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dimis
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posted March 16, 2009 01:09 AM |
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Exactly. That's exactly what I wanted to see.
By the way, thanks to all who participated (either directly or by sending an IM) and thanks to Binabik for pointing out the right approach.
As a side note of the talk where I retracted that example, they referred to what is so called "Cheaters Detection". For example, if the cards had on one side the age of somebody and on the other side the drink they have, and you have the rule
rule: If under 21 then non-alcoholic drinks
As well as the four cards
Beer, 16, 32, Coke
then, everybody will guess right on which cards to turn in order to verify the rule.
It's fairly easy now to justify it, right? 
Well, I just found it interesting and thought I could share it with you. What kind of games our minds can play from time to time. I hope you enjoyed it!
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The empty set
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Laharl
Hired Hero
Demon Prince
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posted March 16, 2009 01:59 AM |
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I hate math, it sux 
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I_AM_NOT_THANATOS_BUT_AM_KNOWN_AS_DEMON_PRINCE_LAHARL_AND_YOU_SUCK_LOL
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Asheera
Honorable
Undefeatable Hero
Elite Assassin
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posted March 16, 2009 02:44 AM |
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Edited by Asheera at 02:44, 16 Mar 2009.
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Please, if you have nothing constructive to say just don't post, Laharl. 
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Laharl
Hired Hero
Demon Prince
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posted March 16, 2009 02:53 AM |
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same for you ashie 
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Stratos
Hired Hero
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posted March 16, 2009 06:22 AM |
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Kids. They just dont have any respect these days like before.
Sounds like sonny boy here likes to start fights. Come here Lahurl. Come sit on my lap you whipper snapper. I will give you a nice spanking. Then I will send you to your room.
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dimis
Responsible
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posted March 16, 2009 06:54 AM |
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Edited by dimis at 06:58, 16 Mar 2009.
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Powersets and empty set
Ok,
Since I can't sleep, I'll give you guys a problem that seems to be more difficult than I originally thought when I posed it to some people.
problem: Write down the powerset of the powerset of the empty set. (no typo in expression)
I don't know if you prefer this variation
problem rephrased: Write down the set of all subsets of the set of all subsets of the empty set. (again no typo)
EDIT: Yes, I know that one answer is 42. I am interested in the other one ...
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Binabik
Responsible
Legendary Hero
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posted March 16, 2009 07:17 AM |
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It's been since like grade 5 since I did sets in school. I never heard of a superset. And I don't have a clue about the second phrasing....unless it's the empty set or the null set. I seem to remember something about the null set being different from the empty set.
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dimis
Responsible
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posted March 16, 2009 07:34 AM |
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Edited by dimis at 07:43, 16 Mar 2009.
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Ok. Well, the empty set is the set that has no members, and here in thread we can denote it as {}. In other words there is nothing between the "{" and "}" that belongs to this set.
The powerset of a set is the set of all subsets of a set.
Now a subset of a set is a set composed by some of its members; where some ranges from 0 up to all the members of the set. For example a subset of {a, b, c} is {a, c}.
I don't know if this makes it any easier, although now there is a hint.
In terms of logic, you can define the subset as
(\forall x) [ x \in A ==> x \in B ]
This means every x that is found in A, is also found in B. If this is the case, then A is a subset of B.
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dimis
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posted March 17, 2009 06:25 PM |
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Edited by dimis at 18:26, 17 Mar 2009.
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Hint
I think the heart of the problem boils down to this sentence (or a similar one) which comes along with a known fact:
fact: Moon is *not* made of cheese.
sentence: If moon is made of cheese, then I am Superman.
Now the question is
question: Is the sentence above TRUE or FALSE ?
If we can answer this correctly, then we can solve the problem.
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The empty set
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TheDeath
Responsible
Undefeatable Hero
with serious business
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posted March 17, 2009 06:43 PM |
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Edited by TheDeath at 18:43, 17 Mar 2009.
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Quote:
question: Is the sentence above TRUE or FALSE ?
If we can answer this correctly, then we can solve the problem.
I don't think we can know if the sentence is true as the "if" statement has never been confirmed... but that can lead to Curry s Paradox, an example of fallible logic.
