https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usMon, 31 Aug 2026 14:39:00 -0400https://pop.com.sg/integral-int-limits-0-pi-sin-x-cos-x-dx.htmlhttps://pop.com.sg/integral-int-limits-0-pi-sin-x-cos-x-dx.html$$\int \limits _0 ^\pi |\sin x + \cos x|\; dx$$ If I divide integral in two parts $\int\limits_0^{\frac{\pi}{2}}{(\sin x + \cos x)\,dx}$ and $\int\limits_{\frac{\pi}{2}}^\pi{(\sin x - \cos x)\,dx......Mon, 31 Aug 2026 18:32:37 -0400https://pop.com.sg/finding-general-formula-for-a-recursion-function.htmlhttps://pop.com.sg/finding-general-formula-for-a-recursion-function.htmlIf we have a recursion relation defined as $a_n = 3a_{n-1}+1$ with $a_1=1$ then find the general formula for $a_n$ in terms of $n$ with a(1) = 1. So far I have: $a_n = 3a_{n-2}+1+1 = 3a_{n-3}+1+1......Mon, 31 Aug 2026 18:27:07 -0400https://pop.com.sg/proving-the-fundamental-period-of-tangent.htmlhttps://pop.com.sg/proving-the-fundamental-period-of-tangent.htmlI'm very new to math and proofs -- so I apologize if my math skills and vocabulary offends you. I have a question that states: Prove that $\pi$ is a fundamental period of the tangent function. I ......Mon, 31 Aug 2026 18:20:27 -0400https://pop.com.sg/probability-of-drawing-5-cards-from-a-deck-of-52-that-will-have-the-sa.htmlhttps://pop.com.sg/probability-of-drawing-5-cards-from-a-deck-of-52-that-will-have-the-sa.htmlA standard deck of cards has 52 members consisting of 4 suits each with 13 members. Five cards are dealt from the randomly mixed deck. What is the probability that all cards are the same suit? E......Mon, 31 Aug 2026 18:13:17 -0400https://pop.com.sg/prove-convergent-sequences-are-bounded.htmlhttps://pop.com.sg/prove-convergent-sequences-are-bounded.htmlI don't understand this one part in the proof for convergent sequences are bounded. Proof: Let $s_n$ be a convergent sequence, and let $\lim s_n = s$. Then taking $\epsilon = 1$ we have: $n > N \...Mon, 31 Aug 2026 18:00:45 -0400https://pop.com.sg/isomorphism-classes-of-abelian-groups-of-certain-order.htmlhttps://pop.com.sg/isomorphism-classes-of-abelian-groups-of-certain-order.htmlworking on a practice question about finite abelian groups and just want to see if I am on the right track: Let $H = <(123)(4567),(8\space 9)(10\space 11),(8\space 11)(9\space 10) > \space......Mon, 31 Aug 2026 18:00:02 -0400https://pop.com.sg/independence-of-a-random-variable-x-from-itself.htmlhttps://pop.com.sg/independence-of-a-random-variable-x-from-itself.htmlIn our lecture on probability, my professor made the comment that "a random variable X is not independent from itself." (Here he was specifically talking about discrete random variables.) I asked h......Mon, 31 Aug 2026 17:44:58 -0400https://pop.com.sg/what-is-the-difference-between-p-a-b-and-p-a-b.htmlhttps://pop.com.sg/what-is-the-difference-between-p-a-b-and-p-a-b.htmlI feel that $p(a,b)$ = the probability that event $a$ and $b$ happen at the same time. $p(a|b)$ = the probability that event $a$ happens due to the event $b$ happens. For me, I think the meanin......Mon, 31 Aug 2026 17:42:39 -0400https://pop.com.sg/asymptotic-approximation-of-the-arctangent.htmlhttps://pop.com.sg/asymptotic-approximation-of-the-arctangent.htmlThat is, I am looking for an algebraic function $f(x)$ that approximates $\arctan x$ for large values of $x$. The approximation could be reasonably modest -- perhaps something like $$\tan (f(x))......Mon, 31 Aug 2026 17:39:52 -0400https://pop.com.sg/use-cylindrical-coordinates-to-find-the-volume-of-a-solid.htmlhttps://pop.com.sg/use-cylindrical-coordinates-to-find-the-volume-of-a-solid.htmlTo Find: use cylindrical coordinates to find the volume of