MathSE ID int64 795 4.87M | Math Question stringlengths 23 305 | Tag stringclasses 2
values |
|---|---|---|
361,118 | Prove that $4x-x^4 \leq 3, x \in \Bbb R$ | Ambiguous |
2,006,621 | How do I calculate the number of subsets of a finite set given its cardinality? | Unambiguous |
62,748 | What is the difference between two related set-builder notations? | Ambiguous |
2,054,106 | Is it true that pointwise convergence with a uniform bound implies convergence of the integrals? | Ambiguous |
742,532 | Why is the right endpoint given as $3\frac{i^2}{n^2}$ for the limit $\lim_{n \to \infty} \sum_{i=1}^{n} f(c_{i}) \Delta x_{i}$? | Ambiguous |
105,575 | What is the difference between Completeness and Soundness in first order logic? | Ambiguous |
1,978,354 | Distribution of $\frac{X+Y}{X-Y}$ | Ambiguous |
788,217 | A hammer and a nail together cost $1.10, and the hammer costs $1.00 more than the nail. What is the cost of the nail? | Unambiguous |
1,788,328 | Find the numerical value of $\sin 10^\circ \sin 50^\circ \sin 70^\circ$. | Ambiguous |
3,944,628 | In the definition of a stopping time, how important are the filtration conditions mentioned earlier? | Ambiguous |
1,079,395 | Inclusion Exclusion Principle (Probability) Problem: What is P(Rain)? | Ambiguous |
1,783,734 | Can you explain the logical basis and validity of the principle of mathematical induction for proving statements over natural numbers? | Unambiguous |
1,896,418 | For a set of $n$ points in the plane where no four are collinear, what is $f(n)$ for the case where no three are collinear? | Ambiguous |
1,214,106 | Find all polynomials $P(X)$ such that $P(x^2-y^2) = P(x-y)P(x+y)$ for all complex numbers $x, y$. | Unambiguous |
700,216 | Different solutions of $x+y+z=10$ where $x$, $y$, $z$ are all positive integers and $x, y, z \leq 10$ | Ambiguous |
3,086,320 | Evaluate $\lim_{x\to 2}\frac{2x-4}{x^2+2x-6}$ and explain why direct substitution giving $0$ differs from the result obtained via L'Hôpital's rule. | Unambiguous |
3,289,735 | Is the arc length of a circle always irrational between two points with rational coordinates? | Unambiguous |
1,762,325 | Is $\sim (\sim p) \rightarrow p$ always true? | Ambiguous |
227,317 | What is the limit of the sequence defined by the iterated cosine function $x_{n+1} = \cos(x_n)$ as $n$ approaches infinity? | Unambiguous |
4,256,389 | Alternating sum involving reciprocals of powers of primes equals $\ln(\frac{\pi}{4})$ | Ambiguous |
1,319,618 | An inequality, comprising of liminf and Sup of a set | Ambiguous |
2,498,847 | Is 1/x a surjective function when the domain and the range are all real numbers? | Ambiguous |
1,894,173 | Does the sum of the reciprocals of all prime numbers converge or diverge? | Unambiguous |
3,247,615 | How to find $P(-2)$ here? | Ambiguous |
602,860 | How to calculate which point is furthest from the origin in the direction of a normalized vector? | Ambiguous |
3,502,297 | How to show $\log(x)$ loses to powers? | Ambiguous |
1,347 | How can Euclidean geometry be axiomatized or derived using criteria from hyperbolic geometry? | Unambiguous |
11,194 | How to solve a linear recurrence relation where the nth term depends on the sum of all previous terms? | Unambiguous |
290,435 | Find five distinct positive integers whose reciprocals sum to 1. | Unambiguous |
4,019,674 | Prove that the value of $\frac{\arcsin(3/5)}{\pi}$ is an irrational number. | Unambiguous |
2,353,213 | Are there any other Lie subgroups of $\textrm{Gl}(n, \mathbb{R})$ and is there a classification for them? | Ambiguous |
1,347 | Studying Euclidean geometry using hyperbolic criteria | Ambiguous |
4,255,331 | If $ \lim_{x\to \infty} (f(x+1)-f(x))=1$, then $ \lim_{x\to \infty} \frac{f(x)}{x}=1$? | Ambiguous |
3,395,798 | Definition of a matrix by matching two vectors | Ambiguous |
