https://liberty.ioCurated star highlights with fresh energy.en-usThu, 27 Aug 2026 19:35:40 -0400https://liberty.io/finding-the-length-of-a-complex-vector-duplicate.htmlhttps://liberty.io/finding-the-length-of-a-complex-vector-duplicate.htmlLet us say the vector $u = (1 - i, 2 + 3i, 5)$. How would I find $\|u\|$? I have tried solving this through: $\sqrt{(1-i)^2 + (2+3i)^2 + 5^2}$ But I ended up needing to find the square root of a ...Thu, 27 Aug 2026 23:23:41 -0400https://liberty.io/calculating-probabilities-over-longer-period-of-time.htmlhttps://liberty.io/calculating-probabilities-over-longer-period-of-time.htmlThere's a great question/answer at: Calculating probabilities over different time intervals This is an awesome answer, but I'd like to ask a related question: What if the period goes the other dire......Thu, 27 Aug 2026 23:16:34 -0400https://liberty.io/proving-int-csc-2-x-dx.htmlhttps://liberty.io/proving-int-csc-2-x-dx.htmlAccording to the standard integral results, it is known that $\int \csc^2 x \, dx = -\cot x$ since $\frac{d}{dx} \cot x = - \csc^2 x$. However, supposing that this is not known, is there a better m......Thu, 27 Aug 2026 23:16:30 -0400https://liberty.io/how-to-excel-at-math.htmlhttps://liberty.io/how-to-excel-at-math.htmlI know this question has been asked at least a hundred times earlier but before you mark this as duplicate please read this cause I seriously need some professional help here. I'm a 15 year old ki......Thu, 27 Aug 2026 23:15:57 -0400https://liberty.io/find-x-as-function-of-y.htmlhttps://liberty.io/find-x-as-function-of-y.htmlI am trying to find $y$ as function of $x$ of: $$y=\frac{x-1}{x+1}$$ I have tried $$y=\frac{x-1}{x+1}\iff (x+1)y=x-1\iff(x+1)y+1=x$$ Which is of course wrong...Thu, 27 Aug 2026 23:15:55 -0400https://liberty.io/what-is-an-acyclic-connected-graph-in-graph-theory.htmlhttps://liberty.io/what-is-an-acyclic-connected-graph-in-graph-theory.htmlI want to know What it is and whether there is a difference in the definition when looking at undirected and directed graphs?...Thu, 27 Aug 2026 23:15:13 -0400https://liberty.io/about-the-definition-of-axiomatizable-theory-and-consistency.htmlhttps://liberty.io/about-the-definition-of-axiomatizable-theory-and-consistency.htmlDefinition: If $A$ is a theory and $B \subseteq A$ then $B$ is a set of axioms for $A$ iff 1) B is recursive and 2) $B \models C$ for all $C \in A$. We say $A$ is axiomatizable iff $A$ has a se......Thu, 27 Aug 2026 23:02:53 -0400https://liberty.io/trapezoidal-pyramid.htmlhttps://liberty.io/trapezoidal-pyramid.htmlIs the following problem correct? If it is correct, please present a solution. Given a trapezoidal pyramid ABCDM. The base $ABCD$ is trapezoid (where $AB\; ||\; CD$). The planes $ADM$ and $BCM$ ......Thu, 27 Aug 2026 22:52:04 -0400https://liberty.io/why-is-the-eccentricity-of-a-parabola-1.htmlhttps://liberty.io/why-is-the-eccentricity-of-a-parabola-1.htmlWhat I know: The standard form for a parabola is $(x-h)^2=4a(y-k)$, if the axis of symmetry of the parabola is vertical, or $(y-k)^2=4a(x-h)$, if the axis of symmetry of the parabola is horizontal......Thu, 27 Aug 2026 22:46:08 -0400https://liberty.io/what-is-division-sum-closed.htmlhttps://liberty.io/what-is-division-sum-closed.htmlWhat is the meaning of "division sum". I have seen the being used a lot but haven't found an explanation defining the term. Understanding from its usage, it seems to be another term for '...Thu, 27 Aug 2026 22:34:59 -0400https://liberty.io/suppose-that-x-and-y-are-irrational-but-x-y-is-rational-prove-that-x-y.htmlhttps://liberty.io/suppose-that-x-and-y-are-irrational-but-x-y-is-rational-prove-that-x-y.htmli was wondering if someone could check my proof $Q= \{a/b , c,d : a,c ∈ \mathbb Z , b,d ∈ N>0\}$ $a/b =x+y$ $a/b -y=x$ proof by contradiction. Let $x-y$ is rational $c/d = x-y$ sub $a/b -y......Thu, 27 Aug 2026 22:32:42 -0400https://liberty.io/find-the-constant-term-of-polynomial.htmlhttps://liberty.io/find-the-constant-term-of-polynomial.htmlThere's fifth degree polynomial, it's first coefficient equals $-7$. $$-7x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0$$ Also: $$W(1)=-2$$ $$W(2)=-4$$ $$W(3)=-6$$ $$W(4)=-8$$ $$W(5)=-10$$ Find the value of ...Thu, 27 Aug 2026 22:30:03 -0400https://liberty.io/power-method-matlab.htmlhttps://liberty.io/power-method-matlab.htmlI am trying to find the largest eigenvalue and its corresponding eigenvector by the Power method .. I did this code and I just want to display the largest eigenvalue and its eigenvector after ......Thu, 27 Aug 2026 