https://liberty.ioCurated star highlights with fresh energy.en-usMon, 24 Aug 2026 15:31:51 -0400https://liberty.io/how-to-find-correlation-coefficient-given-joint-distribution.htmlhttps://liberty.io/how-to-find-correlation-coefficient-given-joint-distribution.htmlRandom variables $X$ and $Y$ follow a joint distribution $$f(x, y) = \left\{ \begin{array}{ll} 2,& 0 < x \leq y < 1,\\ 0,&\text{otherwise}\end{array}\right. $$ Determine the correlat......Mon, 24 Aug 2026 19:22:03 -0400https://liberty.io/a-tourist-in-france-wants-to-visit-12-different-cities-find-the-probab.htmlhttps://liberty.io/a-tourist-in-france-wants-to-visit-12-different-cities-find-the-probab.htmlThe Question : A tourist in France wants to visit 12 different cities. If the route is randomly selected, what is the probability that she will visit the cities in alphabetical order? I thought ...Mon, 24 Aug 2026 19:20:16 -0400https://liberty.io/how-to-simplify-the-integration.htmlhttps://liberty.io/how-to-simplify-the-integration.htmlIf any integration is in form $$\int \frac{1}{1+x^2}dx$$ it easily follows the $\tan^{-1}x$ but How to simplify if we have $$\int \frac{1}{(1+x^2)^2}dx$$...Mon, 24 Aug 2026 19:17:33 -0400https://liberty.io/constructing-a-matrix-such-its-square-is-not-039-0-039-but-its-cube-is.htmlhttps://liberty.io/constructing-a-matrix-such-its-square-is-not-039-0-039-but-its-cube-is.htmli have been asked to construct a matrix A such that $A^2$ is not equal to '0' but, $A^3=0$. how should i proceed. i can only understand that all the eigenvalues for A , $A^2$ and $A^3$ will be 'z......Mon, 24 Aug 2026 19:16:35 -0400https://liberty.io/group-theory-notation.htmlhttps://liberty.io/group-theory-notation.htmlWhat does the notation $(G,.)$ mean in group theory? I have seen in places that $.$ implies the binary operation multiplication on group $G$. But then, why do we show an abelian group as $(G, +)$? ......Mon, 24 Aug 2026 19:10:42 -0400https://liberty.io/usage-of-the-word-formal-ly.htmlhttps://liberty.io/usage-of-the-word-formal-ly.htmlThis is weird. To me, in mathematical contexts, "formally" means something like "rigorously", i.e. the opposite of informally/heuristically. And yet, I very often read papers very the word seems t......Mon, 24 Aug 2026 18:59:47 -0400https://liberty.io/dfa-union-operation-how-to.htmlhttps://liberty.io/dfa-union-operation-how-to.htmlI'm currently looking at deterministic finite automata, and learning how to combine two DFAs using AND or OR. I think I understand how to construct the INTERSECTION (AND) of two DFAs, but I'm at a ......Mon, 24 Aug 2026 18:49:10 -0400https://liberty.io/which-of-the-following-statements-are-true-and-which-are-false.htmlhttps://liberty.io/which-of-the-following-statements-are-true-and-which-are-false.htmlLet $f$ and $g$ be continuous functions such that $f(x) ≤ g(x)$ for all $x ∈ [0, 1]$. Determine which of the following statements are true and which are false: $$ \begin{align} (a) & {}\quad \......Mon, 24 Aug 2026 18:44:39 -0400https://liberty.io/straightforward-logical-techniques-for-solving-matrices.htmlhttps://liberty.io/straightforward-logical-techniques-for-solving-matrices.htmlI'm studying for a Linear Algebra exam, and having difficulty with solving matrices in a timely manner. Our professor provided us with several sample exams to practice with, which begin with a matr......Mon, 24 Aug 2026 18:35:38 -0400https://liberty.io/determining-the-area-of-a-heart-curve-2d.htmlhttps://liberty.io/determining-the-area-of-a-heart-curve-2d.htmlI found out the area of a heart but I am not sure whether it is correct. Can someone please check whether what I have done makes any sense at all. The equation of a heart is shown below: I first ...Mon, 24 Aug 2026 18:27:39 -0400https://liberty.io/chromatic-number-of-a-graph-obtained-from-petersen-s-graph.htmlhttps://liberty.io/chromatic-number-of-a-graph-obtained-from-petersen-s-graph.htmlI am studying on Chromatic numbers of graphs and I came across Petersen’s graph. Petersen’s graph which is an undirected graph with 10 vertices and 15 edges. And it has the chromatic number 3, mea......Mon, 24 Aug 2026 18:16:21 -0400https://liberty.io/determining-the-derivative-of-e-5x-tan-2x.htmlhttps://liberty.io/determining-the-derivative-of-e-5x-tan-2x.htmlDetermine the derivative of $e^{5x}\tan(2x)$ I have no idea, what is this question mean- but what I can do is $$\begin{align*}y&=e^{5x}\\ \frac{dy}{dx}&=e^{5x}(5)\\ \frac{dy}{dx}&=5......Mon, 24 Aug 2026 18:07:38 -0400https://liberty.io/how-to-calculate-e-x-with-a-standard-calculator.htmlhttps://liberty.io/how-to-calculate-e-x-with-a-standard-calculator.htmlIs there a simple method for calculating the $e^x$ ($x\in\mathbb{R}$) with a basic add/subtract/multiply/divide calculator that converges in reasonable time, preferably without having to memorize ...Mon, 24 Aug 2026 18:05:46 -0400https://liberty.io/why-is-this-a-valid-proof-for-the-harmonic-series.htmlhttps://liberty.io/why-is-this-a-valid-proof-for-the-harmonic-series.htmlThe other day, I encountered the Harmonic series which I though is an interesting concept. There were many ways to prove that it's divergent and the one I really liked was rather simple but did the......Mon, 24 Aug 2026 18:03:57 -0400https://liberty.io/number-of-possible-permutations-of-three-pairs-of-socks-one-blue-one-b.htmlhttps://liberty.io/number-of-possible-permutations-of-three-pairs-of-socks-one-blue-one-b.htmlThere are 3 colours of socks; blue, black and white and each colour has two pairs of socks and we need to distribute them to six people and the left leg socks are distinguishable from right leg soc......Mon, 24 Aug 2026 17:59:09 -0400https://liberty.io/finding-the-major-and-minor-axis-vertices-for-an-ellipse-given-two-con.htmlhttps://liberty.io/finding-the-major-and-minor-axis-vertices-for-an-ellipse-given-two-con.htmlI've been googling, searching forums and looking in my old algebra/trig books to try to understand how to find the end points to the major and minor axis of an ellipse given the end points of two ...Mon, 24 Aug 2026 17:54:24 -0400https://liberty.io/how-to-calculate-this-limit-without-l-039-hopital-039-s-rule.htmlhttps://liberty.io/how-to-calculate-this-limit-without-l-039-hopital-039-s-rule.htmlI know that in the very small values of $x$ $$(1+x)^n ≈1+ nx$$ and I can prove it using Taylor series. But I wanted to prove it without any smell of derivative. So... In order to calculate the ...Mon, 24 Aug 2026 17:46:53 -0400https://liberty.io/principia-mathematica-and-axioms.htmlhttps://liberty.io/principia-mathematica-and-axioms.htmlI have been trying to understand how mathematicians achieve the truth through axiomatic systems. Surely, if one searches a bit more, it will get to know how axiomatic systems work, generally. After......Mon, 24 Aug 2026 17:41:49 -0400https://liberty.io/calculate-expected-value-in-dice-game.htmlhttps://liberty.io/calculate-expected-value-in-dice-game.htmlWe play a game in 2 stages: In stage one, we throw a dice until we get number 6, Let N represent the number of times played until we got 6 for the first time. In stage two, we throw N dices (each one ...Mon, 24 Aug 2026 17:37:42 -0400https://liberty.io/what-is-the-expected-area-of-a-polygon-whose-vertices-lie-on-a-circle.htmlhttps://liberty.io/what-is-the-expected-area-of-a-polygon-whose-vertices-lie-on-a-circle.htmlI came across a nice problem that I would like to share. Problem: What is expected value of the area of an $n$-gon whose vertices lie on a circle of radius $r$? The vertices are uniformly distribu......Mon, 24 Aug 2026 17:36:53 -0400https://liberty.io/differentiate-log-gamma-function.htmlhttps://liberty.io/differentiate-log-gamma-function.htmlI am working with the log of negative binomial distribution $NegBin(r,p)$. I need to differentiate the following with respect to $r$ such that at the end, I am NOT left with $r$ in factorial form. ......Mon, 24 Aug 2026 17:30:04 -0400https://liberty.io/writing-out-some-propositions-using-disjunctions-and-conjunctions.htmlhttps://liberty.io/writing-out-some-propositions-using-disjunctions-and-conjunctions.htmlSuppose that the universe of discourse of the atomic formula $P(x,y)$ is $\{1, 2, 3\}$. Write out the following propositions using disjunctions and conjunctions. a) $\exists x \forall y \ P(x,y)$ ...Mon, 24 Aug 2026 17:27:47 -0400https://liberty.io/calculate-the-sum-using-fourier-series-of-big-cos-frac-x-2-big.htmlhttps://liberty.io/calculate-the-sum-using-fourier-series-of-big-cos-frac-x-2-big.htmlI have been given task to evaluate the sum $\dfrac{(-1)^{n-1}}{4n^2-1}$ using Fourier series for function $|\cos{\frac{x}{2}}|$. I used the interval $(-\pi, +\pi)$ for evaluation of the sum and I n......Mon, 24 Aug 2026 17:19:17 -0400https://liberty.io/gradient-of-2-norm-squared.htmlhttps://liberty.io/gradient-of-2-norm-squared.htmlCould someone please provide a proof for why the gradient of the squared $2$-norm of $x$ is equal to $2x$? $$\nabla\|x\|_2^2 = 