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dimis
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posted March 17, 2009 07:48 PM |
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Edited by dimis at 20:00, 17 Mar 2009.
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No TheDeath. I also give you something as a fact which you should use and base your answer. If you want to believe that moon is made of cheese (which also contradicts all previous missions by nasa on moon), then discuss that and try to convince people in the thread that discusses about god. Freedom of speech has its limits ... The statement is either TRUE or FALSE given the fact above.
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The empty set
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Binabik
Responsible
Legendary Hero
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posted March 17, 2009 08:05 PM |
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I've never taken a class in logic and don't know the rules. And I also don't know how software guys would approach this. But in hardware logic it would be the following:
If moon is made of cheese, then I am Superman. = unknown
I am Superman = false
This is interesting because in hardware logic things can work a little differently.
If you have a simple single 2-state input logic gate with a single 2-state output,
and if the input changes state
then the output will also change state
There may be some exceptions to the above statement, but I can't think of any off the top of my head.
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dimis
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posted March 17, 2009 08:31 PM |
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Edited by dimis at 20:33, 17 Mar 2009.
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As I said, this is the heart of the problem at hand.
Given the fact that moon is not made of cheese, will you ever conclude by that statement that I am Superman? Can you say something (based only on the fact and the sentence) about me being or not being Superman? And the answer is NO. So, the statement, is indeed a TRUE statement ==> It leads to no inconsistency. Similar is the case to the statement
If moon is made of cheese, then I am not Superman.
Nothing wrong with this sentence really. Whether I am Superman or not is irrelevant since it will follow as a consequence only if the moon is made of cheese; and by the fact above (moon is *not* made of cheese), I will never conclude by the statement if I am Superman or not.
Finally, It is not true that a 2-state input logic gate has to change output "bit" if you change the input bit. Just think about the gate that always says "YES", or always says "NO"; i.e. does not care about your input. The gate is stubborn (any similarities with people @ HC is coincidental  ) and will always light up (or not) that output bit. I guess after working many years on hardware parts though you tend to neglect cases like these, merely because the input is unimportant; i.e. yields no information, so why bother?
Oh, and hardware is not different from software, so I can not cheat by thinking one version or the other.
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dimis
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posted March 17, 2009 10:48 PM |
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Edited by dimis at 23:09, 17 Mar 2009.
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Gates
Actually I really like your idea about thinking in terms of gates, which "light up" a bulb when something is TRUE.
Let's see then the truth table.
AND gates
Say you have A, B. I believe we all agree with the interpretation of "and". An "and" statement is TRUE only when both parts are true; i.e. we have the following truth table:
._________________
| | | |
| A | B | A and B |
|___|___|_________|
| F | F | F |
|---|---|---------|
| F | T | F |
|---|---|---------|
| T | F | F |
|---|---|---------|
| T | T | T |
|---|---|---------|
The light bulb will actually light up when both A and B are true.
EQUIVALENCE gates
Now let's see something else. What do we really mean when we say "A is equivalent to B" (we denote it as "A <=> B"). Clearly whenever one of them is TRUE, then the other one has to be TRUE. Moreover, if one of them is FALSE the other one is also FALSE, otherwise they are not equivalent.
So, what is the truth table for this connective? Here we go:
._________________
| | | |
| A | B | A <=> B |
|___|___|_________|
| F | F | T |
|---|---|---------|
| F | T | F |
|---|---|---------|
| T | F | F |
|---|---|---------|
| T | T | T |
|---|---|---------|
Note that in the first line when A is FALSE and B is FALSE we have a TRUE.
IMPLICATION/CONSEQUENCE gates
Now let's turn our attention to the strange connective "==>".
How can we form the truth table for this connective? Let's see:
._________________
| | | |
| A | B | A ==> B |
|___|___|_________|
| F | F | ? |
|---|---|---------|
| F | T | ? |
|---|---|---------|
| T | F | F |
|---|---|---------|
| T | T | T |
|---|---|---------|
What we really mean when we say that "A implies B", is that if A is TRUE then B is also TRUE. This justifies the last row above.