22:22:50 -0400https://liberty.io/what-is-the-difference-between-a-linear-n-k-code-and-a-linear-n-k-d-co.htmlhttps://liberty.io/what-is-the-difference-between-a-linear-n-k-code-and-a-linear-n-k-d-co.htmlIn my lecture notes on coding theory, I have the following 2 definitions: Definition 1: Definition 2: Where: $n$ is the number of symbols in each codeword $k$ is the dimension of the vector space ...Thu, 27 Aug 2026 22:18:50 -0400https://liberty.io/for-any-real-function-if-f-0-0-then-f-039-0-0-closed.htmlhttps://liberty.io/for-any-real-function-if-f-0-0-then-f-039-0-0-closed.htmlThis seems a little too trivial but I have been facing many problems in my intro to real analysis books about functions who at first are discontinuous at zero but then "the hole is plugged" How co......Thu, 27 Aug 2026 22:06:00 -0400https://liberty.io/what-is-the-opposite-of-logarithmic-scale.htmlhttps://liberty.io/what-is-the-opposite-of-logarithmic-scale.htmlIf one wants to plot a variable whose value changes drastically, it makes sense to plot its order of magnitude. That way, the smaller values in the plot aren't squished all the way at the bottom. T......Thu, 27 Aug 2026 22:03:06 -0400https://liberty.io/the-class-of-all-classes-not-containing-themselves.htmlhttps://liberty.io/the-class-of-all-classes-not-containing-themselves.htmlIn ZF classes are used informally to resolve Russells Paradox, that is the collection of all sets that do not contain themselves does not form a set but a proper class. But doesn't the same paradox ...Thu, 27 Aug 2026 21:49:46 -0400https://liberty.io/mass-of-wire-using-line-integral.htmlhttps://liberty.io/mass-of-wire-using-line-integral.htmlIf $\rho(x, y)$ is the density of a wire (mass per unit length), then $m = \int \rho(x,y)ds$ is the mass of the wire. Find the mass of a wire having the shape of a semicircle $x = 1 + \cos(t), y = ......Thu, 27 Aug 2026 21:43:27 -0400https://liberty.io/what-is-the-mathematical-symbol-representation-for-the-remainder-opera.htmlhttps://liberty.io/what-is-the-mathematical-symbol-representation-for-the-remainder-opera.htmlIs there an equivalent in mathematical language to the modulo (or mod) function in computing?...Thu, 27 Aug 2026 21:34:29 -0400https://liberty.io/prove-the-riemann-lebesgue-lemma-int-b-af-x-cos-nx-dx-rightarrow-0-as-.htmlhttps://liberty.io/prove-the-riemann-lebesgue-lemma-int-b-af-x-cos-nx-dx-rightarrow-0-as-.htmlI would like to prove the Riemann-Lebesgue lemma, namely that $\int^b_af(x)\cos(nx)dx\rightarrow 0$ as $n\rightarrow \infty$ for any regulated function $f$. The textbook which I'm working from say......Thu, 27 Aug 2026 21:33:16 -0400https://liberty.io/express-frac-ln-left-frac-1-2-right-80000000-as-a-negative-exponent.htmlhttps://liberty.io/express-frac-ln-left-frac-1-2-right-80000000-as-a-negative-exponent.htmlI would like to express $\dfrac{\ln\left(\frac{1}{2}\right)}{80000000}$ as a negative exponent. Using Desmos online calculator I tried this variant: Why are the results different? I expected/goal ......Thu, 27 Aug 2026 21:31:52 -0400https://liberty.io/given-a-simple-graph-and-its-complement-prove-that-either-of-them-is-a.htmlhttps://liberty.io/given-a-simple-graph-and-its-complement-prove-that-either-of-them-is-a.htmlI was tasked to prove that when given 2 graphs $G$ and $\bar{G}$ (complement), at least one of them is a always a connected graph. Well, I always post my attempt at solution, but here I'm totally ......Thu, 27 Aug 2026 21:30:08 -0400https://liberty.io/is-empty-set-element-of-every-set-if-it-is-subset-of-every-set.htmlhttps://liberty.io/is-empty-set-element-of-every-set-if-it-is-subset-of-every-set.htmlThis problem is from Discrete Mathematics and its Applications My question is on 9b. I know that the sign represents an element is a member of. (from book) I know that the O with a slash across i......Thu, 27 Aug 2026 21:25:33 -0400https://liberty.io/pathway-to-low-dimensional-topology.htmlhttps://liberty.io/pathway-to-low-dimensional-topology.htmlI am currently studying Artin Algebra and Topology by Munkres. I have previously studied analysis. After reading few lecture notes on knot theory, I have developed a genuine interest in Low dimensi......Thu, 27 Aug 2026 21:15:30 -0400https://liberty.io/stationary-increments-definition.htmlhttps://liberty.io/stationary-increments-definition.htmlLet $X_t$ be a stochastic process on the reals. I was under the impression the definition of stationary increments was that $X_{t+s}-X_{s}$ has the same distribution as $X_{t}-X_{0}$ for all $s,t$. I ...Thu, 27 Aug 2026 21:03:19 -0400https://liberty.io/meaning-of-the-expression-mu-a-e.htmlhttps://liberty.io/meaning-of-the-expression-mu-a-e.htmlIn the monotone convergence theorem: Let $g_n\geq 0$ be a sequence of measurable functions, such that $g_n \uparrow g\;\; \mu \text{ a.e.