a solid that is enclosed between the surface $r^2+z^2=20$ and the surface $z=r^2$: $$\iiint_R f(x,y,z)\,dv=\iiint_R f(r\cos\theta,r\sin\t......Mon, 31 Aug 2026 17:38:13 -0400https://pop.com.sg/definition-of-maximal-and-minimal-duplicate.htmlhttps://pop.com.sg/definition-of-maximal-and-minimal-duplicate.htmlPossible Duplicate: difference between maximal element and greatest element When I first encountered the terms maximal and minimal, I confused them with maximum and minimum. Many of my classma......Mon, 31 Aug 2026 17:36:44 -0400https://pop.com.sg/difference-between-basis-and-subbasis-in-a-topology.htmlhttps://pop.com.sg/difference-between-basis-and-subbasis-in-a-topology.htmlI was reading Topology from Munkres and got confused by the definition of a subbasis. What is/are the difference between basis and subbasis in a topology?...Mon, 31 Aug 2026 17:29:02 -0400https://pop.com.sg/definition-of-an-induced-matrix-norm.htmlhttps://pop.com.sg/definition-of-an-induced-matrix-norm.htmlCould someone explain the second equality in the definition of a induced matrix norm to me? Let $\| \cdot \|$ be a norm for $\cdot \in \mathbb{R}^n$, then the induced matrixnorm for $A\in \ma......Mon, 31 Aug 2026 17:27:02 -0400https://pop.com.sg/polynomial-whose-roots-adjoin-to-a-field-mathbb-q-sqrt6-to-give-an-ext.htmlhttps://pop.com.sg/polynomial-whose-roots-adjoin-to-a-field-mathbb-q-sqrt6-to-give-an-ext.htmlI want to find a polynomial whose roots adjoin to a field $\mathbb Q(\sqrt6)$ to give an extension field $\mathbb Q(\sqrt2,\sqrt3)$such that $[\mathbb Q(\sqrt2,\sqrt3):\mathbb Q(\sqrt6)]=2$. Is th......Mon, 31 Aug 2026 17:26:20 -0400https://pop.com.sg/finding-eccentricity-from-the-rotating-ellipse-formula.htmlhttps://pop.com.sg/finding-eccentricity-from-the-rotating-ellipse-formula.htmlI see that from a normal ellipse formula, we can acquire the eccentricity via this formula here. However, for this formula (1): $A(x − h)^2 + B(x − h)(y − k) + C(y − k)^2 = 1$ When parameter $......Mon, 31 Aug 2026 17:19:43 -0400https://pop.com.sg/proof-of-arctan-2-pi-2-arctan-1-2.htmlhttps://pop.com.sg/proof-of-arctan-2-pi-2-arctan-1-2.htmlI am studying undergraduate complex analysis, and in my Textbook the author claimed that $$\arctan(2)=\frac{\pi}{2}-\arctan\left(\frac{1}{2}\right)$$ when he was doing an example regarding to ...Mon, 31 Aug 2026 16:58:51 -0400https://pop.com.sg/hyperbolic-curve-and-hyperbola.htmlhttps://pop.com.sg/hyperbolic-curve-and-hyperbola.htmlDef: A hyperbolic curve is an algebraic curve obtained by removing $r$ points from a smooth, proper curve of genus $g,$ where $g$ and $r$ are nonnegative integers such that $2g−2+r > 0.$ How ......Mon, 31 Aug 2026 16:57:47 -0400https://pop.com.sg/solving-right-triangle-given-two-sides.htmlhttps://pop.com.sg/solving-right-triangle-given-two-sides.htmlI have a right triangle whose base has length 40 cm and whose hypotenuse has length 43 cm. How can I determine the height and the measures of the remaining two angles?...Mon, 31 Aug 2026 16:54:44 -0400https://pop.com.sg/what-is-the-exact-definition-of-a-reflexive-relation.htmlhttps://pop.com.sg/what-is-the-exact-definition-of-a-reflexive-relation.htmlWell... The three definitions I got for reflexive, symmetric, and transitive relations are: R is said to be reflexive on A if $\forall x\in A : xRx$. R is symmetric if $\forall x,y\in A : xRy \...Mon, 31 Aug 2026 16:48:33 -0400https://pop.com.sg/determine-the-signs-of-the-partial-derivatives-for-the-function-f-whos.htmlhttps://pop.com.sg/determine-the-signs-of-the-partial-derivatives-for-the-function-f-whos.htmlMy question is how am I supposed to determine what function that is? I