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 Mar23 comment Where did mathematicians learn how to do truth tables? Are you saying there is evidence that some Greek philosophers had the concept of truth functions (functions whose inputs and outputs are something like true and false)_and_ graphical representation of truth tables (a tabular representation)? I don't doubt the first, but I do the latter. Also, the question I have is about the intellectual provenance of the truth-table display in modern mathematics, not the multiple possibly non-influencing reinventions across the world. Mar16 comment Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers @user86418 the derivation after $x+1=2y$ is elementary. Getting to that point is also elementary but missing; just say where it comes from (why +1? Why 2y?) Mar16 comment Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers @user86418: nice... that's the super simple start. But you should add that to the answer to make it complete (i.e. where $x+1 = 2y$ comes from, both $x+1$ and $2y$ equal a side of the square). Mar15 comment Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers @user86418 I'm trying to see how 3/4/5 comes in a flash rather than through hard work. One flash would be as you say, to consider the center of the circle, rather than the square, and the sides of that triangle. But then how does one figure the units. Even if 5 for the radius, I do not see any flash that leads to 3 and 4 for the legs except blind guessing. Where does $x+1 = 2y$ come from, your very first step, without already knowing it's a 3/4/5 triangle? Mar15 comment Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers 'Notice the... (3,4,5) triangle'? Is there a quick observation that leads to that? Like the others, that seems to be an observation after all the calculations for the solution have been done. Mar7 comment Odd number of students in odd number of classes From the intro, there's no way to distinguish girls from boys so by symmetry, I don't see how pt a can possibly be true. Are you leaving out something like $b = \sum b_i > g = \sum g_i$? Feb13 comment Prove and element must be the identity element? In other words, the theorem is that in a group if there is an element a that is the identity for any particular element, then it's the identity element for all elements. Why? There's the symbolic proof, which may suffice, but do you want to know the meaning? Feb9 comment Where did mathematicians learn how to do truth tables? Lost: 1) you should edit your answer to paraphrase or quote the relevant text in those links so we have an idea of what you're talking about (answers on SO are expected to be self-contained). 2) Saying that Wittgenstein met Turing is little evidence of an intellectual connection, especially with respect to what essentially boils down to notation. Feb9 comment Where did mathematicians learn how to do truth tables? Edit your answer instead of trying to comment. Feb9 comment Where did mathematicians learn how to do truth tables? Can you give the details of your comment? Jan19 comment What do mathematicians mean by “equipped” Which is to say that 'equipped' is metaphorical language, not a technical term. Jan6 comment Why is it important to have the closed form of a generating function? A side consideration is simply that it is a general mathematical style that finite is desired more than the infinite. You have a better handle on things if they have been reduced to a finite thing. Ellipses are a sign of inscrutability. Of course, with notation, ellipses can be encapsulated into symbols with iteration, and then progress is measured in terms of reduction in bound variables (removing the iteration). Sure a gf can be more inscrutable, but part of math is having multiple formalisms to operate on. That's general strategy. Yuval's excellent answer gives specifics for ops on gfs. Dec12 comment What's the difference between a bijection and an isomorphism? In short, a bijection (or -jection) is a kind of function on sets. An isomorphism (or -morphism) is a kind of function that is structure preserving. Or rather one term is in the language of set theory, the other is in category theory. Underlying all isomorphisms is some bijection, but a bijection doesn't necessarily preserve structure in the category you care about. If your category is Sets, then they're the same. Dec8 comment Is 128 the only multi-digit power of 2 such that each of its digits is also a power of 2? Don't forget to also rule out 8. Dec3 comment Is it possible to simulate a floor() function with elementary arithmetic? @ChristianSemrau the given ops do not allow complex math (exact, not floating point) or arbitrary roots of unity either. The true answer is either 'no' because of the reason given in the top answer, or 'yes' because the domain is integers and integer division already gives the floor. Dec2 comment Is it possible to simulate a floor() function with elementary arithmetic? Since all your ops are on integers, the output of each op will be integers. The default implementation of integer division and so will be to 'round to 0' which is equivalent to 'floor' on non-negative integers. Which means you don't need a special 'floor' function because you already have it by default. Sep29 comment Why are real numbers useful? Use the binomial distribution. Sep25 comment Understanding why Hall's marriage theorem $\Leftrightarrow$ Dilworth's theorem robertborgersen.info/Presentations/GS-05R-1.pdf gives a number of proofs around all these theorems. Sep25 comment Analytic Proof Of ${n\choose r}={n-1\choose r-1}+{n-1\choose r}$ I would call this an 'algebraic' proof. An 'analytic' proof would more likely be a proof using generating functions. Of course everything uses algebra to some extent, but the major category of a proof is the primary strategy. Sep25 comment Sum of sequence of cubes and summation on the upper index $k^3 = 6{k \choose 3} + 3 k^2 -2k$ (I got this by expanding out ${k \choose 3}$). Now eliminate the $k^2$ using ${k \choose 2}$. Then again for the leftover $k$ term. Now, if you sum the LHS, you sum the RHS. But by the stated thm, you can calculate the sum of the RHS.