},$ i.e. $g_n(\omega) \leq g_{n+1}(\omega)\;\; \mu \text{ a.......Thu, 27 Aug 2026 20:57:57 -0400https://liberty.io/limits-of-points-along-x-axis-of-the-2-variable-function.htmlhttps://liberty.io/limits-of-points-along-x-axis-of-the-2-variable-function.htmlLet $f:\Re ^2\rightarrow\Re$ be a real-valued function such that f(x,y) is defined as, $$f(x,y)= \begin{cases} xsin(1/y),& \text{if $y\neq 0$} \\[2ex] 0,& \text{if $y=0$} \end{cases}$$ The ...Thu, 27 Aug 2026 20:50:14 -0400https://liberty.io/product-of-two-gaussian-pdfs-is-a-gaussian-pdf-but-product-of-two-gaus.htmlhttps://liberty.io/product-of-two-gaussian-pdfs-is-a-gaussian-pdf-but-product-of-two-gaus.htmlThe Product of Two Gaussian Random Variables is not Gaussian distributed: Is the product of two Gaussian random variables also a Gaussian? Also Wolfram Mathworld So this is saying $X \sim N(\mu_1, \...Thu, 27 Aug 2026 20:47:00 -0400https://liberty.io/how-to-prove-this-is-a-rational-number.htmlhttps://liberty.io/how-to-prove-this-is-a-rational-number.htmlI'm not sure how to prove this is a rational number $\frac{q}{m}$, can some one show me? $$\frac{q}{m}=\frac{(\frac{1+\sqrt5}{2})^n - (\frac{1-\sqrt5}{2})^n}{\sqrt5}$$...Thu, 27 Aug 2026 20:37:36 -0400https://liberty.io/give-a-geometric-description-of-the-following-set-of-points.htmlhttps://liberty.io/give-a-geometric-description-of-the-following-set-of-points.htmlGive a geometric description of the following set of points: $x^2 +y^2 + z^2-8x+14y-18z>/= 65 $ So I completed the square and got the set to read: $(x-4)^2+(y+7)^2+(z-9)^2>/= 211 $ Howeve......Thu, 27 Aug 2026 20:30:57 -0400https://liberty.io/covariance-of-two-random-variables.htmlhttps://liberty.io/covariance-of-two-random-variables.htmlI am trying to find the covariance of two random variables but I am not having any lucky. Just for simplicity lets say that my random variables are : X = value rolled by a die Y = 1 if even 0 if ......Thu, 27 Aug 2026 20:26:13 -0400https://liberty.io/why-f-x-sqrt-x-is-a-function.htmlhttps://liberty.io/why-f-x-sqrt-x-is-a-function.htmlWhy $f(x) = \sqrt{x}$ is a function (as I found in my textbook) since for example the square root of $25$ has two different outputs ($-5,5$) and a function is defined as "A function from A to B is ......Thu, 27 Aug 2026 20:25:54 -0400https://liberty.io/user-al-jabra-mathematics-stack-exchange.htmlhttps://liberty.io/user-al-jabra-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Thu, 27 Aug 2026 19:59:52 -0400https://liberty.io/how-does-e-ln-x-e-ln-1-x.htmlhttps://liberty.io/how-does-e-ln-x-e-ln-1-x.htmlI understand the inverse of e^{x} is the natural logarithm. However I don't understand how the following expression is true: $e^{-\ln x} = e^{\ln(1/x)}$ Any assistance is appreciated....Thu, 27 Aug 2026 19:36:24 -0400https://liberty.io/the-volume-of-body-in-polar-coordinates.htmlhttps://liberty.io/the-volume-of-body-in-polar-coordinates.htmlIf we have curve $r=F(\varphi)$ and two polar radii $\varphi=\alpha$ and $\varphi=\beta$ $(\alpha<\beta)$. Then volume of body which is bounded by above polar curve and lines which is revolved a......Thu, 27 Aug 2026 19:32:43 -0400https://liberty.io/how-to-solve-a-system-with-more-equations-than-unkowns.htmlhttps://liberty.io/how-to-solve-a-system-with-more-equations-than-unkowns.htmlIn general, how do you solve a system with more equations than unknowns? I know that if I select the equations to match them with the number of unknowns, there may be zero or many solutions dependi......Thu, 27 Aug 2026 19:26:13 -0400https://liberty.io/to-find-an-extremal-of-the-functional-int-0-1-y-039-2-12-xy-dx.htmlhttps://liberty.io/to-find-an-extremal-of-the-functional-int-0-1-y-039-2-12-xy-dx.htmlI have to find extremal of following : $\int_0^1 [(y')^2 + 12 xy] dx$ with $y(0) = 0$ and $y(1) = 1$. I applied the Euler's equation $\frac{\partial F}{\partial y} - \frac{d}{dx}(\frac{\partial F}{\...Thu, 27 Aug 2026 19:18:25 -0400https://liberty.io/finding-parametric-equations-for-curve-of-intersection.htmlhttps://liberty.io/finding-parametric-equations-for-curve-of-intersection.htmlI am having difficulty finding parametric equations of the curve of intersection of $z = x^2 - y^2$ and cylinder $x^2 + y^2 = 1$ I am aware that the first equation represents a hyperboloid, and the ...Thu, 27 Aug 