https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usSat, 22 Aug 2026 08:36:48 -0400https://pop.com.sg/proof-of-c-a-b-for-a-right-triangle.htmlhttps://pop.com.sg/proof-of-c-a-b-for-a-right-triangle.htmlI am asked to "prove or disprove the following statement using contradiction" --> If a, b, c are the sides of a right traingle and c is the hypotenuse then $c<a+b$. Here is my proposed proof: P......Sat, 22 Aug 2026 12:35:20 -0400https://pop.com.sg/surface-area-of-cone.htmlhttps://pop.com.sg/surface-area-of-cone.htmlthanks for any help. I'm trying to find the surface area of a cone via integration. I know that the parametric equation of a cone is $$x=u\cos(p) \\ y=u\sin(p) \\ z=u$$ So as a vector, $\vec{R} = \...Sat, 22 Aug 2026 12:20:52 -0400https://pop.com.sg/solution-for-exponential-function-039-s-functional-equation-by-using-a.htmlhttps://pop.com.sg/solution-for-exponential-function-039-s-functional-equation-by-using-a.htmllet $f(0)=1$ and $f'(0)=1$. and $f(x+y)=f(x)f(y)$ for $x,y\in R$. How can I found $f(x)$ by using a definition of derivative?...Sat, 22 Aug 2026 12:15:48 -0400https://pop.com.sg/finding-unbiased-point-estimate-of-population-variance.htmlhttps://pop.com.sg/finding-unbiased-point-estimate-of-population-variance.htmlQ.The contents of each of a random sample of 100 cans of a soft drink are measured. The results have a mean of 331.28 ml and a standard deviation of 2.97 ml. Show that an unbiased estimate of the ...Sat, 22 Aug 2026 12:13:32 -0400https://pop.com.sg/definition-of-power-set-uses-strict-subset-duplicate.htmlhttps://pop.com.sg/definition-of-power-set-uses-strict-subset-duplicate.htmlIn the following lecture notes one finds the following: Examples. The following are all examples of σ-algebras. (HW: check this in each case) • Let P(X) denote the collection of all subsets of X, ......Sat, 22 Aug 2026 11:53:42 -0400https://pop.com.sg/how-to-express-sqrt-x-1.htmlhttps://pop.com.sg/how-to-express-sqrt-x-1.htmlHow would one express a solution to $\sqrt{x} =-1$? I just read that a solution to the above equation cannot be expressed in the form of complex numbers, really interested in any additional inform......Sat, 22 Aug 2026 11:52:26 -0400https://pop.com.sg/what-039-s-a-proof-that-the-angles-of-a-triangle-add-up-to-180.htmlhttps://pop.com.sg/what-039-s-a-proof-that-the-angles-of-a-triangle-add-up-to-180.htmlBack in grade school, I had a solution involving "folding the triangle" into a rectangle half the area, and seeing that all the angles met at a point: However, now that I'm in university, I'm not ...Sat, 22 Aug 2026 11:47:23 -0400https://pop.com.sg/how-to-find-the-inverse-cosine-without-a-calculator.htmlhttps://pop.com.sg/how-to-find-the-inverse-cosine-without-a-calculator.htmlHow to find the inverse of: $$\cos(c)=\frac{1}{3}$$ In other words, i'm trying to solve for c and without a calculator. If it's hard or not possible, then how would you go about solving inverses in ...Sat, 22 Aug 2026 11:39:18 -0400https://pop.com.sg/how-does-e-i-x-produce-rotation-around-the-imaginary-unit-circle.htmlhttps://pop.com.sg/how-does-e-i-x-produce-rotation-around-the-imaginary-unit-circle.htmlEuler’s formula states that $e^{i x} = \cos(x) + i \sin(x)$. I can see from the MacLaurin Expansion that this is indeed true; however, I don’t intuitively understand how raising $e$ to the power ......Sat, 22 Aug 2026 11:29:17 -0400https://pop.com.sg/what-is-additive-homomorphic-encryption-and-multiplicative-homomorphic.htmlhttps://pop.com.sg/what-is-additive-homomorphic-encryption-and-multiplicative-homomorphic.htmlExplain in brief about additive homomorphic encryption and multiplicative homomorphic encryption. I have read about them but I could not properly understand. So would you please elaborate additive ...Sat, 22 Aug 2026 11:28:20 -0400https://pop.com.sg/number-of-surjective-functions-from-a-to-b.htmlhttps://pop.com.sg/number-of-surjective-functions-from-a-to-b.htmlAm I on the right track? I am not sure about my reasoning... Number of surjective functions from $A$ to $B$ $$A = \{1,2,3,4\} ; B = \{a,b,c\}$$ We must count the surjective functions, meaning the ...Sat, 22 Aug 2026 11:27:59 -0400https://pop.com.sg/convolution-integral-limits.htmlhttps://pop.com.sg/convolution-integral-limits.htmlThere are 2 types of convolution: The limit of the integral is from minus infinity to plus infinity The limit is from zero to t. When we use the first and when we use the second? $$\int f(\tau)g......Sat, 22 Aug 2026 11:27:48 -0400https://pop.com.sg/definition-of-curl-in-mathbb-r-n.htmlhttps://pop.com.sg/definition-of-curl-in-mathbb-r-n.htmlIn