Kermit1941
12-24-2007, 04:11 PM
Hello. I would love to discuss math in general with anyone who wishes.
Kermit Rose < [email protected] >
Kermit Rose < [email protected] >
View Full Version : Math Discussion
Willie Lumplump
12-24-2007, 09:19 PM
Hello. I would love to discuss math in general with anyone who wishes.Kermit Rose < [email protected] >Oh, goodie! You first.
Zeno Swijtink
12-24-2007, 10:25 PM
Hello. I would love to discuss math in general with anyone who wishes.
Kermit Rose < [email protected] >
I am not sure what you have in mind but here is a logical puzzle:
The people on the Island of Knights and Knaves are either knights or knaves. Knights are honest and virtuous, and whatever they say is true.
Knaves are base and shameful, and whatever they say is false.
A and B. Stranger ask A: Are you a knight or a knave?
A answers indistinctly.
Stranger asks B: What did A say?
B answers: A said he is a knave.
What are A and B?
Kermit Rose < [email protected] >
I am not sure what you have in mind but here is a logical puzzle:
The people on the Island of Knights and Knaves are either knights or knaves. Knights are honest and virtuous, and whatever they say is true.
Knaves are base and shameful, and whatever they say is false.
A and B. Stranger ask A: Are you a knight or a knave?
A answers indistinctly.
Stranger asks B: What did A say?
B answers: A said he is a knave.
What are A and B?
Barry
12-24-2007, 10:40 PM
I am not sure what you have in mind but here is a logical puzzle:
The people on the Island of Knights and Knaves are either knights or knaves. Knights are honest and virtuous, and whatever they say is true.
Knaves are base and shameful, and whatever they say is false.
A and B. Stranger ask A: Are you a knight or a knave?
A answers indistinctly.
Stranger asks B: What did A say?
B answers: A said he is a knave.
What are A and B?Easy! (Mind if I butt in?)
B is a knave and A is a knight!
Logic:
When asked "Are you a knight or a Knave":
A Knight would always answer "I am a knight" because he speaks the truth.
A Knave would always say "I am knight" because they always lie.
So either way "A" would say "Knight"
So if "B" says "A said he was a knave" then "B" is lying and thus a knave and "A" is knight! :thanks:
---
Now I have one for you:
2 men walk up to a judge. The judge asked the first man "Who are you?"
The first man answers "I am Mr. Smith."
The judge then asks Mr Smith, "who is the man next to you?"
Mr. Smith answers:
"Brothers and Sisters I have none,
This man's father is my father's son"
What is the relationship between Mr. Smith and the other man?
The people on the Island of Knights and Knaves are either knights or knaves. Knights are honest and virtuous, and whatever they say is true.
Knaves are base and shameful, and whatever they say is false.
A and B. Stranger ask A: Are you a knight or a knave?
A answers indistinctly.
Stranger asks B: What did A say?
B answers: A said he is a knave.
What are A and B?Easy! (Mind if I butt in?)
B is a knave and A is a knight!
Logic:
When asked "Are you a knight or a Knave":
A Knight would always answer "I am a knight" because he speaks the truth.
A Knave would always say "I am knight" because they always lie.
So either way "A" would say "Knight"
So if "B" says "A said he was a knave" then "B" is lying and thus a knave and "A" is knight! :thanks:
---
Now I have one for you:
2 men walk up to a judge. The judge asked the first man "Who are you?"
The first man answers "I am Mr. Smith."
The judge then asks Mr Smith, "who is the man next to you?"
Mr. Smith answers:
"Brothers and Sisters I have none,
This man's father is my father's son"
What is the relationship between Mr. Smith and the other man?
Zeno Swijtink
12-24-2007, 10:49 PM
Easy! (Mind if I butt in?)
B is a knave and A is a knight!
Logic:
When asked "Are you a knight or a Knave":
A Knight would always answer "I am a knight" because he speaks the truth.
A Knave would always say "I am knight" because they always lie.
So either way "A" would say "Knight"
So if "B" says "A said he was a knave" then "B" is lying and thus a knave and "A" is knight! :thanks:
---
Now I have one for you:
2 men walk up to a judge. The judge asked the first man "Who are you?"