2x$$...Mon, 24 Aug 2026 17:02:39 -0400https://liberty.io/calculate-sin9-closed.htmlhttps://liberty.io/calculate-sin9-closed.htmlI found this question in the olympiad book. But I could not find the solution. The question is to calculate the following real number: $$\sin{9°}$$...Mon, 24 Aug 2026 16:55:08 -0400https://liberty.io/heaviside-step-function-squared.htmlhttps://liberty.io/heaviside-step-function-squared.htmlI have a question about the Heaviside step function $\theta(\xi)$, defined by $$\theta(\xi):=\begin{cases} 1, & \xi\geq0\\ 0, & \xi<0 \end{cases}$$ I need to evaluate the square of the ...Mon, 24 Aug 2026 16:47:44 -0400https://liberty.io/sard-039-s-theorem-for-algebraic-varieties.htmlhttps://liberty.io/sard-039-s-theorem-for-algebraic-varieties.html(One version of) Sard's theorem states that: Theorem (Sard): Given $M$ and $N$ smooth manifolds of dimensions $m$ and $n$ respectively, and a smooth map $f:M\to N$, then the set of singular valu......Mon, 24 Aug 2026 16:40:48 -0400https://liberty.io/the-equation-of-a-line-reflected-about-another-line.htmlhttps://liberty.io/the-equation-of-a-line-reflected-about-another-line.htmlI need to find the equation of this reflected line:...Mon, 24 Aug 2026 16:35:39 -0400https://liberty.io/find-the-parameters-for-which-the-matrix.htmlhttps://liberty.io/find-the-parameters-for-which-the-matrix.html$\left(\begin{matrix} 1 & 2 & -1 \\ 3a & 2 & b \\ 1 & 2& 1 \end{matrix}\right)$ For example I have such a matrix. Can you, please explain how can I find for what values of ......Mon, 24 Aug 2026 16:34:26 -0400https://liberty.io/what-does-it-mean-when-all-the-values-of-a-row-in-a-matrix-are-0.htmlhttps://liberty.io/what-does-it-mean-when-all-the-values-of-a-row-in-a-matrix-are-0.htmlI'm fairly new to linear algebra and I'm trying to make sense of what I'm being taught in class. I'm a little confused as to what happens when all the values of a matrix's row are equal to $0$: doe......Mon, 24 Aug 2026 16:33:59 -0400https://liberty.io/questions-tagged-ramanujan-summation.htmlhttps://liberty.io/questions-tagged-ramanujan-summation.htmlQ&A for people studying math at any level and professionals in related fields...Mon, 24 Aug 2026 16:14:19 -0400https://liberty.io/fixed-point-iteration-method-with-parameters.htmlhttps://liberty.io/fixed-point-iteration-method-with-parameters.htmlWe want to approach the number $\alpha =\sqrt[3]{2}$. The function $f(x) = x^{3} - 2$ has $\alpha$ as a root. Now take a function $g$ so that $\alpha$ is a fixed point, $g(\alpha) = \alpha$. Use $ ......Mon, 24 Aug 2026 16:08:52 -0400https://liberty.io/what-is-the-peano-definition-of-subtraction.htmlhttps://liberty.io/what-is-the-peano-definition-of-subtraction.htmlI came up with this: (S.1) $a - a = 0$ (S.2) $a - b = S(a - S(b))$ This seems to work. At least for $a$$\ge$$b$. Is this the correct or most efficient formulation? Also, does there happen to b......Mon, 24 Aug 2026 16:00:26 -0400https://liberty.io/law-of-iterated-expectations-example.htmlhttps://liberty.io/law-of-iterated-expectations-example.htmlI have a question that I hope can be shown using the LIE. There is a urn which contains 3 marbles (2 white and 1 black). The person who gets to pick the black marble gets to win $100 whereas the ...Mon, 24 Aug 2026 15:41:45 -0400https://liberty.io/why-does-there-seem-to-be-so-much-error-in-the-laws-of-sines-and-cosin.htmlhttps://liberty.io/why-does-there-seem-to-be-so-much-error-in-the-laws-of-sines-and-cosin.htmlI've been computing the angles of a triangle with sides a = 17, b = 6 and c = 15 using the law of cosines to find the first angle and then the law of sines to find the other 2. I follow the convent......Mon, 24 Aug 2026 15:15:09 -0400https://liberty.io/how-should-i-write-down-the-alternating-group-a-3-closed.htmlhttps://liberty.io/how-should-i-write-down-the-alternating-group-a-3-closed.htmlI didn't understand how to write down the alternating group A3. Is this the group consisting of only the even permutations? Also, what familiar group is this isomorphic to?...Mon, 24 Aug 2026 15:14:13 -0400https://liberty.io/recommended-maths-book-for-beginner-to-study-in-computer-science-close.htmlhttps://liberty.io/recommended-maths-book-for-beginner-to-study-in-computer-science-close.htmlI am going to study computer science next year. I am afraid I can't handle the mathematics in the university because I only know some basic mathematics, such as set theory, simple probability, simple ...Mon, 24 Aug 2026 15:03:45 -0400https://liberty.io/ti-83-plus-is-it-possible-to-find-the-eigenvalues-on-this-calculator.htmlhttps://liberty.io/ti-83-plus-is-it-possible-to-find-the-eigenvalues-on-this-calculator.htmlI searched the Internet and can't seem to find a source on how to find the eigenvalues on a TI-83 Plus calculator. Can anyone give me a hand?...Mon, 24 Aug 2026 15:00:16 -0400https://liberty.io/rules-for-modulus-and-multiplication.htmlhttps://liberty.io/rules-for-modulus-and-multiplication.htmlMy question is rather simple as I'm interested about modulus and multiplication, specifically whether it holds that $(a*b)\,mod\,n=(a\,mod\,n)*(b\,mod\,n)$?...Mon, 24 Aug 2026 14:57:39 -0400https://liberty.io/what-does-test-function-mean.htmlhttps://liberty.io/what-does-test-function-mean.htmlI am trying to learn weak derivatives. In that, we call $\mathbb{C}^{\infty}_{c}$ functions as test functions and we use these functions in weak derivatives. I want to understand why these are called ...Mon, 24 Aug 2026 14:54:43 -0400https://liberty.io/derive-fourier-transform-for-summation-of-shifted-dirac-delta-function.htmlhttps://liberty.io/derive-fourier-transform-for-summation-of-shifted-dirac-delta-function.htmlHow to derive this Fourier transform: $$ F\{ \sum_{n=-\infty}^{\infty} \delta (t- nT) \text{ }\} =\omega_o \sum_{n=-\infty}^{\infty} \delta (\omega - n \omega_o)$$ where: $$ \omega_o = 2 \pi / T $$...Mon, 24 Aug 2026 14:43:56 -0400https://liberty.io/proof-for-how-9999999-1-duplicate.htmlhttps://liberty.io/proof-for-how-9999999-1-duplicate.htmlI was just curious for the proof/theorem for a very close decimal (so when you keep adding a decimal) to equal the next integer such as .999999999999....=1. Thank you....Mon, 24 Aug 2026 14:43:23 -0400https://liberty.io/alternating-series-determine-if-it-converges-absolutely-conditionally-.htmlhttps://liberty.io/alternating-series-determine-if-it-converges-absolutely-conditionally-.htmlI am trying to determine to whether the series $\sum _{n=0}^{\infty }\:\left(-1\right)^n\frac{1}{\sqrt{n}\left(ln\left(n\right)^{2021}\right)}$ is conditionally convergent, absolutely convergent, or ...Mon, 24 Aug 2026 14:25:19 -0400https://liberty.io/intuitive-understanding-of-the-first-isomorphism-theorem.htmlhttps://liberty.io/intuitive-understanding-of-the-first-isomorphism-theorem.htmlI've been reading some things about an intuitive understanding of the first isomorphism theorem, and there's just one more part that I do not understand. Specifically, the first isomorphism theorem ...Mon, 24 Aug 2026 14:07:51 -0400https://liberty.io/what-does-formal-mean.htmlhttps://liberty.io/what-does-formal-mean.htmlI know the definition of formal power series, power series and polynomials. But what does the adjective "formal" mean? In google English dictionary, does it mean "9. Of or relating to linguistic or ...Mon, 24 Aug 2026 13:55:16 -0400https://liberty.io/convert-complex-number-to-polar-coordinates-round-2.htmlhttps://liberty.io/convert-complex-number-to-polar-coordinates-round-2.htmlThis is follow up problem to this. I post new question related to previous question since all of the problem were not solved. I have new questions / problems that came after i closed the previous p......Mon, 24 Aug 2026 13:28:25 -0400https://liberty.io/what-is-2-1-1-duplicate.htmlhttps://liberty.io/what-is-2-1-1-duplicate.html$2-1+1$; a fairly straightforward question, but I (well, not me, but Henry Reich) found something strange. Most people would evaluate it as $2+(-1)+1 = 2$; however, this goes against the famed, and ...Mon, 24 Aug 2026 13:27:25 -0400https://liberty.io/polar-form-modulus-argument.htmlhttps://liberty.io/polar-form-modulus-argument.htmlQuestions: a) Express $-4i$ in polar form b) Hence solve the equation $z^2 =-4i$ I've attempted the question and was only able to find $z=4$ using the form $r(\cos x+i\sin x)=z$. Could someone ...Mon, 24 Aug 2026 13:14:46 -0400https://liberty.io/what-is-the-structural-hierarchy-in-mathematics.htmlhttps://liberty.io/what-is-the-structural-hierarchy-in-mathematics.htmlBy structural hierarchy, I mean the mental concept in which things are 'done' in mathematics. At the top, you have mathematics itself, which is a collection of systems, like arithmetic, algebra, ge......Mon, 24 Aug 2026 13:07:54 -0400https://liberty.io/why-is-slope-rise-run.htmlhttps://liberty.io/why-is-slope-rise-run.htmlWhat makes slope rise over run? What makes the standard equation for a line use a slope of rise over run as opposed to run