calculus we saw that how to define curl in $\mathbb{R}^3$. My question is about the generalization; I mean is there any way that one can define curl in $\mathbb{R}^n$? If yes, how can I do it?...Sat, 22 Aug 2026 11:24:06 -0400https://pop.com.sg/is-my-interpretation-of-direct-comparison-test-correct.htmlhttps://pop.com.sg/is-my-interpretation-of-direct-comparison-test-correct.htmlI had been studying convergence and divergence of series in Calculus. This is the problem that I've to prove using direct comparison test. Prove that the series $\sum_{n=1}^{\infty} \frac{1}{2^......Sat, 22 Aug 2026 11:18:28 -0400https://pop.com.sg/why-does-0-0-mean-there-it-a-hole-and-not-an-asymptote.htmlhttps://pop.com.sg/why-does-0-0-mean-there-it-a-hole-and-not-an-asymptote.htmlhttps://www.khanacademy.org/math/algebra2/rational-expressions/rational-function-graphing/v/finding-asymptotes-example At around 8 minutes he says that we want a number which makes only the denomi......Sat, 22 Aug 2026 11:07:57 -0400https://pop.com.sg/log-of-a-negative-number.htmlhttps://pop.com.sg/log-of-a-negative-number.htmlWhy is the log of $x$ equal to the log of the absolute value of $x$ plus $i$ times $\pi$? $$\log(x)=\log(|x|)+i\pi\text{, for }x < 0$$ Where does the $\pi$ come from? Is it from a logarithmic ...Sat, 22 Aug 2026 11:03:54 -0400https://pop.com.sg/what-is-the-relationship-between-graphs-graph-theory-and-logic.htmlhttps://pop.com.sg/what-is-the-relationship-between-graphs-graph-theory-and-logic.htmlI have read that Graph theory is useful to mathematical logic, but I don’t understand how. How exactly does a logic of graphs differ from standard predicate logic? Here is an example from an arti......Sat, 22 Aug 2026 10:57:22 -0400https://pop.com.sg/perfect-solutions-to-the-kissing-number-problem-besides-in-dimensions-.htmlhttps://pop.com.sg/perfect-solutions-to-the-kissing-number-problem-besides-in-dimensions-.htmlThe kissing number problem asks how many n dimensional unit spheres can fit around a central one with no overlapping; a natural question is in what dimensions can this be done so that there is no e......Sat, 22 Aug 2026 10:53:34 -0400https://pop.com.sg/opposite-of-mod-in-a-equation.htmlhttps://pop.com.sg/opposite-of-mod-in-a-equation.htmlI am trying to solve this module equation (due to a cryptography program that I am coding) but cannot figure it out how: X + 7 % 10 = 6 How Do i solve this? Thanks in advance....Sat, 22 Aug 2026 10:52:54 -0400https://pop.com.sg/definition-clarification-of-graph-embeddings.htmlhttps://pop.com.sg/definition-clarification-of-graph-embeddings.htmlRecently I started reading about graph embeddings, but I am unable to grasp its definition from Wikipedia. Can anyone explain this term with an example....Sat, 22 Aug 2026 10:48:37 -0400https://pop.com.sg/figuring-out-the-amount-of-039-straight-edge-039-pieces-in-a-puzzle.htmlhttps://pop.com.sg/figuring-out-the-amount-of-039-straight-edge-039-pieces-in-a-puzzle.htmlI was wondering if there was any set way to determine the number of 'straight edge' pieces in a puzzle, assuming the pieces are all in neat rows and columns? Does the ratio of edge pieces to middle ...Sat, 22 Aug 2026 10:48:20 -0400https://pop.com.sg/what-is-the-largest-prime-number-in-the-denominator-of-a-fraction-that.htmlhttps://pop.com.sg/what-is-the-largest-prime-number-in-the-denominator-of-a-fraction-that.htmlI was studying a Target question for Math League competitions, and after a few hours of pondering, I finally came up with the following method of solving the mentioned problem: For any decimal, it......Sat, 22 Aug 2026 10:46:55 -0400https://pop.com.sg/positive-definiteness-of-the-schur-complement.htmlhttps://pop.com.sg/positive-definiteness-of-the-schur-complement.htmlLet $M$ to be a real-valued symmetric and positive-definite (PD) matrix (also sparse and banded if it helps) $$ M= \begin{bmatrix} A & B\\ B^T & D \end{bmatrix} $$ Under what conditions the ...Sat, 22 Aug 2026 10:24:17 -0400https://pop.com.sg/calculating-the-derivative-of-csc-2-4x.htmlhttps://pop.com.sg/calculating-the-derivative-of-csc-2-4x.htmlI am having problems calculating derivatives of squared trigonometric functions. $$f(x) = \csc^2(4x)$$ Let's see... This is equivalent to $$f(x) = \csc(4x)\cdot \csc(4x)$$ Now, to get the deriv......Sat, 22 Aug 2026 10:17:07 -0400https://pop.com.sg/how-to-explain-what-is-wrong-in-this.htmlhttps://pop.com.sg/how-to-explain-what-is-wrong-in-this.htmlMy friend showed this to me and I instantly know that this is wrong. However, I cannot explain why this is wrong to my friend. Question. Prove $\displaystyle \frac{100-100}{100-100} = 2.