2026 19:16:53 -0400https://liberty.io/correct-interpretation-of-confidence-interval.htmlhttps://liberty.io/correct-interpretation-of-confidence-interval.htmlI think this question may have been asked in one form or another, but I did not find an answer that I understood or thought was satisfactory...so I apologize if it has been explained. My understan......Thu, 27 Aug 2026 19:15:19 -0400https://liberty.io/product-rule-for-gradient.htmlhttps://liberty.io/product-rule-for-gradient.htmlI have two functions scalar functions of vector $\textbf{x}$, $f(\textbf{x})$ and $g(\textbf{x})$. I know the gradient of $f(\textbf{x})$ (i.e. $\triangledown f(\textbf{x})$) and I want to find the ...Thu, 27 Aug 2026 19:08:57 -0400https://liberty.io/how-can-i-simplify-and-verify-the-logical-equivalence-using-these-laws.htmlhttps://liberty.io/how-can-i-simplify-and-verify-the-logical-equivalence-using-these-laws.html∼(p ∨∼q) ∨ (∼p ^ ~ q) ≡ ~p Please help I don't know where to start. These are the laws I need to list in each step when simplifying. Commutative laws: p ∧ q ≡ q ∧ p p ∨ q ≡ q ∨ p Associative l......Thu, 27 Aug 2026 18:55:18 -0400https://liberty.io/linearizing-logarithmic-function.htmlhttps://liberty.io/linearizing-logarithmic-function.htmlI have a given set of data points (y,x) with uncertainties. When I plot those points on a graph, the trendline appears to follow the equation y = c + a*ln(x). I want to be able to find the uncer......Thu, 27 Aug 2026 18:46:28 -0400https://liberty.io/what-is-this-angle-in-a-right-triangle-with-sides-of-length-5-12-and-1.htmlhttps://liberty.io/what-is-this-angle-in-a-right-triangle-with-sides-of-length-5-12-and-1.htmlHow do I find the missing adjacent angle to leg b in a right triangle with the following side lengths: leg a = 5, leg b = 12, and hypotenuse = 13. Thanks...Thu, 27 Aug 2026 18:45:51 -0400https://liberty.io/finding-remainder-using-fermat-039-s-little-theorm.htmlhttps://liberty.io/finding-remainder-using-fermat-039-s-little-theorm.htmlUse Fermat's Little Theorem to find the remainder of $5^{15}$ divided by $1337$. I know Fermat's Little Theorem, but failed to understand how to use it in this. output.944...Thu, 27 Aug 2026 18:45:35 -0400https://liberty.io/proof-of-stars-and-bars-formula.htmlhttps://liberty.io/proof-of-stars-and-bars-formula.htmlI am trying to prove a formula (for ways of distributing $n$ identical balls among $r$ persons when each person may get any number of balls) ${n+r-1}\choose{r-1}$. But I am not able to prove it. I ......Thu, 27 Aug 2026 18:45:01 -0400https://liberty.io/finding-the-parametric-form-of-a-standard-equation.htmlhttps://liberty.io/finding-the-parametric-form-of-a-standard-equation.htmlI need to find the parametric form of $3x - 2y + 10 = 0$. I found that the parametric form for this equation could be : \begin{align}x &= t\\ y &= 5 + \frac{3}{2} t \end{align} I did thi......Thu, 27 Aug 2026 18:39:06 -0400https://liberty.io/finding-domain-of-sqrt-x-12-x-9-x-4-x-1.htmlhttps://liberty.io/finding-domain-of-sqrt-x-12-x-9-x-4-x-1.htmlHow to find the domain of this function $\sqrt{x^{12} - x^9 + x^4 - x + 1}$? Its really hard to find all 12 roots and plot them or use tricks like "wavy curve" to check where the expression in the ...Thu, 27 Aug 2026 18:38:30 -0400https://liberty.io/equilibrium-probability-in-a-markov-chain-closed.htmlhttps://liberty.io/equilibrium-probability-in-a-markov-chain-closed.htmlcan someone please tell me what an equilibrium probability in a markov chain is and what the relation between the equlibrium probability and equilibrium distribution is? Thanks...Thu, 27 Aug 2026 18:31:17 -0400https://liberty.io/correct-graph-of-arccot-x.htmlhttps://liberty.io/correct-graph-of-arccot-x.htmlThere are 2 possible ways of choosing an interval of cot x graph for obtaining the graph of arccot x. Why do we choose the interval (0,$\pi$) instead of the first choice? This may be because the gr......Thu, 27 Aug 2026 18:29:29 -0400https://liberty.io/how-to-compute-a-linear-fractional-transformation-that-maps-a-circle-t.htmlhttps://liberty.io/how-to-compute-a-linear-fractional-transformation-that-maps-a-circle-t.htmlHow to compute a linear fractional transformation that maps a circle to a given circle? For example, let $C_1$ be the circle $|z+(2+i)|=1$ and $C_2$ be the circle $|z-5|=7$. How to find a linear ...Thu, 27 Aug 2026 18:27:55 -0400https://liberty.io/either-john-isn-039-t-stupid-and-he-is-lazy-or-he-is-stupid-he-039-s-s.htmlhttps://liberty.io/either-john-isn-039-t-stupid-and-he-is-lazy-or-he-is-stupid-he-039-s-s.htmlI'm