Moreover, clearly we can not have a valid statement "A implies B" when A is TRUE and B is FALSE. This justifies the row before the last one above.
So the real question is what on earth are those question marks in the first two lines.
Let's examine the very first one when both A and B are FALSE. We 'll get some help by the EQUIVALENCE gate. When we write "A <=> B" we really mean "(A ==> B) and (B ==> A)"; i.e. we can read "A <=> B" starting from either endpoint.  But the truth table for an AND gate tells us that we get a TRUE when both parentheses actually evaluate to TRUE. So, whenever A is FALSE and B is FALSE we should evaluate the "A ==> B" (as well as "B ==> A") as TRUE since this is the only way that we can be consistent with the fact that when both A and B are FALSE the equivalence "A <=> B" holds. Having said that we get the following format on the truth table:
._________________
| | | |
| A | B | A ==> B |
|___|___|_________|
| F | F | T |
|---|---|---------|
| F | T | ? |
|---|---|---------|
| T | F | F |
|---|---|---------|
| T | T | T |
|---|---|---------|
and now we are left with just one question mark which is either TRUE or FALSE.
Note by the way, that the above T actually classifies the sentence
If moon is made of cheese, then I am Superman
as TRUE since both parts are FALSE.
But I think I owe an explanation on the last question mark.
Well, it has a long history ... But the main idea of setting it to T appeared in the previous problems and was very nicely explained by Binabik.
If one side has a vowel, then the other side has an even number.
You did not have to turn a card in which you read the number 4 in order to test the validity of the statement. What we really mean when we say
If one side has a vowel, then the other side has an even number
is that if we read an even number this is may be the result of the other side having a vowel, or may be not. So, setting the question mark equal to FALSE essentially means that whenever I have A, then I also have B (perfectly fine so far), and whenever I don't have A, then I also don't have B. But this is not what we mean. So the question mark has to be set equal to T (TRUE). Hence, we have the truth table:
._________________
| | | |
| A | B | A ==> B |
|___|___|_________|
| F | F | T |
|---|---|---------|
| F | T | T |
|---|---|---------|
| T | F | F |
|---|---|---------|
| T | T | T |
|---|---|---------|
I really hope all this analysis actually helps.
And now back to the problem, what are all the possible subsets of the empty set?
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TheDeath
Responsible
Undefeatable Hero
with serious business
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posted March 18, 2009 02:24 AM |
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Edited by TheDeath at 02:25, 18 Mar 2009.
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I have no idea what you meant or why you got angry with what I said, but I tell you one thing, you are not very open minded on it (but if you have a bone to pick with my logic, go and argue with Curry's Paradox and let me know when you can prove you're right, I'm not holding my breath).
Meanwhile what I meant with my post is that the answer is UNKNOWN. NOT KNOWN. that not clear enough?
I have no idea how you got the FALSE statement, but I can prove it wrong. It says that IF the Moon is made of cheese, then you are Superman. You say it's false. But I say it's unknown. You have NOT verified that IF the Moon is made of cheese you are NOT Superman. You may not be superman right now, but the Moon is definitely NOT made of cheese, so you cannot verify.
Similar logic can be illustrated like this:
IF the Moon is flat, THEN the Earth is flat.
since I have not verified the above, seeing as how I never seen a flat Moon, then how can I possibly come to the conclusion that it is false?
Now as per Curry's paradox, what about: IF this statement is true, the Moon is made of cheese.
(My "common sense" tells me the latter is false, as well as saying also that it is UNKNOWN what you proved with this 'logic' (that also proves the Moon is made of cheese ) as being false... but I could be off track though, can you enlighten me a bit instead of screaming at me again like you should have?)
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dimis
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posted March 18, 2009 03:40 AM |
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Edited by dimis at 04:07, 18 Mar 2009.
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Quote:
I have no idea what you meant or why you got angry with what I said, but I tell you one thing, you are not very open minded on it (but if you have a bone to pick with my logic, go and argue with Curry's Paradox and let me know when you can prove you're right, I'm not holding my breath).
I got angry because your attitude is negative to many things and I thought you wanted to "play" with people's minds once again with something that is not so easily understood.