984,271 | How does the isomorphism between $\mathbb{K} \otimes M_n(\mathbb{C})$ and $\mathbb{K}$ work? | Ambiguous |
1,249,731 | What is the formula for the first order moment (mean vector) of a multivariate Gaussian random vector? | Unambiguous |
1,251,548 | Provide an example of a set in a metric space that is bounded but not totally bounded. | Unambiguous |
4,488,057 | Can a function be represented by a Laurent series expansion in a neighborhood of a non-isolated singularity? | Unambiguous |
217,335 | In how many ways can a total of $1.50 be formed using exactly 50 US coins? | Unambiguous |
329,838 | How to simplify a square root of a number by extracting perfect square factors? | Unambiguous |
1,376,320 | Proving that if $a+b$ and $ab$ are of the same parity, then $a$ and $b$ are even. | Ambiguous |
4,383,384 | How to extract the vector from a rank-$1$ matrix? | Ambiguous |
1,895,699 | I'm stuck on this part of the proof: $\cot C \frac{\sin(A-B)}{\cos A \cos B}$. How do I get to the other side? | Ambiguous |
71,874 | Can I use my powers for good? | Ambiguous |
395,543 | What are the formulas for generating all primitive Pythagorean triples? | Unambiguous |
872,256 | What is the geometric interpretation of finding the $n^{ ext{th}}$ roots of a complex number in the complex plane? | Unambiguous |
2,837,226 | Show that a subset of a complete metric space is complete. | Ambiguous |
538,941 | If $xy$ and $x+y$ are both even integers (with $x,y$ integers), then $x$ and $y$ are both even integers | Ambiguous |
884,228 | How do you calculate the reliability function for a continuous uniform distribution given its lower and upper bounds? | Unambiguous |
3,035,983 | How can I find all pairs of integers $(n, m)$ that satisfy the equation $\frac{150 + n}{15 + n} = m$? | Unambiguous |
2,880,747 | Product of Summation for a single table of values. | Ambiguous |
1,002,480 | How to go from talking about line segments to points to prove $X(A)$ is compact? | Ambiguous |
488,020 | Let $M$ be a matrix, and define $V(M)$ from the signs of vectors in $\ker(M)$. Prove that $V$ is invariant under change of coordinates. | Ambiguous |
859,968 | Is $X/R$ homeomorphic to $Y$ for the broom spaces? | Ambiguous |
2,520,461 | How to minimize $AP/PB$? It doesn't seem doable for some locations. | Ambiguous |
1,496,959 | What are the necessary and sufficient conditions for a sequence of continuous functions to converge uniformly on a compact set? | Unambiguous |
1,279,205 | Can an irrational number be classified as odd or even under the standard definition of parity for integers? | Unambiguous |
4,437,668 | What are some good, elementary and maybe also interesting proofs by induction? | Ambiguous |
2,576,976 | How do I find the inverse of a $2\times2$ matrix if the determinant is zero? | Unambiguous |
1,626,298 | Prove that $2^{\log_4 n} = n^{\log_4 2}$ using logarithmic identities. | Unambiguous |
3,811,446 | What is the value of the infinite series $\sum_{n=1}^{\infty}\frac{1+2+\cdots +n}{n!}$? | Unambiguous |
1,891,318 | Number of bit strings of length 8 that do not contain "$100$"? | Ambiguous |
1,311,558 | What is the formal definition of a set in Zermelo-Fraenkel set theory and how is it intuitively conceptualized? | Unambiguous |
3,960,592 | Finding the equation of hyperbola that passes through $(2,0)$ and has foci points $(2,3)$ and $(1,0)$ | Ambiguous |
3,791,855 | Why does the quadratic equation $x^2-8x+17=0$ have non-real complex solutions? | Unambiguous |
1,668,681 | Provide an example of a 2D planar shape whose center of mass lies on its boundary. | Unambiguous |
219,343 | How can I prove that the propositional logic formula $[(p \to \neg q) \wedge q] \to \neg p$ is a tautology using a truth table or natural deduction? | Unambiguous |
1,141,210 | I am trying to do a set proof and I don't understand where the part highlighted in green comes from. | Ambiguous |