The first man answers "I am Mr. Smith."
The judge then asks Mr Smith, "who is the man next to you?"
Mr. Smith answers:
"Brothers and Sisters I have none,
This man's father is my father's son"
What is the relationship between Mr. Smith and the other man?
Sorry, Barry, not completely right. :(:
B is a knave and A is a knight!
Logic:
When asked "Are you a knight or a Knave":
A Knight would always answer "I am a knight" because he speaks the truth.
A Knave would always say "I am knight" because they always lie.
So either way "A" would say "Knight"
So if "B" says "A said he was a knave" then "B" is lying and thus a knave and "A" is knight! :thanks:
---
Now I have one for you:
2 men walk up to a judge. The judge asked the first man "Who are you?"
The first man answers "I am Mr. Smith."
The judge then asks Mr Smith, "who is the man next to you?"
Mr. Smith answers:
"Brothers and Sisters I have none,
This man's father is my father's son"
What is the relationship between Mr. Smith and the other man?
Sorry, Barry, not completely right. :(:
Valley Oak
12-25-2007, 11:27 AM
Mr. Smith is either the father or an uncle of the other man. There is also a possibility of Mr. Smith being a stepfather or a step uncle or a half uncle.
Edward
Now I have one for you:
2 men walk up to a judge. The judge asked the first man "Who are you?"
The first man answers "I am Mr. Smith."
The judge then asks Mr Smith, "who is the man next to you?"
Mr. Smith answers:
"Brothers and Sisters I have none,
This man's father is my father's son"
What is the relationship between Mr. Smith and the other man?
Edward
Now I have one for you:
2 men walk up to a judge. The judge asked the first man "Who are you?"
The first man answers "I am Mr. Smith."
The judge then asks Mr Smith, "who is the man next to you?"
Mr. Smith answers:
"Brothers and Sisters I have none,
This man's father is my father's son"
What is the relationship between Mr. Smith and the other man?
Valley Oak
12-25-2007, 11:36 AM
Is infinity the square root of infinity?
Edward
Hello. I would love to discuss math in general with anyone who wishes.
Kermit Rose < [email protected] >
Edward
Hello. I would love to discuss math in general with anyone who wishes.
Kermit Rose < [email protected] >
mykil
12-25-2007, 12:44 PM
A and B are just that, A and B LOL the man is his son
Zeno Swijtink
12-25-2007, 06:57 PM
[A Knight and Knave Puzzle]
The people on the Island of Knights and Knaves are either knights or knaves. Knights are honest and virtuous, and whatever they say is true.
Knaves are base and shameful, and whatever they say is false.
A and B. Stranger ask A: Are you a knight or a knave?
A answers indistinctly.
Stranger asks B: What did A say?
B answers: A said he is a knave.
What are A and B?
Kermit Rose send in a correct answer privately, with a new puzzle:
Don't know what A is. But we know that B is a knave.
Now if B had added, that A really is a knave, then we would know that A is a knight.
Here is another one.
In a particular insane asylum, each individual is of one of two types--
a doctor or a patient.
Each individual is also either completely sane (totally accurate in his/her beliefs)
or completely insane (totally inaccurate in his/her beliefs--
that is, each true proposition is believed to be false and
each false proposition is believed to be true).
Everyone is totally honest--all say what they truly believe.
You meet two people in this asylum.
Mr. Jones tells you that Mr. Smith is a doctor on the staff.
Mr. Smith tells you that Mr. Jones is a patient.
You become concerned that this asylum has a problem--
either it's employing an insane doctor or
it has admitted a patient who is totally sane.
Prove that your concern is well founded.
Kermit could not know what A is: there is not enough information to decide one way or another. Anybody can solve his puzzle??
The people on the Island of Knights and Knaves are either knights or knaves. Knights are honest and virtuous, and whatever they say is true.
Knaves are base and shameful, and whatever they say is false.
A and B. Stranger ask A: Are you a knight or a knave?
A answers indistinctly.
Stranger asks B: What did A say?
B answers: A said he is a knave.
What are A and B?
Kermit Rose send in a correct answer privately, with a new puzzle:
Don't know what A is. But we know that B is a knave.
Now if B had added, that A really is a knave, then we would know that A is a knight.