$ A......Sat, 22 Aug 2026 09:59:43 -0400https://pop.com.sg/what-is-the-meaning-of-c-0-0-1-c-0-1-and-c-1-0-1.htmlhttps://pop.com.sg/what-is-the-meaning-of-c-0-0-1-c-0-1-and-c-1-0-1.htmlSome difficulty to understand the symbol what is the meaning of $C^0[0,1]$ , $C[0,1]$ and $C^1[0,1]$ ?...Sat, 22 Aug 2026 09:56:45 -0400https://pop.com.sg/online-visualization-tool-for-planes-spans-in-linear-algebra.htmlhttps://pop.com.sg/online-visualization-tool-for-planes-spans-in-linear-algebra.htmlI would like to visualize planes in 3D as I start learning linear algebra, to build a solid foundation. Surprisingly, I have been unable to find an online tool (website/web app) to visualize planes......Sat, 22 Aug 2026 09:54:08 -0400https://pop.com.sg/how-to-find-the-area-under-a-semicircle-using-integration-closed.htmlhttps://pop.com.sg/how-to-find-the-area-under-a-semicircle-using-integration-closed.htmlHow would I go about finding the area under a semicircle? I know that to use integration the formula is $\int_a^b f(x) \mathrm{d}x.$However, when I put this into my graphing calculator it doesn't w......Sat, 22 Aug 2026 09:32:37 -0400https://pop.com.sg/proving-monge-039-s-theorem-using-menelaus-039.htmlhttps://pop.com.sg/proving-monge-039-s-theorem-using-menelaus-039.htmlMonge's theorem states that for any three circles in a plane, none of which is completely inside one of the others, the intersection points of each of the three pairs of external tangent lines are ...Sat, 22 Aug 2026 09:28:35 -0400https://pop.com.sg/finding-a-joint-moment-generating-function.htmlhttps://pop.com.sg/finding-a-joint-moment-generating-function.htmlPlease consider the following problem and my solution to it: Problem: Let $(X,Y)$ be a continues bivariate r.v. with joint pdf \begin{eqnarray*} f_{XY}(x,y) &=& \begin{cases} e^{-(x+y)} &a......Sat, 22 Aug 2026 09:10:25 -0400https://pop.com.sg/norm-of-gradient-in-gradient-descent.htmlhttps://pop.com.sg/norm-of-gradient-in-gradient-descent.htmlThis question discusses the size of gradient in gradient descent. Some examples were pointed to show it is not necessarily the case that gradient will decrease, for example, $f(x) = \sqrt{|x|}$ or ......Sat, 22 Aug 2026 09:07:56 -0400https://pop.com.sg/how-does-the-reduced-cone-look-like.htmlhttps://pop.com.sg/how-does-the-reduced-cone-look-like.htmlI have to find in terms of standard spaces (that is interval, circle, disk, sphere, cone, etc.) the reduced cone (i.e. cone in category of pointed spaces) $\operatorname{Cone}(*)$ where $\{*\}$ is ......Sat, 22 Aug 2026 09:03:59 -0400https://pop.com.sg/binomial-distribution-word-problem.htmlhttps://pop.com.sg/binomial-distribution-word-problem.htmlA student gets at least 8 hours of sleep 45% of the nights; the sleeping schedule is independent from night to night. Let Xi indicate whether the student gets at least 8 hours of sleep during the n......Sat, 22 Aug 2026 09:00:50 -0400https://pop.com.sg/what-is-the-symbol-for-coincident-in-geometry.htmlhttps://pop.com.sg/what-is-the-symbol-for-coincident-in-geometry.htmlI am looking for a symbol to say that one geometrical figure coincides with another without writing the phrase "is coincident with." For example, the altitude $a$ of an equilateral triangle coinci......Sat, 22 Aug 2026 08:55:19 -0400https://pop.com.sg/get-polynomial-function-from-3-points.htmlhttps://pop.com.sg/get-polynomial-function-from-3-points.htmlI need to understand how to define a polynomial function from 3 given points. Everything I found on the web so far is either too complicated or the reversed way around. (how to get points with a gi......Sat, 22 Aug 2026 08:54:41 -0400https://pop.com.sg/how-does-squaring-both-sides-of-an-equation-lead-to-extraneous-solutio.htmlhttps://pop.com.sg/how-does-squaring-both-sides-of-an-equation-lead-to-extraneous-solutio.htmlLet’s say I have $x = x + 1$, which is a false statment for real $x$; why can I solve for real $x$ when I square both sides of the equation, giving $x^2=(x+1)^2$?...Sat, 22 Aug 2026 08:50:43 -0400https://pop.com.sg/convert-y-2-4-x-1-to-a-polar-equation.htmlhttps://pop.com.sg/convert-y-2-4-x-1-to-a-polar-equation.htmlI'm trying to convert the rectangular cartesian equation $$ y^2 = 4(x + 1) $$ to a polar equation. After replacing $y = r \sin \theta$ and $x = r \cos \theta$, I get $$ r^2 \sin^2 \theta = 4(r \......Sat, 22 Aug 2026 08:40:36 -0400https://pop.com.sg/what-is-the-perimeter-of-a-sector.htmlhttps://pop.com.sg/what-is-the-perimeter-of-a-sector.htmlI don't understand