Does this make sense?: Prove A∪(A∩B)=A. Let’s prove by counterexample; if A∪(A∩B)≠A, then A⊂(A∩B) would have to hold. If A⊂B, then A∩B=A. We are left with A∪A≠A. We know that A∪A=A, and we are left with a contradiction. Therefore, A∪(A∩B)=A.
@Committingtoachallenge but it is a solved problem ! a good algorithm exists for arriving at a solution ! so it is not a previously unencountered problem ..
I guess the mark of a good theorem is that it has an interesting proof. I do not care to read through mindless computations. And we have proofs of some really major statements which are so incomprehensible that they only really serve to show we can't disprove the statement...
It's more of an online class I am taking. I input the answer in. If it's wrong it says "Sorry! Incorrect!" if it's right, then it says it's correct and loads the solution.
yeah, that's what it looks like in generating function form
ohh
i see
x^3 in my thread, 2 in the original problem statement.
It's "The third kid receives "0, 2, or 5 candies"
Determine how many ways I can distribute 80 candies to 3 kids, such that:
The first kid receives an arbitrary number of candies (possibly 0). The second kid receives an even positive number of candies. The third kid receives 0, 2, or 5 candies. Every candy is distributed.
"The generating function representing all possible allotments to the first kid is 1+x+x^2+x^3+\cdots = \frac 1{1-x}. The generating function for the second kid is x^2+x^4+x^6+\cdots = \frac {x^2}{1-x^2}. The generating function for the third kid is 1+x^2+x^5. Let P(x) be the product of these three functions. Then the coefficient of x^n in the power series expansion of P(x) is the number of ways of distributing exactly n candies to the three kids. Specifically, we want the coefficient of x^{80}.
@HatMan i have had sleep paralysis induced hallucinations my entire life and i always see a weird silhouette of a man wearing a hat or something on or around his head in my hallucinated state. but i know for a fact it is an hallucination since other properties of the dream state are wrong and my parents explained to me the science behind it and your brain when interrupted from REM sleep is in a paralised state due to neurotransmitters, and part of your brain responsible for dream fabricati-
-on is the same part of your brain active during hallucinations
@HatMan the reported cases of the hat man are after awakening? what other scenario am i to refer to?
@HatMan it is post dream though, it is when my awareness is awake, but my body is still in paralysis and my dream related areas of my brain are still active
i know that $(x,y)\mapsto(y,x)$ means that we just reverse the components of the 2$-tuples$, but would you write $f:X\to Y$ for $f(x)=x^2$ as $x\mapsto x^2$?
we we use the $\mapsto$ symbol to show $x\mapsto f(x)$?
so if i had $f:X\to Y, g: Y\to Z,g(f(x)),$ where $f(x)=x+1$ and $g(x)=x^2$, would it be fine to write $x\mapsto (x+1) \mapsto (x+1)^2$ or should i write $x\mapsto [y=(x+1)] \mapsto [z=y^2]$
@AlexanderGruber what are the seven mathematical seas? topology, analysis, calculus, geometry, statistics, algebra, number theory?
actually i think number theory is a type of algebra :\
@Integrator If they have not posted or commented on meta, then you can't tell very much about their opinions on the things important to moderation. More importantly, I think that participation in meta is required for moderation.
@Integrator I see at least two, but I don't want to say anything more.
@Integrator reputation is only important to moderation as it somewhat indicates a devotion to the site and some experience with the way things work here. However, mathematical knowledge is not a requirement for moderation.
This is a nice problem but I think this is not a problem for MO.
Anyway, the coset trick mentioned by @Alain Valette is nice.
As another way to approach a solution, consider the function $f : [0,1]\longrightarrow \Bbb{R}$ with
$f(x) = \limsup_n \frac{x_1+x_2+\cdots+x_n}{n}$ where $0.x_1x_2\c...
@Khallil: Yeah. And he's rather short, like between 165 and 170 cm, and also not built like a truck, just a little wider shoulders than the usual guy.
@Khallil: A friend I went to school with is currently playing in third league and he's probably the best player I've ever played with. So, yeah. Looking forward to a real challenge. :P
This is a nice problem but I think this is not a problem for MO.
Anyway, the coset trick mentioned by @Alain Valette is nice.
As another way to approach a solution, consider the function $f : [0,1]\longrightarrow \Bbb{R}$ with
$f(x) = \limsup_n \frac{x_1+x_2+\cdots+x_n}{n}$ where $0.x_1x_2\c...