344,268 | How do I solve a rational inequality where the variable $x$ appears in the denominator? | Unambiguous |
1,501,369 | How to prove a function is one to one given $f(f(m)+f(n))=m+n$? | Ambiguous |
2,207,598 | Prove or disprove: if $A\times A = B\times B$ then $A=B$. | Ambiguous |
738,236 | How to calculate the area of a triangle ABC when given three position vectors $a, b$, and $ c$ in 3D? | Ambiguous |
4,285,591 | Does the expression $\frac{1}{1-hL}$ have an upper bound for $h > 0$ and $L > 0$ such that $hL < 1$? | Unambiguous |
4,080,901 | Using a block matrix to show that $A$ and $B$ commute | Ambiguous |
3,582,088 | Solving for polynomial $f(x)$ satisfying $f(x)f(y)=f(x)+f(y)+f(xy)-2 \forall x\in \mathbb{R}, f(1)\ne 1, f(3)=10$ | Ambiguous |
3,729,349 | If a series $\sum a_n$ converges and the sequence of its terms $(a_n)$ is monotonically increasing, why must the terms be non-positive? | Unambiguous |
2,494,260 | List mathematical theorems, conjectures, or concepts that are named after geographic locations. | Unambiguous |
3,265,216 | How to perform horizontal and vertical rescaling of a function $f(x)$ in algebra? | Unambiguous |
114,560 | Sum of normal bundle and tangent bundle. | Ambiguous |
1,001,101 | Let $f:\mathbb{R}\to\mathbb{R}$ be an arbitrary function (not necessarily continuous). Prove there exists an interval $(a,b)$ and $c>0$ such that for every $x\in(a,b)$, there is a sequence $x_n\to x$ with $|f(x_n)|\le c$. | Unambiguous |
4,564,068 | What is the formula for 0 + 1 + 3 + 6 + 10 + 15 ... ? | Ambiguous |
227,211 | Is there a real-valued function $f: \mathbb{R} \to \mathbb{R}$ such that $f(f(x)) = f(x)$ for all $x$ and $f(0) = 1$? | Unambiguous |
1,381,335 | How to tell which function fits my series to make it a Riemann Sum? | Ambiguous |
217,315 | How do you tell the difference between the minus signs? | Ambiguous |
1,432,379 | What is the minimum of $F(x,y,z)=\frac{1}{x+y}+\frac{1}{x+z}-\frac{1}{x+y+z}$? | Ambiguous |
217,315 | In algebra, what is the difference between the unary minus operator used to indicate a negative number and the binary minus operator used for subtraction? | Unambiguous |
3,711,681 | How to compute the matrix derivative of the scalar function $f(X) = \operatorname{tr}(B X X^T A)$ with respect to the matrix $X$? | Unambiguous |
1,435,003 | How to find $P(X_i=1)$ to get the expected number of empty boxes? | Ambiguous |
1,562,459 | Prove that the sequence $(1 + n^{-2})^n$ converges to 1 as $n$ approaches infinity. | Unambiguous |
1,304,823 | Can someone explain the parts of this equation? | Ambiguous |
2,657,952 | What's the relationship between mathematical logic and just plain old "logic"? The notation differs, do the concepts? The rules? | Ambiguous |
982,093 | Proof for $\lim_{n \to \infty}\left(1+\frac{a_n}{n}\right)^n = e^a$? | Ambiguous |
251,033 | What are the search terms for the largest subset with no 3 element progressions? | Ambiguous |
2,766,082 | How do you simplify the algebraic expression $\frac{10x\Delta x + 5(\Delta x)^2}{\Delta x}$ to get $10x + 5\Delta x$, assuming $\Delta x \neq 0$? | Unambiguous |
608,477 | What can you tell me about $4^{n+1}+5^{2n-1}$ and 21? | Ambiguous |
1,329,254 | Solve the ODE $\frac{d^2y}{dt^2}+10\frac{dy}{dt} = -5t^2+t^4\cosh(t)$ using the Fast Fourier Transform method. | Unambiguous |
3,422,194 | Compute the sum of the infinite series $\sum_{k=1}^{\infty} \frac{1}{k(k+2)}$ using partial fraction decomposition. | Unambiguous |
244,424 | If rank x < rank y, does it follow that x is an element of y? | Ambiguous |
687,510 | Provide an example of a Pigeonhole Principle problem involving points placed inside a square area. | Unambiguous |
2,863,657 | Simple Ratios Shortcut - Why does it work? | Ambiguous |
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