reading How To Prove it by Daniel J. Velleman, and I came across the following example which sounds odd to me. Either John isn't stupid and he is lazy, or he is stupid. He's Stupid. Ther......Thu, 27 Aug 2026 18:20:04 -0400https://liberty.io/how-to-factorize-x-2-10yz-2xz-2xy-3y-2-3z-2.htmlhttps://liberty.io/how-to-factorize-x-2-10yz-2xz-2xy-3y-2-3z-2.htmlHow to factorize $$x^2+10yz-2xz-2xy-3y^2-3z^2$$ It is expanded and we should make them into parts and factorize each part individually. the last answer is $$(x+y-3z)(x-3y+z)$$ but how to get it ?...Thu, 27 Aug 2026 18:17:00 -0400https://liberty.io/is-there-a-general-formula-for-the-sum-of-a-quadratic-sequence.htmlhttps://liberty.io/is-there-a-general-formula-for-the-sum-of-a-quadratic-sequence.htmlI tried Googling "formula for sum of quadratic sequence", which did not give me anything useful. I just want an explicit formula for figuring out a sum for a quadratic sequence. For example, how wo......Thu, 27 Aug 2026 18:12:13 -0400https://liberty.io/variance-of-conditional-variance.htmlhttps://liberty.io/variance-of-conditional-variance.htmlIf we have two random variables X and Y which share a joint pdf and pmf (there is a discrete and a continuous scenario), how do we calculate: Var[Var[X|Y]] - I looked at the Theory of Total Varian......Thu, 27 Aug 2026 18:09:18 -0400https://liberty.io/proof-of-caratheodory-039-s-theorem-for-convex-sets-using-radon-039-s-.htmlhttps://liberty.io/proof-of-caratheodory-039-s-theorem-for-convex-sets-using-radon-039-s-.htmlI am self-studying some discrete geometry / convex analysis. Many descriptions of Caratheodory's Theorem for convex sets mention that Radon's Lemma can be used to simplify the proof, but I haven't......Thu, 27 Aug 2026 17:51:52 -0400https://liberty.io/finding-the-solution-set-for-an-absolute-value-equation-algebraically.htmlhttps://liberty.io/finding-the-solution-set-for-an-absolute-value-equation-algebraically.htmle.g. the equation $|x-2|=3$, geometrically it's easy to solve drawing the number line. I get the solution set {-1,5} which I understand. but I was looking at some notes I found online of how to so......Thu, 27 Aug 2026 17:49:02 -0400https://liberty.io/show-that-if-h-and-k-are-subgroups-of-one-abelian-group-g-then-g-1-hk-.htmlhttps://liberty.io/show-that-if-h-and-k-are-subgroups-of-one-abelian-group-g-then-g-1-hk-.htmlHere's the problem: Show that if $H$ and $K$ are subgroups of one abelian group $G$, then $G_1=\{hk\mid h\in H\text{ and } k\in K\}$ is a subgroup of $G$ I know that "abelian" means tha......Thu, 27 Aug 2026 17:41:25 -0400https://liberty.io/fitch-system-or-alternatives.htmlhttps://liberty.io/fitch-system-or-alternatives.htmlIs there a good resource for how to understand the rules of a Fitch proof system, or an alternative? I'm currently trying to get a sense or how things are proved in natural deduction, which differs......Thu, 27 Aug 2026 17:34:09 -0400https://liberty.io/icosahedral-symmetry-as-permutation-group.htmlhttps://liberty.io/icosahedral-symmetry-as-permutation-group.htmlHopefully an easy question: the icosahedral group of order 60 (orientation preserving symmetries of a regular icosahedron) is isomorphic to the alternating group on 5 points. In terms of the icosa......Thu, 27 Aug 2026 17:29:04 -0400https://liberty.io/when-to-use-the-contrapositive-to-prove-a-statment.htmlhttps://liberty.io/when-to-use-the-contrapositive-to-prove-a-statment.htmlMy question tries to address the intuition or situations when using the contrapositive to prove a mathematical statement is an adequate attempt. Whenever we have a mathematical statement of the fo......Thu, 27 Aug 2026 17:27:11 -0400https://liberty.io/is-my-understanding-of-antisymmetric-and-symmetric-relations-correct.htmlhttps://liberty.io/is-my-understanding-of-antisymmetric-and-symmetric-relations-correct.htmlSo I'm having a hard time grasping how a relation can be both antisymmetric and symmetric, or neither. Are my examples correct? symmetric & antisymmetric R ={(1,1),(2,2),(3,3)} not symmetr......Thu, 27 Aug 2026 17:18:12 -0400https://liberty.io/using-method-of-moments-on-binomial-distribution-closed.htmlhttps://liberty.io/using-method-of-moments-on-binomial-distribution-closed.htmlHi, I am stuck on this particular question. Any help? Thanks....Thu, 27 Aug 2026 17:17:09 -0400https://liberty.io/recurrence-relation-merge-sort.htmlhttps://liberty.io/recurrence-relation-merge-sort.htmlWe know the recurrence relation for normal merge sort. It is T(n) = 2T(n/2) + n. After solving it we can get T(n) = cnlogn. I would