over rise? What would the standard equation of a line look like if m was ...Mon, 24 Aug 2026 13:06:28 -0400https://liberty.io/square-root-of-x-3.htmlhttps://liberty.io/square-root-of-x-3.htmlI understand the concept behind the expression $\sqrt{x^2} = |x|$. So, then why is the square root of $x^3$ NOT equal to $|x|\sqrt{x}$? Specifically, I can write $\sqrt{x^3}$ as $\sqrt{x^2\time......Mon, 24 Aug 2026 13:02:05 -0400https://liberty.io/prove-by-induction-involving-calculus-closed.htmlhttps://liberty.io/prove-by-induction-involving-calculus-closed.htmlProve by induction. Let $$f(x)={ e }^{ { x }^{ 2 } }$$ Then for every $n\in \mathbb{N}$ there exists a polynomial ${ P }_{ n }(x)$ of degree $n$ such that $$\frac { { d }^{ n } }{ { dx }^{ n } }......Mon, 24 Aug 2026 12:57:40 -0400https://liberty.io/prove-that-there-are-infinitely-many-n-in-mathbb-n-such-that-6n-1-6n-1.htmlhttps://liberty.io/prove-that-there-are-infinitely-many-n-in-mathbb-n-such-that-6n-1-6n-1.htmlLet m be a integer. Then if $6n+1$ is a composite number we have that $\operatorname{lcd}(6n+1,m)$ is not just $1$, because then $6n+1$ would be prime. Also this is for $6n-1$. Now I must find all ......Mon, 24 Aug 2026 12:55:38 -0400https://liberty.io/super-perfect-numbers.htmlhttps://liberty.io/super-perfect-numbers.htmlA super-perfect number is a number with $\sigma(\sigma (n))=2n$. How can I prove that every even super perfect number is from the form $n=2^k$ when $2^{k+1}-1$ is prime. I tried every way please ......Mon, 24 Aug 2026 12:55:33 -0400https://liberty.io/what-really-is-an-indeterminate-form.htmlhttps://liberty.io/what-really-is-an-indeterminate-form.htmlWe can apply l’Hôpital’s Rule to the indeterminate quotients $ \dfrac{0}{0} $ and $ \dfrac{\infty}{\infty} $, but why can’t we directly apply it to the indeterminate difference $ \infty - \infty $ ......Mon, 24 Aug 2026 12:55:18 -0400https://liberty.io/how-to-parameterize-an-ellipse.htmlhttps://liberty.io/how-to-parameterize-an-ellipse.htmlI need to parameterize the ellipse $\frac{x^2}{2}+y^2=2$, so this is how I proceed: I know that $a=2$ and $b=1$ (where $a$ and $b$ are the axis of the ellipse), so I parameterize as: \begin{cases}......Mon, 24 Aug 2026 12:55:07 -0400https://liberty.io/prove-that-a-planar-graph-has-four-coloring.htmlhttps://liberty.io/prove-that-a-planar-graph-has-four-coloring.htmlThere is a theorem which says that every planar graph can be colored with five colors. It can also be colored with four colors. How can I prove that any planar graph with max degree of $4$, has a f......Mon, 24 Aug 2026 12:51:12 -0400https://liberty.io/how-many-non-isomorphic-trees-with-four-vertices-are-there-what-are-th.htmlhttps://liberty.io/how-many-non-isomorphic-trees-with-four-vertices-are-there-what-are-th.htmlAs of now I have 4 assuming I got this right. I apologize for formatting because I have no idea how I'd display this in text but here I go. a b c d are my points. - means there is a connection betw......Mon, 24 Aug 2026 12:48:10 -0400https://liberty.io/circle-equation-transformations.htmlhttps://liberty.io/circle-equation-transformations.htmlIf we have a circle of the form $x^2 + y^2 = 4$ it has the middle point $(0,0)$ and radius $2$. If we want to move it $5$ to the right, the equation becomes $(x-5)^2 + y^2 = 4.$ What happens if we ......Mon, 24 Aug 2026 12:45:46 -0400https://liberty.io/what-is-the-difference-between-a-limit-and-derivative.htmlhttps://liberty.io/what-is-the-difference-between-a-limit-and-derivative.htmlWhen you get the derivative of a point, arn't you just getting the limit at that point? I'm not quite sure why they need to be named differently when they seem to be doing the same thing....Mon, 24 Aug 2026 12:40:56 -0400https://liberty.io/how-to-determine-if-a-given-function-is-a-valid-cdf-pmf-or-pdf.htmlhttps://liberty.io/how-to-determine-if-a-given-function-is-a-valid-cdf-pmf-or-pdf.htmlStudying for a statistics exam. I have come across this problem: and it presents to me some important and extremely basic questions (I have a LONG way to go before I'm prepared for this exam). Can ...Mon, 24 Aug 2026 12:25:21 -0400https://liberty.io/how-to-get-polynomial-equation-from-points.htmlhttps://liberty.io/how-to-get-polynomial-equation-from-points.htmlHow can i get polynomial equation from points? I have data in excel and i want to get polynomial equation from it. I can get polynomial equation using excel using trendline but only to order of 6. I ...Mon, 24 Aug 2026 12:22:37 -0400https://liberty.io/how-to-draw-a-graph-f-x-y-in-x-y-plane.htmlhttps://liberty.io/how-to-draw-a-graph-f-x-y-in-x-y-plane.htmlIs there possible way to draw function $\vec{f}: R^2 \to R^2 $ such as like $\vec{f}(x,y) = \sin x+ \sin y, y+\sin x\ $ and other $\vec{f}(x,y)$ in $xy$-corordinate instead of $xyz$ plane. Is it o......Mon, 24 Aug 2026 12:20:55 -0400https://liberty.io/incomparable-elements-in-a-poset.htmlhttps://liberty.io/incomparable-elements-in-a-poset.htmlThe problem I am working on is, Find two incomparable elements in these posets. a) $(P(\{0,1,2\}),⊆)$ b) $(\{1,2,4,6,8\},|)$ For a, I said that $R \subseteq p(\{0,1,2,3\}) \times p(\{0,1,2,3\})$, ...Mon, 24 Aug 2026 12:07:26 -0400https://liberty.io/factoring-a-4th-degree-trinomial.htmlhttps://liberty.io/factoring-a-4th-degree-trinomial.htmlI am trying to factor $3x^4-8x^3+16$, but I have no idea how to even start. I put into Wolfram Alpha, and it said that the answer was $(x-2)^2 (3 x^2+4 x+4)$. How would you factor something like th......Mon, 24 Aug 2026 11:55:26 -0400https://liberty.io/what-do-free-variable-and-leading-variables-mean.htmlhttps://liberty.io/what-do-free-variable-and-leading-variables-mean.htmlWhat do the leading variables and free variables in a matrix mean? I have the system below and am trying to understand which are which. I searched a lot for this, please help me ! $$w + x + y + z =......Mon, 24 Aug 2026 11:55:13 -0400https://liberty.io/subtraction-by-addition.htmlhttps://liberty.io/subtraction-by-addition.htmlCan anyone explain to me "subtraction by addition" in a visual way? The steps say: Take the "complement" of the number we are subtracting Add it to to the number we are subtracting from Disc......Mon, 24 Aug 2026 11:54:39 -0400https://liberty.io/verifying-trigonometric-identities-2-cos-2x-1-frac-1-tan-2x-1-tan-2x.htmlhttps://liberty.io/verifying-trigonometric-identities-2-cos-2x-1-frac-1-tan-2x-1-tan-2x.htmlVerify that $$ 2\cos^2x-1 = \frac{1-\tan^2x}{1+\tan^2x}$$...Mon, 24 Aug 2026 11:50:14 -0400https://liberty.io/angles-in-rhombus-when-an-equilateral-triangle-is-inscribed-in-it-clos.htmlhttps://liberty.io/angles-in-rhombus-when-an-equilateral-triangle-is-inscribed-in-it-clos.htmlWhen one inscribs an equilateral triangle in a rhombus, all the corners are multiples of 30 degrees. I can see this, but I can't proof it. Question: How can I proof that the angle ADC is 120 degr......Mon, 24 Aug 2026 11:43:06 -0400https://liberty.io/proving-3-n-geq-3n-using-mathematical-induction.htmlhttps://liberty.io/proving-3-n-geq-3n-using-mathematical-induction.htmlSo I have to prove that $3^n$ is greater than or equal to $3n$ using induction. The base case is a not a problem, but I can't seem to figure out where to go for $(n-1)$. I've tried saying: $$3^n=3\......Mon, 24 Aug 2026 11:27:52 -0400https://liberty.io/example-of-hypergraph.htmlhttps://liberty.io/example-of-hypergraph.htmlDefinition: A hypergraph $\Gamma=(V,\mathcal{E})$ is a set of vertices $V$ and a collection $\mathcal{E}$ of subsets of $V$ such that for every $E\in \mathcal{E}$, we have $|E|\geq 2$. The members ......Mon, 24 Aug 2026 11:26:37 -0400https://liberty.io/identifying-self-intersection-points-in-one-parametric-graph.htmlhttps://liberty.io/identifying-self-intersection-points-in-one-parametric-graph.htmlMy question for you is how to identify self-intersection points in a parametric curve of the form x = f(t), y = g(t). The specific problem asks for the t values of the intersection where $x = \sin(......Mon, 24 Aug 2026 11:25:45 -0400https://liberty.io/generalized-inverse-of-transpose-closed.htmlhttps://liberty.io/generalized-inverse-of-transpose-closed.htmlHow can we show that if $G$ is generalized Inverse of $A$ , then $G^T$ is generalized Inverse of $A^T$...Mon, 24 Aug 2026 11:09:46 -0400https://liberty.io/notion-of-cusp-in-algebraic-geometry.htmlhttps://liberty.io/notion-of-cusp-in-algebraic-geometry.htmlThis relates to problem 3.14(b) of Hartshorne, wherein I am required to show that the projection of the twisted cubic curve (viewed as a subset of $\mathbb P^3$) from the point $(0,0,1,0)$ onto the ...Mon, 24 Aug 2026 10:56:58 -0400https://liberty.io/write-csc-x-in-terms-of-sec-x.htmlhttps://liberty.io/write-csc-x-in-terms-of-sec-x.htmlI was working on trig homework and came across a question that I didn't understand how to even begin to approach. It asked us to use trigonometric identities to write $\csc(x)$ in terms of $\sec(x......Mon, 24 Aug 2026 10:52:08 -0400https://liberty.io/the-composite-function-limit-theorem-proof.htmlhttps://liberty.io/the-composite-function-limit-theorem-proof.htmlLet us define two functions:- $$f(x)$$ $$g(x)$$ And now let us try to find the limit :- $$\lim_{x \to c} f(g(x))$$ Assuming that all the limits mentioned below exist we can say:- $$ \lim_{x \to c}g......Mon, 24 Aug 2026 10:42:54 -0400https://liberty.io/a-quadratic-trinomial-that-generates-only-prime-numbers-of-the-form-4m.htmlhttps://liberty.io/a-quadratic-trinomial-that-generates-only-prime-numbers-of-the-form-4m.htmlIt is known that Euler's polynomials $\,n^2+n+p\,$ ($p\,$ prime) represent a prime for $\,n=0,\,...,\,p-2\,$ if and only if the field $\,Q (\sqrt{1-4p})\,$ has class number $\,h=1$. The best trino......Mon, 24 Aug 2026 10:42:23 -0400https://liberty.io/probability-to-guess-at-least-one-answer-correctly.htmlhttps://liberty.io/probability-to-guess-at-least-one-answer-correctly.htmlA multiple choice test consists of three problems. For each problem, there are five choices , one of which is correct .One student comes totally unprepared and decides to answer by sheer guessing .......Mon, 24 Aug 2026 10:39:44 -0400https://liberty.io/why-do-we-need-span-in-linear-algebra.htmlhttps://liberty.io/why-do-we-need-span-in-linear-algebra.htmlIn my linear algebra course in university we started learning about span and I was curious what is it good for? and if someone know, how does it relate to 3D graphics? Thank you....Mon, 24 Aug 2026 10:38:48 -0400https://liberty.io/logarithms-with-a-fraction-as-base.htmlhttps://liberty.io/logarithms-with-a-fraction-as-base.htmlHow does one solve a logarithmic expression where the base is a fraction? In my example I am trying to solve the following: $$ n^{\log_\frac{3}{2}(1)} \tag{1} $$ This is related to using the "ma......Mon, 24 Aug 2026 10:36:58 -0400https://liberty.io/find-height-of-irregular-trapezoid-with-known-angles-and-surface-area.htmlhttps://liberty.io/find-height-of-irregular-trapezoid-with-known-angles-and-surface-area.htmlKNOWN: Length DC Alpha Beta Surface S NEEDED: Height h For an algorithm, I require a way to solve this for any trapezoid. Sort of like this question (Given a known isosceles Trapezoid find heig......Mon, 24 Aug 2026 10:36:44 -0400https://liberty.io/write-an-equation-as-a-single-power-grade-11-math-function.htmlhttps://liberty.io/write-an-equation-as-a-single-power-grade-11-math-function.html$$\frac{10^{-4/5} \cdot 10^{1/15}}{10^{2/3}}$$ The answer is $10^{-7/5}$, which seems impossible to me. I get: $10^{-4/5} \cdot 10^{-11/15}$. I see where the numerator $7$ comes from but the ...Mon, 24 Aug 2026 10:33:14 -0400https://liberty.io/what-will-be-the-formula-for-trace-t.htmlhttps://liberty.io/what-will-be-the-formula-for-trace-t.htmlLet $M_n(K)$ denote the space of all $n\times n$ matrices with entries in a field $K$. Fix a non-singular matrix $A=(A_{ij})\in M_n(K)$, and consider the linear map $T:M_n(K)\to M_n(K)$ given by: $......Mon, 24 Aug 2026 10:26:22 -0400https://liberty.io/interchange-of-expected-value-and-summation.htmlhttps://liberty.io/interchange-of-expected-value-and-summation.htmlHey guys I'm studying Statistics and I'm currently at unbiased estimators. I'm trying to find the estimators for different problems. My question is : I know that the following formula holds $$E (\......Mon, 24 Aug 2026 10:25:54 -0400https://liberty.io/what-is-the-mathematical-symbol-for-the-sum-of-numbers.htmlhttps://liberty.io/what-is-the-mathematical-symbol-for-the-sum-of-numbers.htmlFor example, when $n=5$, what is the symbol for $5+4+3+2+1$?...Mon, 24 Aug 2026 10:13:14 -0400https://liberty.io/long-division-differs-from-calculator.htmlhttps://liberty.io/long-division-differs-from-calculator.htmli have been helping my daughter with her long division homework. One of the questions was 172/6 she worked out the answer using the long division tree, and upon checking it found the answer differ......Mon, 24 Aug 2026 10:08:50 -0400https://liberty.io/what-is-so-special-about-the-special-orthogonal-group.htmlhttps://liberty.io/what-is-so-special-about-the-special-orthogonal-group.htmlI am trying to learn a bit about (geometrical Euclidean) group theory and am now wondering "Wat is so special about the Special Orthogonal group?" Or what is the difference between the groups $O(......Mon, 24 Aug 2026 10:04:04 -0400https://liberty.io/clopen-subsets-of-the-cantor-set.htmlhttps://liberty.io/clopen-subsets-of-the-cantor-set.htmlIn our Topology course, we have