Here is another one.
In a particular insane asylum, each individual is of one of two types--
a doctor or a patient.
Each individual is also either completely sane (totally accurate in his/her beliefs)
or completely insane (totally inaccurate in his/her beliefs--
that is, each true proposition is believed to be false and
each false proposition is believed to be true).
Everyone is totally honest--all say what they truly believe.
You meet two people in this asylum.
Mr. Jones tells you that Mr. Smith is a doctor on the staff.
Mr. Smith tells you that Mr. Jones is a patient.
You become concerned that this asylum has a problem--
either it's employing an insane doctor or
it has admitted a patient who is totally sane.
Prove that your concern is well founded.
Kermit could not know what A is: there is not enough information to decide one way or another. Anybody can solve his puzzle??
Zeno Swijtink
12-26-2007, 03:28 PM
Kermit Rose send in a new puzzle:
In a particular insane asylum, (1) each individual is of one of two types--
a doctor or a patient.
(2) Each individual is also either completely sane (totally accurate in his/her beliefs) or completely insane (totally inaccurate in his/her beliefs--
that is, each true proposition is believed to be false and
each false proposition is believed to be true).
Everyone is totally honest--all say what they truly believe.
You meet two people in this asylum.
(3) Mr. Jones tells you that Mr. Smith is a doctor on the staff.
(4) Mr. Smith tells you that Mr. Jones is a patient.
You become concerned that this asylum has a problem--
either it's employing an insane doctor or it has admitted a patient who is totally sane.
Prove that your concern is well founded.
In my proof I continue numbering the statements for easy reference:
To prove: The hospital is employing an insane doctor or it has admitted a patient who is totally sane.
Jones is telling the truth (5) or not telling the truth (6):
If Jones is telling the truth (5)
Then, with (3), Mr Smith is a doctor on the staff (7)
Smith is sane (8) or not sane (9)
If Mr Smith is sane (8), hence (with 2) is telling the truth, then Jones, a patient (with 4), told a truth, hence is sane (with 2), that is, the hospital it has admitted a patient (Jones) who is totally sane.
If Smith is insane (9), the hospital is employing an insane doctor (with 7).
Or Mr. Jones is not telling the truth (6).
Then (with 3) Mr. Smith is not a doctor on the staff (11), hence is (with 1) a patient (12)
If Smith is sane (8), has admitted a patient who is totally sane.
If he is insane (9), then whatever he says is false (with 2) hence, with (4) Jones is not a patient, hence a doctor (with 1), who told a lie, hence is insane (with 2), hence the hospital is employing an insane doctor (Jones).
Hence in all cases, the hospital is employing an insane doctor or it has admitted a patient who is totally sane.
Note 1. I am using the law of the excluded middle (A or not-A), so the proof is not intuitionistically valid.
Note 2. Being a knave as defined in the earlier puzzle is not the same as being totally insane: whatever a knave believes is false but a knave does not have necessarily have a (false) opinion about all facts. And in fact, in the proof I only used the knave part of being totally insane.
In a particular insane asylum, (1) each individual is of one of two types--
a doctor or a patient.
(2) Each individual is also either completely sane (totally accurate in his/her beliefs) or completely insane (totally inaccurate in his/her beliefs--
that is, each true proposition is believed to be false and
each false proposition is believed to be true).
Everyone is totally honest--all say what they truly believe.
You meet two people in this asylum.
(3) Mr. Jones tells you that Mr. Smith is a doctor on the staff.
(4) Mr. Smith tells you that Mr. Jones is a patient.
You become concerned that this asylum has a problem--
either it's employing an insane doctor or it has admitted a patient who is totally sane.
Prove that your concern is well founded.
In my proof I continue numbering the statements for easy reference:
To prove: The hospital is employing an insane doctor or it has admitted a patient who is totally sane.
Jones is telling the truth (5) or not telling the truth (6):
If Jones is telling the truth (5)
Then, with (3), Mr Smith is a doctor on the staff (7)
Smith is sane (8) or not sane (9)
If Mr Smith is sane (8), hence (with 2) is telling the truth, then Jones, a patient (with 4), told a truth, hence is sane (with 2), that is, the hospital it has admitted a patient (Jones) who is totally sane.
