year ago one year ago

1. No-data Group Title

I've been working on this but it seem I can't do anymore. =/

2. No-data Group Title

The original expression is this: $\left(P\Rightarrow R\right)\Rightarrow\left[\left(P\Rightarrow Q\right)\wedge\left(Q\Rightarrow R\right)\right]$

3. No-data Group Title

I see that it is like prove the transitivy propertie for the implication operator. I made a truth table for this and it turned out to be false, but I want to get to the same conclussion without using truth tables.

4. No-data Group Title

This is the work I've done so far.

5. abb0t Group Title

Well, tautology is a proposition that is always true. That's all I know. That's all I know.

6. Rohangrr Group Title

good reasoning @No-data

7. No-data Group Title

Thank you @Rohangrr

8. No-data Group Title

@brinethery Do you have any idea?

9. wio Group Title

Are you sure it's true? It looks like the converse is true.

10. wio Group Title

$\left(P\Rightarrow R\right)\Rightarrow\left[\left(P\Rightarrow Q\right)\wedge\left(Q\Rightarrow R\right)\right]$Suppose $$P \gets T\quad Q\gets F\quad R\gets T$$ $(T\implies T)\implies [(T\implies F)\wedge (F \implies T)] \\ T\implies (F\wedge T) \\ T\implies F \\ F$

11. Edutopia Group Title

I feel like q is irrelvent and its just If P then R

12. wio Group Title

$\left(P\Rightarrow R\right)\Rightarrow\left[\left(P\Rightarrow Q\right)\wedge\left(Q\Rightarrow R\right)\right]$I think it should be $\left[\left(P\Rightarrow Q\right)\wedge\left(Q\Rightarrow R\right)\right] \Rightarrow \left(P\Rightarrow R\right)$

13. Edutopia Group Title

^^ makes alot of sense

14. wio Group Title

There is nothing wrong with using a counter example to show something is false. Trying to prove it without a counter example seems pointless to me.

15. No-data Group Title

I did not say it was true

16. No-data Group Title

Mmm I don't understand the expression after "Suppose" what those arrows mean?

17. No-data Group Title

Ok I got the counter example, is a row on my truth table and its correct. Thank you @wio

18. No-data Group Title

I was just trying to show that the expression is not a tautology.

19. No-data Group Title

The expression you wrote saying you thought it should be is, in fact, a tautology.

20. wio Group Title

Imagine if I have you the proposition $$P$$ and claimed it was a tautology. How would you prove it is wrong?

21. wio Group Title

There is no boolean algebra to prove it, you just give a counter example, right?

22. No-data Group Title

Do you think that is not possible to find a way to show this using only theorems and definitions?

23. wio Group Title

Which theorems?

24. wio Group Title

How would you show that $$P$$ in and of itself is not a tautology?

25. No-data Group Title

There is no way to know. Unless you tell me that P can take the values T and F.

26. wio Group Title

I mean, I have nothing against trying to do things a particular way, but what are the rules really? How would you show $$P$$ is not a tautology without a counter example.

27. No-data Group Title

and then, by definition of P. P is not a tautology

28. wio Group Title

So you'd say by definition of proposition, a proposition is not a tautology? Then how about $$P\wedge Q$$?

29. No-data Group Title

You meant that proposition is not a tautology.

30. No-data Group Title

As far as I know a tautology is a predicate that is always true for any value of their parameters.

31. No-data Group Title

$P\wedge Q$ is a predicate that can be true or false. So it's not a tautology.

32. No-data Group Title

33. wio Group Title

You're claiming it can be false... The point is to get you to prove it without counterexample.

34. wio Group Title

I'm just curious as to the rules of the game.

35. No-data Group Title

I think I'm not using a counter example and that is a valid proof.

36. Edutopia Group Title

A tautology is a restatement within the same premiss like all trees are made of wood

37. wio Group Title

What if I say "is a predicate that can be true or false.", about the original predicate? Said it about $\left(P\Rightarrow R\right)\Rightarrow\left[\left(P\Rightarrow Q\right)\wedge\left(Q\Rightarrow R\right)\right]$Would simply saying that be a valid proof that it is not a tautology?