like to know the recurrence relation for K way merge sort i.e. in......Thu, 27 Aug 2026 17:11:30 -0400https://liberty.io/integral-from-infinity-to-infinity.htmlhttps://liberty.io/integral-from-infinity-to-infinity.htmlMy physics professor today wrote on the blackboard: $$ \int_{\infty}^{\infty} f(x) dx = 0 $$ for every function $f$. And the proof he gave was: $$ \int_{\infty}^{\infty} f(x) dx = \int_{\infty}^{a......Thu, 27 Aug 2026 17:11:11 -0400https://liberty.io/a-parallelogram-and-a-circle.htmlhttps://liberty.io/a-parallelogram-and-a-circle.htmlThe perimeter of a parallelogram $ABCD$ is $30$ $cm$. If $D$ and $M$ which is the midpoint of $\overline{DC}$ lie on the circle with diameter $\overline{AB}$, what is the perimeter of $ABMD$? We k......Thu, 27 Aug 2026 16:58:59 -0400https://liberty.io/why-is-two-the-only-even-number-that-is-prime.htmlhttps://liberty.io/why-is-two-the-only-even-number-that-is-prime.htmlThe other prime numbers are all odd numbers such as $5, 11, 127,$ and $37$. So, why is $2$ the only prime even number there is? Is it because it only has 1 and itself that way, even though it's ......Thu, 27 Aug 2026 16:58:54 -0400https://liberty.io/relaxed-menage-problem-sitting-arrangement-for-couples.htmlhttps://liberty.io/relaxed-menage-problem-sitting-arrangement-for-couples.htmlThe original problem is this: How many ways are there to seat n couples around a circular table such that no couple sits next to each other? For an answer, I looked at the solution here: https://...Thu, 27 Aug 2026 16:55:17 -0400https://liberty.io/in-classical-logic-why-is-p-rightarrow-q-true-if-p-is-false-and-q-is-t.htmlhttps://liberty.io/in-classical-logic-why-is-p-rightarrow-q-true-if-p-is-false-and-q-is-t.htmlProvided we have this truth table where "$p\implies q$" means "if $p$ then $q$": $$\begin{array}{|c|c|c|} \hline p&q&p\implies q\\ \hline T&T&T\\ T&F&F\\ F&T&T\\ F&......Thu, 27 Aug 2026 16:50:59 -0400https://liberty.io/how-do-you-rotate-a-vector-by-a-unit-quaternion.htmlhttps://liberty.io/how-do-you-rotate-a-vector-by-a-unit-quaternion.htmlGiven a 3-variable right-handed vector v that is a translation measured in local space and a unit quaternion representing an orientation from local to world space, how do you use the quaternion to ......Thu, 27 Aug 2026 16:48:09 -0400https://liberty.io/finding-limit-of-multivariable-function-using-the-squeeze-theorem.htmlhttps://liberty.io/finding-limit-of-multivariable-function-using-the-squeeze-theorem.htmlIn my textbook (Stewart's Calculus), the video tutor solutions for some problems use the squeeze theorem to determine the limit of a function. For example: Find $$\lim_{(x, y) \to (0, 0)} \frac{x......Thu, 27 Aug 2026 16:38:44 -0400https://liberty.io/proof-of-the-continuous-image-of-a-connected-set-is-connected.htmlhttps://liberty.io/proof-of-the-continuous-image-of-a-connected-set-is-connected.htmlNone of the existing questions is exactly answering my question so I'm posting a new question, but feel free to refer me to some already answered question! In Rudin Theorem 4.22, we know that ......Thu, 27 Aug 2026 16:38:21 -0400https://liberty.io/proving-the-inverse-if-any-of-a-lower-triangular-matrix-is-lower-trian.htmlhttps://liberty.io/proving-the-inverse-if-any-of-a-lower-triangular-matrix-is-lower-trian.htmlThe inverse of a non-singular lower triangular matrix is lower triangular. Construct a proof of this fact as follows. Suppose that $L$ is a non-singular lower triangular matrix. If $b \in \mathbb{R......Thu, 27 Aug 2026 16:15:36 -0400https://liberty.io/factorising-fourth-power-polynomial-with-5-terms-duplicate.htmlhttps://liberty.io/factorising-fourth-power-polynomial-with-5-terms-duplicate.htmlI've searched all over the internet and cannot seem to factorise this polynomial. $x^4 - 2x^3 + 8x^2 - 14x + 7$ The result should be $(x − 1)(x^3 − x^2 + 7x − 7)$ What are the steps to get to that ...Thu, 27 Aug 2026 16:05:11 -0400https://liberty.io/what-does-the-ln-or-1n-mean-in-this-equation.htmlhttps://liberty.io/what-does-the-ln-or-1n-mean-in-this-equation.htmlWhat does the ln mean here? It's from this page. https://en.wikipedia.org/wiki/Universally_unique_identifier#Collisions Finally, what could I have done to find out for myself? (What did I miss?) I......Thu, 27 Aug 2026 15:54:57 -0400https://liberty.io/calculate-sides-of-right-triangle-with-hypotenuse-and-area-or-perimete.htmlhttps://liberty.io/calculate-sides-of-right-triangle-with-hypotenuse-and-area-or-perimete.htmlI'm trying to find if it is possible to find the lengths of the base and height