If Smith is insane (9), the hospital is employing an insane doctor (with 7).
Or Mr. Jones is not telling the truth (6).
Then (with 3) Mr. Smith is not a doctor on the staff (11), hence is (with 1) a patient (12)
If Smith is sane (8), has admitted a patient who is totally sane.
If he is insane (9), then whatever he says is false (with 2) hence, with (4) Jones is not a patient, hence a doctor (with 1), who told a lie, hence is insane (with 2), hence the hospital is employing an insane doctor (Jones).
Hence in all cases, the hospital is employing an insane doctor or it has admitted a patient who is totally sane.
Note 1. I am using the law of the excluded middle (A or not-A), so the proof is not intuitionistically valid.
Note 2. Being a knave as defined in the earlier puzzle is not the same as being totally insane: whatever a knave believes is false but a knave does not have necessarily have a (false) opinion about all facts. And in fact, in the proof I only used the knave part of being totally insane.
Willie Lumplump
12-26-2007, 07:28 PM
Is infinity the square root of infinity? EdwardAt last something that I know the answer to. Yes.
Willie Lumplump
12-26-2007, 07:31 PM
Hello. I would love to discuss math in general with anyone who wishes.Kermit Rose < [email protected] >I kind of doubt that anyone here can keep up with you mathematically, but there are some real heavyweights in a Yahoo group called Sciencechatforum. They have a whole section devoted exclusively to mathematics.
Zeno Swijtink
12-26-2007, 07:31 PM
At last something that I know the answer to. Yes.
Now is 2 to the power infinity equal to infinity?
Now is 2 to the power infinity equal to infinity?
Willie Lumplump
12-27-2007, 01:06 PM
Now is 2 to the power infinity equal to infinity?I would think so. You're still dealing with countable numbers, isn't that so?
Kermit1941
12-27-2007, 01:23 PM
I would think so. You're still dealing with countable numbers, isn't that so?
hmmm...
I had assumed that infinity meant non-finite, and corresponding to an intuition of being greater than any finite number.
This concept of infinity applies to both countable transfinite numbers and uncountable transfinite numbers.
There are simple proofs that the transfinite number
2 to the aleph-null is not countable.
The simplest such proof ,
Cantor's diagonal proof,
proves that the set of real numbers between 0 and 1 is uncountable.
( We independently prove that this set has cardinality 2 to the aleph-null.)
So, 2^ countable transfine cardinal = uncountable transfinite cardinal.
hmmm...
I had assumed that infinity meant non-finite, and corresponding to an intuition of being greater than any finite number.
This concept of infinity applies to both countable transfinite numbers and uncountable transfinite numbers.
There are simple proofs that the transfinite number
2 to the aleph-null is not countable.
The simplest such proof ,
Cantor's diagonal proof,
proves that the set of real numbers between 0 and 1 is uncountable.
( We independently prove that this set has cardinality 2 to the aleph-null.)
So, 2^ countable transfine cardinal = uncountable transfinite cardinal.
Willie Lumplump
12-28-2007, 10:43 PM
hmmm...
I had assumed that infinity meant non-finite, and corresponding to an intuition of being greater than any finite number.
This concept of infinity applies to both countable transfinite numbers and uncountable transfinite numbers.
There are simple proofs that the transfinite number
2 to the aleph-null is not countable.
The simplest such proof ,
Cantor's diagonal proof,
proves that the set of real numbers between 0 and 1 is uncountable.
( We independently prove that this set has cardinality 2 to the aleph-null.)
So, 2^ countable transfine cardinal = uncountable transfinite cardinal.Have I told you about my bugs yet?
I had assumed that infinity meant non-finite, and corresponding to an intuition of being greater than any finite number.
This concept of infinity applies to both countable transfinite numbers and uncountable transfinite numbers.
There are simple proofs that the transfinite number
2 to the aleph-null is not countable.
The simplest such proof ,
Cantor's diagonal proof,
proves that the set of real numbers between 0 and 1 is uncountable.
( We independently prove that this set has cardinality 2 to the aleph-null.)
So, 2^ countable transfine cardinal = uncountable transfinite cardinal.Have I told you about my bugs yet?