been studying the anti-equivalance of categories between zero-dimensional compact Hausdorff spaces and Boolean algebras. The Cantor set has come up a lot, and I hav......Mon, 24 Aug 2026 10:00:05 -0400https://liberty.io/what-are-some-good-rigorous-precalculus-textbooks-for-self-study.htmlhttps://liberty.io/what-are-some-good-rigorous-precalculus-textbooks-for-self-study.htmlBooks that: has an introduction to proof, logic, and topics like sets and groups. Books can prepare you for rigorous calculus texts like Spivak and Apostol. Question 1. I've been looking at the......Mon, 24 Aug 2026 09:58:26 -0400https://liberty.io/limit-of-1-x-as-x-approaches-infinity-duplicate.htmlhttps://liberty.io/limit-of-1-x-as-x-approaches-infinity-duplicate.htmlWhy is $\lim_{x\to\infty} \frac{1}{x}$ equal to $0$?...Mon, 24 Aug 2026 09:52:31 -0400https://liberty.io/volume-of-an-ellipsoid-using-cylindrical-polar-coordinates.htmlhttps://liberty.io/volume-of-an-ellipsoid-using-cylindrical-polar-coordinates.htmlI've been working on a question about finding the volume of an ellipsoid $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1.$$ This is fine if I consider rescaling the axes to give a spher......Mon, 24 Aug 2026 09:50:06 -0400https://liberty.io/suppose-t-is-a-linear-transformation-such-that-t-1-1-1-0-1-2-t-1-0-1-1.htmlhttps://liberty.io/suppose-t-is-a-linear-transformation-such-that-t-1-1-1-0-1-2-t-1-0-1-1.htmlWhat is $T(x,y,z)$ What I tried: $$T(x,y,z)= \begin{pmatrix} 1&1&1|0&1&2\\ 1&0&1|1&1&1\\ 0&0&1|1&2&3\\ \end{pmatrix} $$ Which reduces to \begin{pm......Mon, 24 Aug 2026 09:42:11 -0400https://liberty.io/approximating-logs-and-antilogs-by-hand.htmlhttps://liberty.io/approximating-logs-and-antilogs-by-hand.htmlI have read through questions like Calculate logarithms by hand and and a section of the Feynman Lecture series which talks about calculation of logarithms. I have recognized neither of them useful......Mon, 24 Aug 2026 09:41:22 -0400https://liberty.io/what-is-the-definition-of-double-sequence-a-mn-being-convergent-to-l.htmlhttps://liberty.io/what-is-the-definition-of-double-sequence-a-mn-being-convergent-to-l.htmlWhat is the definition of double sequence $a_{mn}$ being convergent to $l$? I have this definition. Definition: The double sequence $(a_{m,n})^∞_{m,n=1}$ is said to Converge to the real number $......Mon, 24 Aug 2026 09:39:15 -0400https://liberty.io/what-is-the-column-space-in-a-5-times-5-invertible-matrix.htmlhttps://liberty.io/what-is-the-column-space-in-a-5-times-5-invertible-matrix.htmlIf $A$ is any $5 \times 5$ invertible matrix, then what is its column space? Why? I'm totally lost with column space. Any ideas?...Mon, 24 Aug 2026 09:36:49 -0400https://liberty.io/oriented-atlas-on-a-circle.htmlhttps://liberty.io/oriented-atlas-on-a-circle.htmlI'm trying to find an oriented atlas on the circle $S^1$, i.e., I want to find an atlas for $S^1$ such that for any two overlapping charts $(U,s)$ and $(V,t)$ of the atlas, the derivative $d s/d......Mon, 24 Aug 2026 09:35:58 -0400https://liberty.io/dual-of-h-1-0-h-1-or-h-0-1.htmlhttps://liberty.io/dual-of-h-1-0-h-1-or-h-0-1.htmlI have a problem related to dual of Sobolev space $H^1_0$. By definition, the dual of $H^1_0$ is $H^{-1}$, which contains $L^2$ as a subspace. However, from Riesz representation theorem, dual of a ...Mon, 24 Aug 2026 09:28:45 -0400https://liberty.io/is-this-the-correct-way-to-say-an-integral.htmlhttps://liberty.io/is-this-the-correct-way-to-say-an-integral.htmlFor this structure: Is the correct way to verbally say this (such that a screen reader would read it aloud correctly), "The integral over the interval from negative infinity to positive infinity o......Mon, 24 Aug 2026 09:27:12 -0400https://liberty.io/partial-fractions-proper-fraction-with-a-perfect-square-factor.htmlhttps://liberty.io/partial-fractions-proper-fraction-with-a-perfect-square-factor.htmlI think I have figured why the following is the case, I need however a confirmation that it is indeed so. Consider: $\displaystyle{\frac{x-1}{(x+1)(x-2)^2} \equiv \frac{A}{x+1} + \frac{Bx+C}{(x-2......Mon, 24 Aug 2026 09:25:57 -0400https://liberty.io/domain-and-range-of-a-multivariable-function.htmlhttps://liberty.io/domain-and-range-of-a-multivariable-function.htmlI have this exercise $$z={2x\over y+5}$$, and I am supposed to obtain the domain and range. I understand that the domain is all the pair of $(x,y)$ except $y=-5$ , then the exercise said that the r......Mon, 24 Aug 2026 09:25:43 -0400