this. So we have: \begin{align} r &= 12 \color{gray}{\text{ (radius of circle)}} \\ d &= 24 \text{ (r}\times2) \color{gray}{\text{ (diameter of circle)}} \\ c &= 24\......Sat, 22 Aug 2026 08:37:00 -0400https://pop.com.sg/symbols-for-odd-and-even.htmlhttps://pop.com.sg/symbols-for-odd-and-even.htmlLet $A$ be a sequence of letters $\langle a,b,c,d,e,f \rangle$. I want to create two subsequences, one with the values with odd index and other with the values with even index: $A_\mathrm{odd} = \l......Sat, 22 Aug 2026 08:33:11 -0400https://pop.com.sg/context-free-grammar-for-palindrome-over-all-terminals.htmlhttps://pop.com.sg/context-free-grammar-for-palindrome-over-all-terminals.htmlI've found various examples of context-free grammars for palindromes but they all seem to hardcode the rules to be of the form "terminal Statement (same) terminal" For example, if $T \...Sat, 22 Aug 2026 08:32:49 -0400https://pop.com.sg/what-is-the-square-root-of-infinity-and-what-is-infinity-2.htmlhttps://pop.com.sg/what-is-the-square-root-of-infinity-and-what-is-infinity-2.htmlIn rigorous mathematics, how can one formalize the notion of a "square root of infinity", as well as that of "infinity squared"?...Sat, 22 Aug 2026 08:27:23 -0400https://pop.com.sg/what-is-the-best-calculus-book-for-a-strong-introdutory-course.htmlhttps://pop.com.sg/what-is-the-best-calculus-book-for-a-strong-introdutory-course.htmlI'm teaching for a small group of future undergraduate science students. I'm from Brazil, so here, we don't see any calculus in school, Just definition of function and some properties of most known ...Sat, 22 Aug 2026 08:26:03 -0400https://pop.com.sg/coin-toss-probability-using-bayes-theorem.htmlhttps://pop.com.sg/coin-toss-probability-using-bayes-theorem.htmlIn a box, there are the same number of two kinds of coins: the fair coins (50% chance for head) and the biased coins (70% chance for head). One person randomly selected a coin and tosses it twice. ......Sat, 22 Aug 2026 08:23:09 -0400https://pop.com.sg/2-easy-gre-questions.htmlhttps://pop.com.sg/2-easy-gre-questions.htmlI've been having trouble with these two questions. The first is simple interest, the second is rate. I'm sure they're easy but I can't focus on getting the solution because I'm terrible at focusing......Sat, 22 Aug 2026 08:19:15 -0400https://pop.com.sg/does-sum-limits-x-1-infty-sin-x-converge.htmlhttps://pop.com.sg/does-sum-limits-x-1-infty-sin-x-converge.htmlI received a task to find out whether the following series converges: $$\sum_{x=1}^\infty\sin(x)$$ On first look it seems simple, but as I keep thinking about it, there's not a single lemma or ...Sat, 22 Aug 2026 08:19:04 -0400https://pop.com.sg/what-is-the-minimum-possible-exterior-surface-area-of-an-open-glassed-.htmlhttps://pop.com.sg/what-is-the-minimum-possible-exterior-surface-area-of-an-open-glassed-.htmlThis is a question from Khan Academy I do not understand. An open-topped glass aquarium with a square base is designed to hold $32$ space cubic feet of water. What is the minimum possible exterior ...Sat, 22 Aug 2026 08:13:46 -0400https://pop.com.sg/finding-integer-solutions-to-y-2-x-3-2.htmlhttps://pop.com.sg/finding-integer-solutions-to-y-2-x-3-2.htmlI have the equation: $$y^2=x^3-2$$ It seems to be deceivingly simple, yet I simply cannot crack it. It is obviously equivalent to finding a perfect cube that is two more than a perfect square, and a ...Sat, 22 Aug 2026 08:11:45 -0400https://pop.com.sg/terminal-value-formula-in-the-discounted-cash-flow-dcf-valuation.htmlhttps://pop.com.sg/terminal-value-formula-in-the-discounted-cash-flow-dcf-valuation.htmlI have just started with Finance and have a physics background. However, I am struggling to understand the formula for the terminal value in a discounted cash flow valuation (DCF valuation) accordi......Sat, 22 Aug 2026 08:06:34 -0400https://pop.com.sg/disconnected-connected-graphs.htmlhttps://pop.com.sg/disconnected-connected-graphs.htmlDetermine whether the statements below are true or false. If the statement is true, then prove it; and if it is false, give a counterexample. (a) Every disconnected graph has a vertex of degree 0.......Sat, 22 Aug 2026 08:06:19 -0400https://pop.com.sg/finding-the-number-of-surfaces-containing-a-curve-given-its-parametriz.htmlhttps://pop.com.sg/finding-the-number-of-surfaces-containing-a-curve-given-its-parametriz.htmlIf there is a parametrized curve: $$r(t) = t^2\,i + ln(t)\,j + \frac{1}{t}\,k$$ then as I understand it, to find surfaces that contain the curve, you can solve for $t$ and find three equations, each ...Sat, 22 Aug 2026 08:00:20 -0400https://pop.com.sg/is-there-something-like-a-normal-distribution-model-for-discrete-proba.htmlhttps://pop.com.sg/is-there-something-like-a-normal-distribution-model-for-discrete-proba.htmlIf one wants to find the probability that a continuous random variable will fall within a range of $a \leq X \leq b$, based on a mean value $\mu$, and a deviation of $\sigma$, he would integrate the ...Sat, 22 Aug 2026 07:59:51 -0400https://pop.com.sg/finding-moment-of-inertia-rotational-inertia-i-using-integration.htmlhttps://pop.com.sg/finding-moment-of-inertia-rotational-inertia-i-using-integration.htmlI just came back from my Introduction to Rotational Kinematics class, and one of the important concepts they described was Rotational Inertia, or Moment of Inertia. It's basically the equivalent of......Sat, 22 Aug 2026 07:50:01 -0400https://pop.com.sg/conditional-introduction-rule.htmlhttps://pop.com.sg/conditional-introduction-rule.htmlIn the derivation (the image below) the author shows that given the premise $\neg S \land \neg J$, the conclusion is $S \implies J$. All these deductive maneuver for concluding implications I find ...Sat, 22 Aug 2026 07:48:29 -0400https://pop.com.sg/how-can-we-make-11-non-isomorphic-graphs-on-4-vertices.htmlhttps://pop.com.sg/how-can-we-make-11-non-isomorphic-graphs-on-4-vertices.htmlHow can we draw all the non-isomorphic graphs on $4$ vertices ? But it is mentioned that $ 11 $ graphs are possible....Sat, 22 Aug 2026 07:47:34 -0400https://pop.com.sg/how-are-asymptotes-actually-defined-in-rigorous-mathematics.htmlhttps://pop.com.sg/how-are-asymptotes-actually-defined-in-rigorous-mathematics.htmlThis question is coming from the analytic geometry viewpoint. Please ignore the viewpoint of algebraic geometry here, unless that viewpoint is somehow able to handle non-algebraic curves like $x \m......Sat, 22 Aug 2026 07:29:24 -0400https://pop.com.sg/find-the-value-of-the-expression.htmlhttps://pop.com.sg/find-the-value-of-the-expression.htmlThe expression $ax^2 + bx + 1$ takes the values $1$ and $4$ when $x$ takes the values $2$ and $3$ respectively. How can we find the value of the expression when $x$ takes the value of $4$?...Sat, 22 Aug 2026 07:16:04 -0400https://pop.com.sg/determining-whether-cartesian-first-order-tensor.htmlhttps://pop.com.sg/determining-whether-cartesian-first-order-tensor.htmlI'm struggeling to understand when we call a certain tuple a vector or first order Cartesian tensor $\mathbf v$. Riley, Hobson and Riley (3rd) states that when you apply a rotation, the new compone......Sat, 22 Aug 2026 07:13:08 -0400https://pop.com.sg/why-is-the-determinant-of-a-symplectic-matrix-1.htmlhttps://pop.com.sg/why-is-the-determinant-of-a-symplectic-matrix-1.htmlSuppose $A \in M_{2n}(\mathbb{R})$. and$$J=\begin{pmatrix} 0 & E_n\\ -E_n&0 \end{pmatrix}$$ where $E_n$ represents identity matrix. if $A$ satisfies $$AJA^T=J.$$ How to figure out $$\det......Sat, 22 Aug 2026 07:09:18 -0400https://pop.com.sg/the-interior-of-a-connected-set-in-mathbb-r-k.htmlhttps://pop.com.sg/the-interior-of-a-connected-set-in-mathbb-r-k.htmlIs the interior of a connected set in $\mathbb R^k$ connected?...Sat, 22 Aug 2026 07:02:22 -0400https://pop.com.sg/intuitively-what-is-the-difference-between-eigendecomposition-and-sing.htmlhttps://pop.com.sg/intuitively-what-is-the-difference-between-eigendecomposition-and-sing.htmlI'm trying to intuitively understand the difference between SVD and eigendecomposition. From my understanding, eigendecomposition seeks to describe a linear transformation as a sequence of three b......Sat, 22 Aug 2026 07:02:16 -0400https://pop.com.sg/i-have-a-d3-three-sided-die-and-i-m-trying-to-get-the-highest-possible.htmlhttps://pop.com.sg/i-have-a-d3-three-sided-die-and-i-m-trying-to-get-the-highest-possible.htmlAs the title says, if I roll a 2 on a d3 and am allowed to reroll it, is there an advantage to re-rolling? It seems like the choice doesn’t matter, or is purely preferential but what is wrong with ...Sat, 22 Aug 2026 07:01:38 -0400https://pop.com.sg/empty-and-trivial-graph-diestel-039-s-book.htmlhttps://pop.com.sg/empty-and-trivial-graph-diestel-039-s-book.htmlThe number of vertices of a graph $G$ is its order, written as $|G|$; For the empty graph $(\varnothing, \varnothing)$ we simply write $\varnothing$. A graph of order $0$ or $1$ is called trivial. ......Sat, 22 Aug 2026 06:54:17 -0400https://pop.com.sg/closure-of-integers.htmlhttps://pop.com.sg/closure-of-integers.htmlFellow Mathers, is the fact that the Integers are closed under addition and