|
Grex > Tutoring > #14: Algebra, Geometry, Calculus, Trig, all that good stuff |  |
|
| Author |
Message |
| 25 new of 132 responses total. |
rcurl
|
|
response 24 of 132:
| 
Jul 5 18:14 UTC 1997 |
touche....
|
senna
|
|
response 25 of 132:
|
Jul 6 02:43 UTC 1997 |
It *does* strike me as rather interesting that while math laws and physics
never change, and never will, the text books teaching their governance have
to be replaced on a regular basis.
|
i
|
|
response 26 of 132:
|
Jul 6 16:06 UTC 1997 |
At the high undergraduate & graduate level (at least in math), most of the
educational fashion experts are out of their depth, and its common to see
standard texts run 10-15 years between minor-revision editions. I doubt
that selling more has anything to do with the revisions - I've seen prof.'s
teach classes from an older edition than what the class has, and far more
texts are kept for future reference by students at that level.
|
valerie
|
|
response 27 of 132:
|
Jul 6 20:49 UTC 1997 |
This response has been erased.
|
anderyn
|
|
response 28 of 132:
|
Jul 6 21:20 UTC 1997 |
Hhhhm. A *lot* of the books we review are reissues of books published
thirty years ago, and a lot more are revisions of books which are
anywhere from twenty to ten years old. OF course, there are books
which come out every year or tow with new revisions, but that's
uncommon (we usually get every %$!@ book that isn't too basic, so
I see a lot of math textbooks)...
|
i
|
|
response 29 of 132:
|
Jul 8 01:45 UTC 1997 |
Re: #27
I went through (mainly theoretical) math from about '85 to '89. Sub-senior
level texts certainly turned over faster, even in math. Computer-oriented
math texts seemed to be replaced by a new generation about every 10 weeks
for a while...
|
tsty
|
|
response 30 of 132:
|
Jul 8 09:02 UTC 1997 |
hells bells, newtonian mechanics hasn't changed in 300 years! amazing,
it's still 'new' to students.
i did get a teacher to try teaching math as a language, translatable
to/from english. about a year later, she said she was ahving much
better success with er students as a result. i smiled.
|
aruba
|
|
response 31 of 132:
|
Jul 8 17:16 UTC 1997 |
Re #30: That's a great idea, teaching people to translate between math and
English. I saw a low of college freshmen who had no clue how to do that.
Doesn't anyone have a problem? I suppose I could dig some out, but most of
my good ones have been used up in previous math items.
|
aruba
|
|
response 32 of 132:
|
Jul 10 02:13 UTC 1997 |
Ok, all I have at present are logic problems, but here goes. I'll go through
a chapter in Raymond Smullyan's book "The Lady or the Tiger?", which has a lot
of good stuff in it.
-------------------
Many of you are familiar with Frank Stockton's story "The Lady or the Tiger?",
in which the prisoner must choose between two rooms, one of which contains a
lady and the other a tiger. If he chooses the former, he marries the lady; if
he chooses the latter, he (probably) gets eaten by the tiger.
The king of a certain land had also read the story, and it gave him an idea.
"Just the perfect way to try my prisoners!" he said one day to his minister.
"Only, I won't leave it to chance; I'll have signs on the doors of the rooms,
and in each case I'll tell the prisoner certain facts about the signs. If the
prisoner is clever and can reason logically, he'll save his life - and win a
nice bride to boot!"
"Excellent idea!" said the minister.
THE TRIALS OF THE FIRST DAY
On the first day, there were three trials. In all three, the king explained
to the prisoner that each of the two rooms contained either a lady or a tiger,
but it could be that there were tigers in both rooms, or ladies in both rooms,
or then again, maybe one room contained a lady and the other room a tiger.
The First Trial
"Suppose both rooms contain tigers," asked the prisoner. "What do I do
then?"
"That's your hard luck!" replied the king.
"Suppose both rooms contain ladies?" asked the prisoner.
"Then, obviously, that's your good luck," replied the king. "Surely you
could have guessed the answer to that!"
"Well, suppose one room contains a lady and the other a tiger, what happens
then?" asked the prisoner.
"In that case, it makes quite a difference which room you choose, doesn't
it?"
"How do I know which room to choose?" asked the prisoner.
The king pointed to the signs on the doors of the rooms:
I II
IN THIS ROOM THERE IN ONE OF THESE ROOMS
IS A LADY, AND IN THERE IS A LADY, AND
THE OTHER ROOM IN ONE OF THESE ROOMS
THERE IS A TIGER THERE IS A TIGER
"Is it true, what the signs say?" asked the prisoner.
"One of them is true," replied the king, "but the other one is false."
If you were the prisoner, which door would you open (assuming, of course,
that you preferred the lady to the tiger)?
----------------------
(Of course I expect a full explanation of your answer!)
|
remmers
|
|
response 33 of 132:
|
Jul 10 02:18 UTC 1997 |
Well, I => II, so if I were true, so would II. Hence if one of
them is true and the other false, I must be the false one. So
II must have the lady, and I'd go with that.
|
mcnally
|
|
response 34 of 132:
|
Jul 10 05:55 UTC 1997 |
John's logic is hard to argue with.. Can we proceed to the second day's
test?
|
aruba
|
|
response 35 of 132:
|
Jul 10 07:23 UTC 1997 |
Very nice, John.