of a right triangle with only the hypotenuse and the area (or the perimeter) of the triangle. I would have just figured ...Thu, 27 Aug 2026 15:53:12 -0400https://liberty.io/what-does-ll-mean.htmlhttps://liberty.io/what-does-ll-mean.htmlI saw two less than signs on this Wikipedia article and I was wonder what they meant mathematically. http://en.wikipedia.org/wiki/German_tank_problem EDIT: It looks like this can use TeX commands......Thu, 27 Aug 2026 15:50:57 -0400https://liberty.io/notation-for-repeated-division.htmlhttps://liberty.io/notation-for-repeated-division.htmlFor addition, there is $\sum$ For multiplication, there is $\prod$ What about for division? Something like ${/}_{i=1}^n a_i = a_1 \div a_2\ \div\ ...\ \div\ a_n$?...Thu, 27 Aug 2026 15:47:35 -0400https://liberty.io/finding-formula-for-nth-partial-sum.htmlhttps://liberty.io/finding-formula-for-nth-partial-sum.htmlI need to find a formula for the nth partial sum of the series: $ 2 + \frac 23 + \frac 29 + \frac {2}{27} + .... + \frac {2}{3^{n-1}} + ... $ then I need to use the formula to find the series' su......Thu, 27 Aug 2026 15:41:06 -0400https://liberty.io/related-rates-question-a-man-starts-walking-north-at-4-ft-s-from-a-poi.htmlhttps://liberty.io/related-rates-question-a-man-starts-walking-north-at-4-ft-s-from-a-poi.htmlA man starts walking north at 4 ft/s from a point P. Five minutes later a woman starts walking south at 5 ft/s from a point 500 ft due east of P. At what rate are the people moving apart 15 minutes......Thu, 27 Aug 2026 15:29:16 -0400https://liberty.io/checking-for-validity-using-truth-table-shortcut.htmlhttps://liberty.io/checking-for-validity-using-truth-table-shortcut.htmlI have the following: p ⊃ (q ≡ r) (q ≡ r) ⊃ p ∴ [(~p v q) · ( q ⊃ p) ≡ r I'm trying to check for validity so I created a truth table to check: My understanding is that the argument is ...Thu, 27 Aug 2026 15:28:09 -0400https://liberty.io/what-is-category-theory-useful-for.htmlhttps://liberty.io/what-is-category-theory-useful-for.htmlOkay, so I understand what calculus, linear algebra, combinatorics and even topology try to answer (update: this is not the case in hindsight), but why invent category theory? In Wikipedia it says ......Thu, 27 Aug 2026 15:11:46 -0400https://liberty.io/integration-of-fundamental-solution-of-laplace-039-s-equation.htmlhttps://liberty.io/integration-of-fundamental-solution-of-laplace-039-s-equation.htmlI am currently reading Evan's PDE and am getting hung up on many of the more "technical details". This question may be very basic (multivariable calculus). I am given that the fundamental solution of ...Thu, 27 Aug 2026 15:03:25 -0400https://liberty.io/how-to-find-the-derivative-of-the-function-f-x-2x-3-4-x-2-x-1-5.htmlhttps://liberty.io/how-to-find-the-derivative-of-the-function-f-x-2x-3-4-x-2-x-1-5.htmlCan someone please tell me how to find the derivative of this function? I've been working on this one problem since yesterday and I still can't find the answer... $$f(x) = (2x-3)^4 (x^2 + x + 1)^5......Thu, 27 Aug 2026 15:01:23 -0400https://liberty.io/what-is-a-smooth-surface.htmlhttps://liberty.io/what-is-a-smooth-surface.htmlWhat is a smooth surface in terms of tangents and normals? I read in a book that surfaces are smooth if its surface normals depend continuously on the points of that surface. I did not understand t......Thu, 27 Aug 2026 14:49:46 -0400https://liberty.io/what-exactly-does-it-mean-for-a-function-to-be-well-behaved.htmlhttps://liberty.io/what-exactly-does-it-mean-for-a-function-to-be-well-behaved.htmlOften in my studies (economics) the assumption of a "well-behaved" function will be invoked. I don't exactly know what that entails (I think twice continuously differentiability is one of......Thu, 27 Aug 2026 14:49:28 -0400https://liberty.io/what-039-s-the-correct-definition-of-conjugate-and-do-we-identify-them.htmlhttps://liberty.io/what-039-s-the-correct-definition-of-conjugate-and-do-we-identify-them.htmlWhat does a "conjugate" in math mean? I know the definition says if we have $x+y$, then $x-y$ is its conjugate. So, $5+ \sqrt{5}$ and $5-\sqrt{5}$ are conjugate of one another. Are $8$ and $2$ ...Thu, 27 Aug 2026 14:41:26 -0400https://liberty.io/trace-norm-properties.htmlhttps://liberty.io/trace-norm-properties.htmlLet $\|A\|_1=\operatorname{trace}(\sqrt{A^* A})$. I already proved that for arbitrary unitary matrices $U$ and $V$, $\|UAV^*\|_1=\|A\|_1$ and $\|A\|_1=\sigma_1+\dots+\sigma_k$. Now I would like to ......Thu, 27 Aug 2026 14:40:27 -0400https://liberty.io/sd-of-a-bernoulli-trial.htmlhttps://liberty.io/sd-of-a-bernoulli-trial.htmlWhy