Quote:
Meanwhile what I meant with my post is that the answer is UNKNOWN. NOT KNOWN. that not clear enough?
It was clear right from the start. What does not seem clear is that this is not an answer. It has to be either true or false. It is also irrelevant whether the moon in reality is made of cheese or not since I am giving this to you as a fact. I could have very well said:
fact: 2 < 3.
sentence 1: if (2 >= 3) then God exists.
or even
sentence 2: if (2 >= 3) then God does not exist.
BOTH sentences (the entire thing) are TRUE since the condition is NOT met. Just check the truth table above for the implication. Once the condition is not met, it is irrelevant what follows. You already know that since you program. IF statements can be there whether the condition is met or not. The point is, that you reach a conclusion (execute the "then" part) only when the condition is met. Hence, when the condition is not met, the "IF ... THEN ..." statements are perfectly valid regardless of whether it is true or not the "then" part. That is what I am saying; see my arguments on the post about gates above.
Quote:
I have no idea how you got the FALSE statement, but I can prove it wrong. It says that IF the Moon is made of cheese, then you are Superman. You say it's false.
In fact I say that the entire statement (the whole sentence) is true.
Quote:
But I say it's unknown. You have NOT verified that IF the Moon is made of cheese you are NOT Superman.
I don't have to check whether I am Superman or not, since the moon is not made of cheese. This is given.
Quote:
You may not be superman right now, but the Moon is definitely NOT made of cheese, so you cannot verify.
Read carefully what you wrote: "the Moon is definitely NOT made of cheese" which is exactly the FACT that I gave you; i.e. we both agree that "moon is made of cheese" evaluates to FALSE. Now, when the condition is FALSE in an "if ... then ..." statement, then the entire statement (the whole sentence) is TRUE. That's all I said. You can check my previous posts as well as the analysis on truth tables which serves on justifying this reasoning. The "if A then B" (also written down as "A ==> B") is nothing more than a boolean function: A is either 0 (F) or 1 (T), and B is either 0 (F) or 1 (T). There are four cases (shown in the truth table above) and the last column shows the output of such a function. This output is what characterizes the entire statement. And in both cases where A evaluates to FALSE (0), the entire statement is TRUE.
Quote:
Similar logic can be illustrated like this:
IF the Moon is flat, THEN the Earth is flat.
since I have not verified the above, seeing as how I never seen a flat Moon, then how can I possibly come to the conclusion that it is false?
I say it again since I think this is the problematic point: Say you are given (i.e. HOLDS) "The moon is *not* flat". Then the sentence "the moon is flat" evaluates to FALSE. Therefore the entire sentence "IF the Moon is flat, THEN the Earth is flat." evaluates to TRUE (i.e. it's a perfectly valid statement).
On the other hand, if you have a system and a sentence in that system that says "Moon is flat" (<-- i.e. this is a true statement for your system), and you add the rule above "IF the Moon is flat, THEN the Earth is flat." (say this is your axiom in that system) then you get as a consequence that indeed the Earth is flat. That's perfectly alright, however it is quite susceptible whether this system has any real use.
Quote:
Now as per Curry's paradox, what about: IF this statement is true, the Moon is made of cheese.
(My "common sense" tells me the latter is false, as well as saying also that it is UNKNOWN what you proved with this 'logic' (that also proves the Moon is made of cheese ) as being false... but I could be off track though, can you enlighten me a bit instead of screaming at me again like you should have?)
I believe it is clear how you handle cases like these. In fact in statements "if A then B", the interesting part comes up when A is true.
Check again the problem with the cards that we were turning. We had to turn "A" and "7", since "A" makes the condition "if it is a vowel" evaluate to true, and "7" falsifies the "then it is even" part. And in fact, the cheater would be someone who wrote a vowel on one side and "7" on the other side, so we turn the card with the "7" to check the integrity of the rule. Similar is the case here, and it all boils down to the "axioms" you accept. So, if the sentence
IF this statement is true, the Moon is made of cheese.
is a rule (axiom) in your system, then indeed Moon is made of cheese, because trivially as an axiom in your system the entire rule is true and the if part is satisfied.