multiplication... ∀ a,b ∈ Z , a + b = c where c ∈ Z ∀ a,b ∈ Z , ab = c where c ∈ Z ...universal axioms, theorems, depen......Sat, 22 Aug 2026 06:50:13 -0400https://pop.com.sg/dense-sequence-definition.htmlhttps://pop.com.sg/dense-sequence-definition.htmlA subset $A$ of a space $X$ is called dense if every point $x$ in $X$ either belongs to $A$ or is a limit point of $A$ ; that is, the closure of $A$ constitutes the whole set $X$. That the definiti......Sat, 22 Aug 2026 06:45:56 -0400https://pop.com.sg/2-norm-vs-operator-norm.htmlhttps://pop.com.sg/2-norm-vs-operator-norm.htmlI have read that we define the "2-norm" of a matrix as $$\max_i \,{|\sigma_i|},$$ which I have also heard called the "operator norm" (here $\sigma_i$ are the singular values). Also we have the......Sat, 22 Aug 2026 06:33:32 -0400https://pop.com.sg/riemann-sum-given-partition-and-sample-points.htmlhttps://pop.com.sg/riemann-sum-given-partition-and-sample-points.htmlCalculate the Riemann sum $R(f,p,c)$ for the function $f(x)=3x^2+2x$. Partition $p={0,1,2.5,3.2,5}$ and sample points $c={0.5,2,3,4.5}$. Am able to find a Riemann sum whereby partitions have been ...Sat, 22 Aug 2026 06:33:27 -0400https://pop.com.sg/what-is-the-difference-between-cluster-point-and-limit-point.htmlhttps://pop.com.sg/what-is-the-difference-between-cluster-point-and-limit-point.htmlIn the book of real mathematical analysis written by Pugh, there is a sentence about these two and condensation: Thus, S limits at p, clusters at p, or condenses at p according to whether each Mr(p......Sat, 22 Aug 2026 06:30:31 -0400https://pop.com.sg/making-change-for-a-dollar-and-other-number-partitioning-problems.htmlhttps://pop.com.sg/making-change-for-a-dollar-and-other-number-partitioning-problems.htmlI was trying to solve a problem similar to the "how many ways are there to make change for a dollar" problem. I ran across a site that said I could use a generating function similar to the one quo......Sat, 22 Aug 2026 06:24:03 -0400https://pop.com.sg/what-does-lg-x-mean-is-it-log-2-x-or-log-10-x-in-binary-trees.htmlhttps://pop.com.sg/what-does-lg-x-mean-is-it-log-2-x-or-log-10-x-in-binary-trees.htmlI'm a bit confused, $\log_{10} x = \log x $ right? I believe I've read somewhere that $\log_{2} x = \lg x$ but some people say $\lg = \log$. So what does $\lg$ really stand for? specifically when t......Sat, 22 Aug 2026 06:14:56 -0400https://pop.com.sg/definition-of-the-ordered-triple-a-b-c-according-to-kuratowski-039-s-s.htmlhttps://pop.com.sg/definition-of-the-ordered-triple-a-b-c-according-to-kuratowski-039-s-s.htmlCan someone give Kuratowski's definition of the ordered triple $(a,b,c)$ assuming $A \times B \times C$ is rewritten as $(A \times B) \times C$, please? I noticed there is already an answered quest......Sat, 22 Aug 2026 06:03:49 -0400https://pop.com.sg/fourier-transform-of-sinc-function-confused-about-operatorname-sinc-pi.htmlhttps://pop.com.sg/fourier-transform-of-sinc-function-confused-about-operatorname-sinc-pi.htmlI am confused about the fourier transform of the $\operatorname{sinc}$ function. First I don't know if $$\operatorname{sinc} (x) = \frac{\sin(\pi x)}{(\pi x)}$$ or $$\operatorname{sinc} (\pi x......Sat, 22 Aug 2026 06:03:39 -0400https://pop.com.sg/order-of-operations-for-multiplying-three-matrices.htmlhttps://pop.com.sg/order-of-operations-for-multiplying-three-matrices.htmlIf I have a $1\times 2$ matrix $A$, a $2\times 2$ matrix $B$, and a $2\times 2$ matrix $C$, and am asked to calculate ABC, is there a specific order in which I have to carry out the multiplication? ...Sat, 22 Aug 2026 05:52:56 -0400https://pop.com.sg/find-the-area-of-a-segment-of-a-circle-given-radius-and-central-angle.htmlhttps://pop.com.sg/find-the-area-of-a-segment-of-a-circle-given-radius-and-central-angle.htmlFind the Area of a segment of a circle if the central angle of the segment is $105^\circ$ degrees and the radius is $70$. Formulas I have: Area of a non-right angle triangle= $\frac{1}{2}a b \sin......Sat, 22 Aug 2026 05:39:35 -0400https://pop.com.sg/a-semicircle-has-a-radius-of-2-m-determine-the-dimensions-of-a-rectang.htmlhttps://pop.com.sg/a-semicircle-has-a-radius-of-2-m-determine-the-dimensions-of-a-rectang.html$y^2 + x^2 = 4$ $A(x) = 2xy$ (make base of semicircle = $2x$) plug it in: $A(x) = (\space 2\sqrt{(4-y^2)}\space)\cdot y$ Final derivative: $$\begin{align} A'(x) & = \frac{-2y^2}{\sqrt{4-y^2......Sat, 22 Aug 2026 05:29:42 -0400https://pop.com.sg/understanding-when-a-graph-represents-a-function.htmlhttps://pop.com.sg/understanding-when-a-graph-represents-a-function.htmlWhich of