38. wio Group Title

I don't think saying "is a predicate that can be true or false." is valid without a counter example.

39. wio Group Title

Or maybe there is another way, but I'm not sure

40. No-data Group Title

Is not evident that that predicate can be true or false. Its not a valid proof.

41. No-data Group Title

and $P\wedge Q$is by definition true or false.

42. wio Group Title

Sure but then what makes something 'evident'?

43. wio Group Title

Is there some standard form you want it to be in?

44. No-data Group Title

I wanted to say that there is a definition by saying "evident"

45. wio Group Title

What about $$P\wedge Q \wedge R$$?

46. No-data Group Title

I accepted you method buy I feel like you're trying to say that there is no other method to prove this problem.

47. wio Group Title

What I am saying is that if there is another way to prove this problem, then there needs to be a way to prove very simple predicates.

48. No-data Group Title

let $P\wedge Q\Leftrightarrow S$$S\wedge R$

49. wio Group Title

I mean prove without counter example

50. No-data Group Title

so it's not a tautology.

51. No-data Group Title

I know that, I already proved it using other method. I just wanted to know if it was possible to do it the way I'm trying.

52. wio Group Title

Using your logic that $$\wedge$$ doesn't give a tautology, then couldn't we say $$\neg P\vee P$$ isn't a tautology?

53. Edutopia Group Title

ITs a Syllogism if the expanded form is the antecedent, but because its the consequent it is not tautological

54. wio Group Title

Well it's just an interesting thing to think about. I would like to find a way to prove false without counterexample but it seems hard to understand what is allowed.

55. Edutopia Group Title

q's relation to p and r

56. No-data Group Title

This is not a tautology:$P\wedge Q$, not just the $\wedge$ symbol.

57. No-data Group Title

Sorry I did not understand that @Edutopia

58. No-data Group Title

I'm just beginning with this logic things lol.

59. Edutopia Group Title

my comp is sllooow, i ment to say: q's relation to p and r would have to be in the premiss for it to be tautological

60. No-data Group Title

ahh that is right Edutopia.

61. wio Group Title

You know what's interesting about proving something is true, is that you are showing it is true for all cases. If you are proving something is false, you only need to show one case is false. To prove that it is false for all cases can't always be done and doesn't need to be done. I think to even say $$P$$ is not a tautology, you are implicitly calling upon the counter example just to claim "$$P$$ can be false".

62. wio Group Title

But maybe that is a semantic argument? It's just my thought.

63. Edutopia Group Title

in IF (IF P THEN R) THEN BOTH (IF P THEN Q) AND ( IF Q THEN R) the relation of Q to P and R is not established, imagine the argument where P and R have to do with Physics and Q is someones opinion on abortion.

64. No-data Group Title

Yeah it's interesting @wio and I agree your counterexamples are a good tool. I just think they are not elegant.

65. wio Group Title

When I first saw $\left(P\Rightarrow R\right)\Rightarrow\left[\left(P\Rightarrow Q\right)\wedge\left(Q\Rightarrow R\right)\right]$The $$Q$$ made me think it's likely not a tautology right away. Since it wasn't something inconsequential like $$Q\vee \neg Q$$.

66. No-data Group Title

I'm not sure, but I think it's valid in math.If$x=y$then$x+a=y+a$

67. wio Group Title

That is the addition property of equality. $$Q$$ was not being appended to both sides of the implication.

68. Edutopia Group Title

yes, but in math its all a valid argument because you are using numbers

69. No-data Group Title

I know it just look similar. Thank you guys. It is nice to be able to discuss this kind of things with other people. Thank you for you r time.

70. Edutopia Group Title

a=a and so forth

71. brinethery Group Title

@No-data I have no idea :-). I'm so stupid.

72. brinethery Group Title

Is this for linear algebra?

73. No-data Group Title

It is propositional logic. You're not stupid at all @brinethery. I was curious about how databases work so I picked a book about math applied to databases and this is a problem from the first chapter. Logic and Set Theory are the foundation of DB systems.