If it is not an axiom and you want to evaluate its integrity, then unfortunately you need to evaluate the sentence "the Moon is made of cheese". When you evaluate this, then you can decide about the integrity of the whole sentence. This seems like Russel's paradox, so I guess that's our next problem.
Can we give an end to the "powerset of the powerset of the empty set" problem?
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TheDeath
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posted March 18, 2009 04:08 AM |
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Ok thanks for replying to that, however, I still don't see how it can lead to a true statement. Don't get me wrong, I understand what you mean, but I think that the IF is there for a reason -- i.e even if you know that the IF is false, that doesn't mean that "IF 1=0 THEN anything" is true (where 1=0 is false).
It doesn't make sense. Why would a statement be true, i.e why would "IF 1=0 THEN I exist!" be TRUE? Mind you, this could be true -- after all, it says that "IF 1=0 THEN I exist", but it doesn't mean that "IF 1<>0 then I do not exist", so I can just as easily exist in both scenarios. (as you can see that I exist right now, when 1<>0)
Why would ALSO the statement "IF 1=0 THEN I do not exist" also be TRUE at the same time? How much sense does it make? You can, literally, put anything after the statement and make it true, or am I wrong?
Just because the condition is false?
I think the biggest fault is advocating an either TRUE or FALSE to a statement without actually verifying -- i.e no middle ground of "unknown", in this case.
Sorry for not going to the "set" problem it's 5 AM here and I can't think that well right now
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dimis
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posted March 18, 2009 04:43 AM |
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Edited by dimis at 05:28, 18 Mar 2009.
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Quote:
You can, literally, put anything after the statement and make it true, or am I wrong?
That's the fine line here and it depends on what you mean. When we say that the entire sentence is true (can think of it as valid), we do not mean that the "then" part necessarily holds; i.e. I am not saying that the "then ..." part is TRUE. I am referring to the whole thing. So, yes, if A evaluates to FALSE, then it does not matter to what B evaluates in a statement of the form:
A ==> B
The entire statement is true trivially. And the justification comes from the truth table, because in both cases for B, when A evaluates to FALSE, the outcome of this function is TRUE. That's all.
And here is a program that evaluates the entire sentence "A ==> B":
_____________________________________________________
IF (A is true and B is true) THEN return true;
ELSE IF (A is false) THEN return true;
ELSE return false;
_____________________________________________________
However, the above program has a flaw. And the problem is that it asks for the evaluation of B, even when A is not true. So, the correct version is:
_____________________________________________________
IF (A is false) THEN return true;
ELSE IF (A is true and B is true) THEN return true;
ELSE return false;
_____________________________________________________
since it evaluates B only when it is necessary. Actually it has an even simpler form (since A is trivially true in the "else if" statement):
_____________________________________________________
IF (A is false) THEN return true;
ELSE IF (B is true) THEN return true;
ELSE return false;
_____________________________________________________
and actually this last program justifies the logical equivalence between "A ==> B" and "(not A) or B".
Quote:
Just because the condition is false?
Exactly.
Quote:
I think the biggest fault is advocating an either TRUE or FALSE to a statement without actually verifying -- i.e no middle ground of "unknown", in this case.
Well, that's the entire point. It evaluates either to TRUE or FALSE. We agree on that right? But if the condition is not met (i.e. A evaluates to FALSE), then, the truth table forces you to "accept" the statement. So it really boils down to accepting (we as humans) the truth table for statements of the form "A ==> B". Probably I haven't written something down clearly on the "Gates" post above. Just point me the cases which seem problematic or vague there. I believe this is very fundamental.
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dimis
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posted March 22, 2009 08:01 AM |
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Logicomix
By the way,
I recently heard about the forthcoming publication of the following comics-book.
I am excited and can't wait till it reaches the stores. Here's the link:
Logicomix
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The empty set
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Ecoris
Promising
Supreme Hero
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posted March 22, 2009 11:16 AM |
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This seems interesting, even though one knows how the "quest for the Foundations of Mathematics" ends. Apparently there's much more to it than that.
btw: {Ø,{Ø}}
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