The Second Trial
And so, the first prisoner saved his life and made off with the lady. The
signs on the doors were then changed, and new occupants of the rooms were
selected accordingly. This time the signs read as follows:
I II
AT LEAST ONE OF THESE A TIGER IS IN
ROOMS CONTAINS A LADY THE OTHER ROOM
"Are the statements on the signs true?" asked the second prisoner.
"They are either both true or both false", replied the king.
Which room should the prisoner pick?
|
toking
|
|
response 36 of 132:
|
Jul 10 15:06 UTC 1997 |
II cause if I is true then their is a lady in one room and if I is
true the II is also true
if I is false then both rooms must contain a tiger and if I is false
then II is also false, meaning that I doesn't have a tiger in it, which
would be impossible, because I being false says so.
|
aruba
|
|
response 37 of 132:
|
Jul 10 15:31 UTC 1997 |
Right:
Both false => (I) both rooms contain tigers and (II) room I does not
contain a tiger, two statements which are clearly in conflict. So they can't
both be false, which means (according to the king) they must both be true.
Both true => (I) either there is a lady in room I or there is a lady in
room II, and (II) there is not a lady in room I. So there must be a lady in
room II.
|
aruba
|
|
response 38 of 132:
|
Jul 10 15:41 UTC 1997 |
The Third Trial
In this trial, the king explained that, again, the signs were either both true
or both false. Here are the signs:
I II
EITHER A TIGER IS IN A LADY IS IN
THIS ROOM OR A LADY IS THE OTHER ROOM
IN THE OTHER ROOM
Does the first room contain a lady or a tiger? What about the other
room?
|
aruba
|
|
response 39 of 132:
|
Jul 10 15:47 UTC 1997 |
BTW the symbol "=>" that remmers used in #33 and I used in #37 means
"implies". So "A => B" can be read "A implies B", or in other words,
"If A is true then B must be true as well".
|
aruba
|
|
response 40 of 132:
|
Jul 10 16:23 UTC 1997 |
Here's some more notation, just so we're all talking the same language:
A & B means "A is true and B is true"
A | B means "either A is true or B is true (or both are true)"
!A means "A is false"
A <=> B means "A implies B and B implies A", which is the same as saying
"A is true if and only if B is true"
L(a) means "there is a lady in room a"
T(a) means "there is a tiger in room a"
|
drew
|
|
response 41 of 132:
|
Jul 10 23:42 UTC 1997 |
I am assuming either...or to mean exclusive-or (XOR).
If room I has a cat, then II is false. In that case, if there were a lady
in room II, then the either/or statement (TRUE XOR TRUE) would evaluate FALSE.
If room I has a lady, then II is true. In order for I to evaluate TRUE, then,
there must also be a lady in room II. (Kitty_in_I XOR Lady_in_II ==
FALSE XOR TRUE == TRUE).
So they're either both chicks or both cats, and it makes no difference which
door you pick.
|
nsiddall
|
|
response 42 of 132:
|
Jul 11 05:53 UTC 1997 |
Both signs can't be false. For sign I to be false there must be a
lady in room I and a tiger in in room II. In that case sign II must be true.
Since they cannot both be false, they must be true. If sign II is true, the
first part of sign I cannot be true. Therefore the truth of sign I must be
fulfilled by the second part, meaning there is a lady in room II. Guaranteed
ladies in both rooms.
|
aruba
|
|
response 43 of 132:
|
Jul 11 06:01 UTC 1997 |
You should assume that "either A or B" is an *inclusive-or* - that is,
"either A or B" means "either A is true or B is true or both are true."
(That's the way mathematicians and logicians use the word "or", as a rule,
and that's definitely the way Smullyan means it here. (I can tell from his
solution.))
Care to have another go, drew?
|
aruba
|
|
response 44 of 132:
|
Jul 11 06:13 UTC 1997 |
Nathaniel slipped in, and is quite correct. Which means we should move on
to...
THE SECOND DAY
"Yesterday was a fiasco," said the king to his minister. "All three
prisoners solved their puzzles! Well, we have five trials coming up
today, and I think I'll make them a little tougher."
"Excellent idea!" said the minister.
Well, in each of the tirals of this day, the king explained that in the
lefthand room (Room I), if a lady is in it, then the sign on the door is
true, but if a tiger is in it, the sign is false. In the righthand room
(Room II), the situation is the opposite: a lady in the room means the
sign on the door is false, and a tiger in the room means the sign is true.
Again, it is possible that both rooms contain ladies or both rooms contain
tigers, or that one room contains a lady and the other a tiger.
THE FOURTH TRIAL
After the king explained the above rules to the prisoner, he pointed to
the two signs:
I II
BOTH ROOMS BOTH ROOMS
CONTAIN LADIES CONTAIN LADIES
Which room should the prisoner pick?
|
remmers
|
|
response 45 of 132:
|
Jul 11 13:24 UTC 1997 |
If room (II) contained a tiger, then by the king's condition,
sign (II) would be true and room (II) would contain a lady, a
contradiction. Hence room (II) cannot contain a tiger and so
must contain a lady. The prisoner should pick room (II).
|
aruba
|
|
response 46 of 132:
|
Jul 11 18:44 UTC 1997 |
Quite right. (And room I must contain a tiger.) I'll post the next one
tonight.
|
aruba
|
|
response 47 of 132:
|
Jul 12 05:29 UTC 1997 |
The Fifth Trial
The same rules apply, and here are the signs:
I II
AT LEAST ONE ROOM THE OTHER ROOM
CONTAINS A LADY CONTAINS A LADY
|
srw
|
|
response 48 of 132:
|
Jul 12 07:26 UTC 1997 |
View hidden response.
|