the following is the graph of a function where $y = f(x)$? I got the answer for the above question is A function has one output for each input in its domain. Each $x$-value on this graph ...Sat, 22 Aug 2026 05:29:25 -0400https://pop.com.sg/deriving-the-cosine-formula-using-vectors.htmlhttps://pop.com.sg/deriving-the-cosine-formula-using-vectors.htmlHow did they go from 2b⋅c to -2bccosA? Where did they get the negative sign from?...Sat, 22 Aug 2026 05:21:12 -0400https://pop.com.sg/what-really-is-diagonalization.htmlhttps://pop.com.sg/what-really-is-diagonalization.htmlIf I have a square matrix $A$ representing a linear transformation $T:V\rightarrow V$ w.r.t the basis, $B=\{v_1,v_2..,v_n\}$ and $A$ is Hermitian. So we have $Av_n=\lambda_{n}v_n$ where $\{v_1,v_......Sat, 22 Aug 2026 05:16:43 -0400https://pop.com.sg/how-to-make-continued-fractions-of-any-number.htmlhttps://pop.com.sg/how-to-make-continued-fractions-of-any-number.htmlI recently found an continued fraction representation of $\pi$, and I wondered how can I make an continued fraction that converges into a number? The MAIN question is: how do you make a continued ...Sat, 22 Aug 2026 05:10:54 -0400https://pop.com.sg/if-ab-i-then-ba-i.htmlhttps://pop.com.sg/if-ab-i-then-ba-i.htmlIf $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$. I do not understand anything more than the following. Elementary row operations. Linear ...Sat, 22 Aug 2026 05:01:34 -0400https://pop.com.sg/why-does-the-area-of-a-rhombus-with-same-lengths-as-a-square-has-a-dif.htmlhttps://pop.com.sg/why-does-the-area-of-a-rhombus-with-same-lengths-as-a-square-has-a-dif.htmla rhombus can be created by dragging the four sides of the square to a certain angle. so how does the area of the new formed rhombus differ from the square.it can be proven mathematically but how d......Sat, 22 Aug 2026 05:01:08 -0400https://pop.com.sg/inductive-definition-of-a-set.htmlhttps://pop.com.sg/inductive-definition-of-a-set.htmlI'm a beginner in set theory and I have doubt regarding mathematical induction. I came across the following examples. Example 1: Find the set given by the following definition: 1) $ 3 \in P $ 2......Sat, 22 Aug 2026 04:55:46 -0400https://pop.com.sg/what-is-the-scientific-notation-of-zero.htmlhttps://pop.com.sg/what-is-the-scientific-notation-of-zero.htmlThis question was asked here, where the answer uses this description. The last line reads: "The special case of $0$ does not have a unique representation in scientific notation, i.e., $0=0×10^0=......Sat, 22 Aug 2026 04:50:32 -0400https://pop.com.sg/how-to-draw-hasse-diagram.htmlhttps://pop.com.sg/how-to-draw-hasse-diagram.htmlHow would you draw a Hasse diagram of the divisibility relation? when A = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15} Any help would be appreciated, thank you....Sat, 22 Aug 2026 04:48:43 -0400https://pop.com.sg/what-is-the-mathematical-formula-for-square-root.htmlhttps://pop.com.sg/what-is-the-mathematical-formula-for-square-root.htmlI wasn't at school when we were learning this, and I've forgot how to calculate a square root on paper using a formula? Can anyone please help me? What is the formula? I need this to write an alg......Sat, 22 Aug 2026 04:48:20 -0400https://pop.com.sg/can-all-maths-be-represented-in-geometry.htmlhttps://pop.com.sg/can-all-maths-be-represented-in-geometry.htmlI'm aware that many maths problems can be expressed and understood in geometry, for example, complex numbers can be expressed in 2 dimensions, that can be useful for some problems. Maths started in ...Sat, 22 Aug 2026 04:32:58 -0400https://pop.com.sg/how-to-use-eigenvalues-and-eigenvector-to-solve-systems-of-linear-equa.htmlhttps://pop.com.sg/how-to-use-eigenvalues-and-eigenvector-to-solve-systems-of-linear-equa.htmlA simple system of algebraic (not differential) equations such as: $$ 3 x + 4 y - 8 z = 23 $$ $$ -2 x + 8 y - 11 z = -32 $$ $$ -4 x + 9 y - 32 z = - 8 $$ can be written as: $$\mathbf{A} \vec x = ......Sat, 22 Aug 2026 04:32:31 -0400https://pop.com.sg/eulerian-graph-with-odd-even-vertices-edges.htmlhttps://pop.com.sg/eulerian-graph-with-odd-even-vertices-edges.htmlFind an Eulerian graph with an even/odd number of vertices and an even/odd number of edges or prove that there is no such graph (for each of the four cases). I came up with the graphs shown below ......Sat, 22 Aug 2026 04:31:34 -0400https://pop.com.sg/second-derivative-of-xy-with-respect-to-x.htmlhttps://pop.com.sg/second-derivative-of-xy-with-respect-to-x.html$$\frac{d^2}{dx^2}xy$$ I know it equals zero but I don't know the in-between steps. I'm