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Jul
1
comment Ball with euclidean metric - mistake in book?
You're right, instead of $x^2+y^2\lt1$ it should be $x^2+y^2\lt1^2.$
Jun
28
answered Find the sum of the first $3n$ terms of a geometric series given the sum of the first $n$ terms is $48$ and the sum of first $2n$ terms is $60$
Jun
25
comment What is the combinatorial proof for the formula of S(n,k) - Stirling numbers of the second kind?
I don't think you mean "$k$ is the number of partitions" because $S(n,k)$ is the number of partitions.
Jun
22
comment Complexity of $\binom{n}{2}$
$$\frac{n^n}{e^n}\times e^{n-2}\ne n^n$$
Jun
22
comment Applying Inclusion-Exclusion principle
@Pradeep Your answer is "correct" for distinguishable passengers, assuming that at least one passenger gets off at each floor. A translation error, perhaps?
Jun
22
revised Why can only those younger than 40 years old win the Fields Medal?
spelling
Jun
21
comment A Knight and Knave Problem
@RossMillikan The puzzle states clearly and explicitly of the so-called liars that "they can lie or tell the truth". You may be right to complain that calling them "liars" is misleading, or that it's nonstandard terminology for "puzzles like this". Still, the problem is what it is. It's about 42 people who will always tell the truth, and 27 unreliable sorts who may say anything.
Jun
21
awarded  Enlightened
Jun
21
awarded  Nice Answer
Jun
21
comment How to prove it's possible to place $8$ non-attacking rooks on a chessboard with $7$ cells cut out?
@DanielFischer The problem is an easy application of Hall's marriage theorem, but of course the direct inductive proof is much simpler.
Jun
20
revised Does the Gale-Shapley stable marriage algorithm give at least one person his or her first choice?
added 9 characters in body; edited title
Jun
19
comment Product of disjoint cycles and product of transpositions
If the permutation is written as a product of cycles--disjoint (commuting) or not--it suffices to write each cycle as a product of transpositions. By the way I've never understood why the factorization $(1\ 2\ 3\ 4\ 5\ 6\ 7)=(1\ 7)(1\ 6)(1\ 5)(1\ 4)(1\ 3)(1\ 2)$ is so much more popular than $(1\ 2\ 3\ 4\ 5\ 6\ 7)=(1\ 2)(2\ 3)(3\ 4)(4\ 5)(5\ 6)(6\ 7).$ There must be some big advantage to using the former; I wish I knew what it was.
Jun
19
comment Product of disjoint cycles and product of transpositions
$\alpha=(2\ 1)(4\ 5)(5\ 3)$
Jun
18
comment Simple examples of non-isomorphic but elementarily equivalent structures.
Your uncountable example is needlessly complicated. Two infinite sets, with no relations other than equality, are elementarily equivalent. So just take two infinite sets of different cardinalities, say, one countable and one uncountable.
Jun
18
comment An simple example to show that every countably compact space needn't be compact
If you know about order topologies and ordinal numbers, the set $[0,\omega_1)$ of all countable ordinals with the order topology is a nice easy example of a countably compact space which is not compact.
Jun
16
comment Prove that $2^{mn}$ is always greater than or equal to $m^n$
As an alternative to induction, Cantor's diagonal argument proves that $2^m\gt m.$
Jun
16
comment A construction of sigma-algebras - surely not new, right?
@NoahSchweber I figured he was talking about the fact that the class of Borel sets has cardinality no greater than $c,$ The fact that the cardinality is at least $c$ is, as you point out, trivial.
Jun
16
comment Measure Theory by Halmos, theorem B, page 22
Of course you can't prove that "the class of all finite union[s] of elements of $E$ is a ring of sets." Halmos doesn't say that the class of all finite unions of elements of $E$ is a ring. He says that the class of all sets which may be covered by a finite union of sets of $E$ is a ring.
Jun
16
revised Prove that $ |X| \leq |Y| $ if $d(x) \geq d(y) \forall x,y \in E $ in a bipartite graph
added 1 character in body
Jun
15
comment Prove that $ |X| \leq |Y| $ if $d(x) \geq d(y) \forall x,y \in E $ in a bipartite graph
I edited an explanation of $|X|=\sum_{xy\in E}\frac1{\text d(x)}$ into my answer. But what do you think of my interpretation of the question?