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What does (y or ¬ x) ^x imply? Surely it implies Y.
(edited 12 years ago)
Original post by Blutooth

as (¬Y)^Y is a contradiction (X OR ¬Y) ^Y must = X^Y


I'm by no means an expert but, surely, (¬Y)^Y is the empty set. Where is the contradiction?
Try combining the first and third expressions. the combine the simplified first expression with the second
(edited 12 years ago)
Original post by ben-smith
I'm by no means an expert but, surely, (¬Y)^Y is the empty set. Where is the contradiction?


As they are propositions it is a contradiction...e.g. let Y be the proposition that I am a cat...I cannot be both a cat and not a cat simultaneously...

Spoiler

Original post by ben-smith
I'm by no means an expert but, surely, (¬Y)^Y is the empty set. Where is the contradiction?


Why do you think y^(¬Y) would be the empty set?


Let us assume that y is the empty set, are you telling me that (¬Y) is also the empty set? Surely not. (¬Y) in this case would be something other than the empty set? not the empty set and the empty set is a contradiction (and not the empty set). :smile:
(edited 12 years ago)
Original post by TheMagicMan
As they are propositions it is a contradiction...e.g. let Y be the proposition that I am a cat...I cannot be both a cat and not a cat simultaneously...

Spoiler



errr, well don't view them as propositions :wink:
I guess my point is that if it were a contradiction then the whole initial statement would be a contradiction in which case we aren't really doing anything. Anyway, it was a minor point (if one at all).
No idea why someone negged you. Let me try and balance it out.
Original post by ben-smith
errr, well don't view them as propositions :wink:
I guess my point is that if it were a contradiction then the whole initial statement would be a contradiction in which case we aren't really doing anything. Anyway, it was a minor point (if one at all).
No idea why someone negged you. Let me try and balance it out.


If we look at it from a set point of view, aren't we just going to use the fact that taking the union of a set A with the empty set is equivalent to A
Original post by ben-smith
errr, well don't view them as propositions :wink:
I guess my point is that if it were a contradiction then the whole initial statement would be a contradiction in which case we aren't really doing anything. Anyway, it was a minor point (if one at all).
No idea why someone negged you. Let me try and balance it out.

Sorry, I negged him by accident, but I dont have any rep points so it wont affect your score!
Original post by Blutooth
Why do you think y^(¬Y) would be the empty set?


Let us assume that y is the empty set, are you telling me that (¬Y) is also the empty set? Surely not. (¬Y) in this case would be something other than the empty set? not the empty set and the empty set is a contradiction (and not the empty set). :smile:


Well, my thinking is that ¬Y is the complement of Y and so they are disjoint and therefore, by definition, their Y^(¬Y)= ∅.
Maybe we have different interpretations of what this means, I mean it in the way this does.
Original post by TheMagicMan
If we look at it from a set point of view, aren't we just going to use the fact that taking the union of a set A with the empty set is equivalent to A


That's what I was trying to say!
Original post by ben-smith
Well, my thinking is that ¬Y is the complement of Y and so they are disjoint and therefore, by definition, their Y^(¬Y)= ∅.
Maybe we have different interpretations of what this means, I mean it in the way this does.


There are different ways of interpreting the problem, I was just teasing you. But seriously, if you can tell me what ∅^(¬∅) equals, I will be impressed as I haven't the foggiest. :cool:
(edited 12 years ago)
Reply 3631
Avatar for gff
gff
5
Here we go, a fun question. :tongue:



[*] The curves A,B,CA, B, C and DD are defined in the plane as follows.

A={(x,y):x2−y2=xx2+y2}\displaystyle A = \left\{ (x,y) : x^2 - y^2 = \frac{x}{x^2 + y^2}\right\}

B={(x,y):2xy+yx2+y2=3}\displaystyle B = \left\{ (x, y) : 2xy + \frac{y}{x^2 + y^2} = 3 \right\}

C={ (x,y):x3−3xy2+3y=1}\displaystyle C = \{ \ (x, y) : x^3 - 3xy^2 + 3y = 1 \}

D={ (x,y):3x2y−3x−y3=0}\displaystyle D = \{ \ (x,y) : 3x^2y - 3x - y^3 = 0 \}


Prove that A∩B=C∩DA \cap B = C \cap D.
hhmmmm

Nobody really knows what is happening as notation is far from standard.

¬ denotes a negation. ^c (for instance let A be a set, write A^c) denotes the complement of the set A. etc.

Do you have a reference for your notation or an explanation?
Reply 3633
Avatar for gff
gff
5
I was hoping it's gonna stay for a while... :tongue: +rep
Original post by gff
I was hoping it's gonna stay for a while... :tongue: +rep


Here's a nice question:

i)Take 4 quantities aia_i, 1≤i≤41\leq i \leq 4.

We play a game according to these rules: Player A adds a total of 1 to the quantities, apportioning it however he likes. Player B then reduces any two consecutive quantities, or the first and the last, to 0. They then alternate 'turns'. Assume Player B plays optimally. Player A wins if he makes any one of the quantities greater than a certain value xx. Player B's goal is to stop Player A from winning. The game continues indefinitely. What is the maximum value of xx such that Player A can win?

ii) Using the same rules, what is the maximum value of xx if there are 5 quantities?

iii) And if there are nn quantities?

I have a very ugly and lengthy solution for iii) so bonus points for elegance on that one
(edited 12 years ago)
Original post by TheMagicMan
~Quantity question^

Before I potentially waste a lot of time doing it all wrong, is the answer to i)

Spoiler

Original post by Llewellyn
Before I potentially waste a lot of time doing it all wrong, is the answer to i)

Spoiler



I think perhaps I have been somehow unclear in the game rules...how did you get that?
Original post by TheMagicMan
I think perhaps I have been somehow unclear in the game rules...how did you get that?

It's probably my interpretation of the rules that is the problem, I'll get back to this later though.
Is the answer to ii, 2?
Reply 3639
Avatar for Slumpy
Slumpy
16
Original post by TheMagicMan
Here's a nice question:

i)Take 4 quantities aia_i, 1≤i≤41\leq i \leq 4.

We play a game according to these rules: Player A adds a total of 1 to the quantities, apportioning it however he likes. Player B then reduces any two consecutive quantities, or the first and the last, to 0. They then alternate 'turns'. Assume Player B plays optimally. Player A wins if he makes any one of the quantities greater than a certain value xx. Player B's goal is to stop Player A from winning. The game continues indefinitely. What is the maximum value of xx such that Player A can win?

ii) Using the same rules, what is the maximum value of xx if there are 5 quantities?

iii) And if there are nn quantities?

I have a very ugly and lengthy solution for iii) so bonus points for elegance on that one


After a little bit of work, I'm guessing at

Spoiler

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