using it to prove that Newton's Laws work in any frame of reference. So, say, two guys start from the same......Sat, 22 Aug 2026 04:31:18 -0400https://pop.com.sg/consensus-theorem-legal-to-use-redundant-terms-to-find-more-redundant-.htmlhttps://pop.com.sg/consensus-theorem-legal-to-use-redundant-terms-to-find-more-redundant-.htmlWhen using the Consensus Theorem in Boolean algebra to minimize an expression, is it a legal move to find and add a redundant term to the expression and then use that term to find more redundant te......Sat, 22 Aug 2026 04:15:25 -0400https://pop.com.sg/what-is-a-pullback-of-a-metric-and-how-does-it-work.htmlhttps://pop.com.sg/what-is-a-pullback-of-a-metric-and-how-does-it-work.htmlThe term "metric" is familiar, but not the idea of a pullback on it. I have tried to find intuitive, beginner-friendly explanations of this concept without success. Your attempts would be appreciat......Sat, 22 Aug 2026 04:14:50 -0400https://pop.com.sg/limit-of-x-ln-x.htmlhttps://pop.com.sg/limit-of-x-ln-x.htmlI am trying to solve $$\lim \limits_{x \to 0}x\ln{x}$$ which according to WolframAlpha (and Wikipedia) equals $0$. I managed to solve it by substituting such that $y = \dfrac{1}{x}$ and then using L'...Sat, 22 Aug 2026 04:02:32 -0400https://pop.com.sg/solving-x-3-y-3-xy-61-in-integers.htmlhttps://pop.com.sg/solving-x-3-y-3-xy-61-in-integers.htmlI'm trying to solve the following equation: $$x^3-y^3=xy+61$$ I got: $$(x-y)(x^2+xy+y^2)=xy+61$$ But I can't go any further. I am looking for a solution in integers. I need some hints to proce......Sat, 22 Aug 2026 03:59:19 -0400https://pop.com.sg/how-is-the-derivative-of-the-cdf-of-a-random-variable-x-its-pdf.htmlhttps://pop.com.sg/how-is-the-derivative-of-the-cdf-of-a-random-variable-x-its-pdf.htmlFor a continuos random variable $X$, if we have its p.d.f. $f(x)$, then the cumulative densitity function (c.d.f.) of $X$, $F(x)$, is $$F(x) = \int_{-\infty}^{x}f(t)dt$$ We also have $$F'(x) = \fra......Sat, 22 Aug 2026 03:51:03 -0400https://pop.com.sg/what-is-wrong-with-the-proof-for-ln-2-frac-1-2-ln-2.htmlhttps://pop.com.sg/what-is-wrong-with-the-proof-for-ln-2-frac-1-2-ln-2.htmlI have got a question which is as follows: Is $\ln(2)=\frac{1}{2}\ln(2)$?? The following argument seems suggesting that the answer is yes: We have the series $1-\frac{1}{2}+\frac{1}{3}-\fra......Sat, 22 Aug 2026 03:51:01 -0400https://pop.com.sg/variance-of-negative-binomial-distribution-without-moment-generating-s.htmlhttps://pop.com.sg/variance-of-negative-binomial-distribution-without-moment-generating-s.htmlGiven the discrete probability distribution for the negative binomial distribution in the form $$P(X = r) = \sum_{n\geq r} {n-1\choose r-1} (1-p)^{n-r}p^r$$ It appears there are no derivations o......Sat, 22 Aug 2026 03:50:23 -0400https://pop.com.sg/reducible-and-irreducible-polynomials-are-confusing-me.htmlhttps://pop.com.sg/reducible-and-irreducible-polynomials-are-confusing-me.htmlThe definition claims that a polynomial in a field of positive degree is a reducible polynomial when it can be written as the product of $2$ polynomials in the field with positive degrees. Other wi......Sat, 22 Aug 2026 03:38:17 -0400https://pop.com.sg/logic-behind-pisano-period.htmlhttps://pop.com.sg/logic-behind-pisano-period.htmlI just came to know about Pisano periods, and it's amazing application in computing modulus of large Fibonacci numbers. I know that a Pisano period periodically starts at $0,1$, but I haven't been ......Sat, 22 Aug 2026 03:16:12 -0400https://pop.com.sg/find-the-three-consecutive-numbers.htmlhttps://pop.com.sg/find-the-three-consecutive-numbers.htmlI have my problem here. I can't solve it on my own. I need a solution to prove my answer. Does anybody know this? thanks "Find the three consecutive numbers that have the property that the square ......Sat, 22 Aug 2026 03:15:31 -0400https://pop.com.sg/how-to-prove-that-sin-180-circ-theta-sin-theta.htmlhttps://pop.com.sg/how-to-prove-that-sin-180-circ-theta-sin-theta.htmlMi question is: How to prove $$\sin(180^\circ-\theta)=\sin\theta$$ ? Here, sine is defined for any angle such as 'alpha' This is the question mi college teacher asked me to derive it but i could ......Sat, 22 Aug 2026 03:14:25 -0400https://pop.com.sg/finding-taking-off-speed-help.htmlhttps://pop.com.sg/finding-taking-off-speed-help.htmlAn athlete jumps a horizontal distance of $7.18$m. He was airborne for $2.29$ seconds. The acceleration due to gravity is taken as $9.81$ms$^{-2}$. Assume that air resistance is negligible. Calcula......Sat, 22 Aug 2026 03:04:53 -0400