nothing is mentioned about the standard deviation of a Bernoulli trial ? Does it even make sense if I try to visualize it ?...Thu, 27 Aug 2026 14:29:43 -0400https://liberty.io/euclidean-norm-and-matrix-question.htmlhttps://liberty.io/euclidean-norm-and-matrix-question.htmlIf $A$ is a real $n \times n$ matrix and $x \in \mathbb{R}^n$ then is it true that $|Ax| \le |A||x|$? Here $|x|$ is the usual euclidean norm for vectors and $|A| = \sqrt{\Sigma_{i,j} a_{ij}^2}$...Thu, 27 Aug 2026 14:28:32 -0400https://liberty.io/notation-is-a-monotonic-function-of.htmlhttps://liberty.io/notation-is-a-monotonic-function-of.htmlThe symbols $\sim$ and $\propto$ are used to denote direct proportionality of two quantities. Sometimes, a weaker statement is enough or desired, for example when one can only assume that an incre......Thu, 27 Aug 2026 14:22:10 -0400https://liberty.io/trigonometric-expression-for-1-cos-alpha-cos-2-alpha-cdots-cos-n-alpha.htmlhttps://liberty.io/trigonometric-expression-for-1-cos-alpha-cos-2-alpha-cdots-cos-n-alpha.htmlThis question is not a duplicate because I am asked here to use the fact that $1 + \cos \alpha + \cos 2 \alpha + \cdots + \cos n \alpha = Re (1 + z + z^{2} + \cdots + z^{n})$, where the question t......Thu, 27 Aug 2026 14:12:14 -0400https://liberty.io/how-frac-1-x-is-integrable-if-it-is-not-a-continuous-function.htmlhttps://liberty.io/how-frac-1-x-is-integrable-if-it-is-not-a-continuous-function.htmlIf function is not continuous then it means it's not differentiable and that would mean that it should not be integrable as differentiation is the reverse process of integration. I am talking about ...Thu, 27 Aug 2026 14:04:28 -0400https://liberty.io/ackermann-039-s-function-is-mu-recursive.htmlhttps://liberty.io/ackermann-039-s-function-is-mu-recursive.htmlIn my book there is the following proof that Ackermann's function is $\mu$-recursive: We propose to show that Ackermann's funcition is $\mu$-recursive. The first part of the job is to devise a sc......Thu, 27 Aug 2026 13:46:41 -0400https://liberty.io/prove-that-cot-2-pi-7-cot-2-2-pi-7-cot-2-3-pi-7-5.htmlhttps://liberty.io/prove-that-cot-2-pi-7-cot-2-2-pi-7-cot-2-3-pi-7-5.htmlProve that $\cot^2{(\pi/7)} + \cot^2{(2\pi/7)} + \cot^2{(3\pi/7)} = 5$ . I am sure this is derived from using roots of unity and Euler's complex number function, but I am very uncomfortable in these ...Thu, 27 Aug 2026 13:43:09 -0400https://liberty.io/definition-of-degree-matrix-of-directed-graph.htmlhttps://liberty.io/definition-of-degree-matrix-of-directed-graph.htmlim sorry if this is a trivial question. As the title stated, how is the degree matrix of a directed graph defined? I looked up wikipedia of degree matrix but there was no defitnition of it. Instead......Thu, 27 Aug 2026 13:35:35 -0400https://liberty.io/how-many-axis-of-symmetry-of-the-cube-are-there.htmlhttps://liberty.io/how-many-axis-of-symmetry-of-the-cube-are-there.htmlIn my final mathematics test, I have a bonus question: How many axis of symmetry of the cube are there? The teacher gives me the definition: Definition: If we rotate a 3-dimension object around......Thu, 27 Aug 2026 13:31:01 -0400https://liberty.io/sum-of-divergent-and-convergent-series.htmlhttps://liberty.io/sum-of-divergent-and-convergent-series.html$\sum_{n=1}^{\infty}x_n$ is a convergent series and $\sum_{n=1}^{\infty}y_n$ is a divergent series. Prove their sum diverges. My attempt: Suppose $\sum_{n=1}^{\infty}x_n + y_n$ converges. Since ......Thu, 27 Aug 2026 13:30:28 -0400https://liberty.io/what-is-the-meaning-of-the-term-distance-preserving.htmlhttps://liberty.io/what-is-the-meaning-of-the-term-distance-preserving.htmlI've been looking for a definition of this term in everyday words but I can't come across one, the use of the word is in the context of Isometrics....Thu, 27 Aug 2026 13:24:22 -0400https://liberty.io/what-is-the-use-of-calculus.htmlhttps://liberty.io/what-is-the-use-of-calculus.htmlI know this may seem like a really broad question, but I will narrow it down. I really want to know the purpose of some of the things my teacher is emphasizing in my calc class. For example why it so ...Thu, 27 Aug 2026 13:15:38 -0400https://liberty.io/real-world-examples-of-continuous-uniform-distribution-on-0-1.htmlhttps://liberty.io/real-world-examples-of-continuous-uniform-distribution-on-0-1.htmlCan someone give me real world examples of uniform distribution on [0,1] of a continuous random variable, because I could not make out one....Thu, 27 Aug 2026 13:14:09 -0400