think I could handle the partial derivatives after that.. I couldn't find an example showing how to identify this graph anywhe......Mon, 31 Aug 2026 16:38:19 -0400https://pop.com.sg/does-det-a-b-det-a-det-b-hold.htmlhttps://pop.com.sg/does-det-a-b-det-a-det-b-hold.htmlWell considering two $n \times n$ matrices does the following hold true: $$\det(A+B) = \det(A) + \det(B)$$ Can there be said anything about $\det(A+B)$? If $A/B$ are symmetric (or maybe even of th......Mon, 31 Aug 2026 16:37:19 -0400https://pop.com.sg/find-angle-in-triangle-with-incenter.htmlhttps://pop.com.sg/find-angle-in-triangle-with-incenter.htmlIn a triangle $ABC$ if $AD$, $BE$, $CF$ are the angle bisectors of $\angle A$, $\angle B$ $\angle C$ respectively with incenter $I$ and $\angle A=120^\circ$ Find $\angle DFI$ It's $30^\circ$. If $......Mon, 31 Aug 2026 16:33:22 -0400https://pop.com.sg/how-to-prove-sin-x-sin-y-2-sin-left-frac-x-y-2-right-cos-left-frac-x-y.htmlhttps://pop.com.sg/how-to-prove-sin-x-sin-y-2-sin-left-frac-x-y-2-right-cos-left-frac-x-y.htmlI am stuck at one task, it's to proof the following equation using addition theorems. From a draft I understood that the equation is correct or true. But that's it. What's a good way to explain it ...Mon, 31 Aug 2026 16:30:46 -0400https://pop.com.sg/multivariable-calculus-hard-problems-with-solutions.htmlhttps://pop.com.sg/multivariable-calculus-hard-problems-with-solutions.htmlI'm practicing for my multivariable calculus exam and I'm having some trouble mostly because I have no way of knowing if my solutions are correct or not. For example, a typical problem goes like t......Mon, 31 Aug 2026 16:23:44 -0400https://pop.com.sg/does-the-existence-of-orthocenter-proof-also-work-for-obtuse-triangles.htmlhttps://pop.com.sg/does-the-existence-of-orthocenter-proof-also-work-for-obtuse-triangles.htmlI need to prove the existence of an orthocenter for an obtuse triangle. I tried proving the existence of an orthocenter, meaning a point where the heights of $\triangle ABC$, where $[AB]=c, [AC]=b......Mon, 31 Aug 2026 16:05:19 -0400https://pop.com.sg/proving-the-klein-4-group-is-abelian.htmlhttps://pop.com.sg/proving-the-klein-4-group-is-abelian.htmlIf ker $f = \{ (1), (12)(34), (14)(23), (13)(24) \}$, which is the Klein 4 group, how do I prove that it is an Abelian group? Do I just show that each element in the ker $f$ is commutative to prove......Mon, 31 Aug 2026 15:58:18 -0400https://pop.com.sg/simple-diffy-q-problem.htmlhttps://pop.com.sg/simple-diffy-q-problem.htmlSo as a fun project, I'm trying to work my way through Kreyzig's "Advanced Engineering Mathematics". But I've gotten to a really simple problem: $$xy' = 2y$$ where I know the solution is $x^2$ but......Mon, 31 Aug 2026 15:58:17 -0400https://pop.com.sg/general-solution-for-a-nonhomogeneous-differential-equation.htmlhttps://pop.com.sg/general-solution-for-a-nonhomogeneous-differential-equation.htmlI have to find the general solution of this eq : $$y''-4y'+5y=e^{2s} $$ I have found the general solution of the homogeneous part of this eq. $$Y_h= e^{2s} ( C_1 \cos s - C_2 \sin s ) $$ I ...Mon, 31 Aug 2026 15:57:41 -0400https://pop.com.sg/can-someone-explain-the-concept-of-fundamental-domain-to-me.htmlhttps://pop.com.sg/can-someone-explain-the-concept-of-fundamental-domain-to-me.htmlHi for a Geometries course we're dealing with fundamental and I don't quite understand the definition of a fundamental domain. The definition in my book is that a fundamental domain: is a region......Mon, 31 Aug 2026 15:51:25 -0400https://pop.com.sg/why-are-there-3-solutions-in-this-trigonometric-equation-sin-theta-0.htmlhttps://pop.com.sg/why-are-there-3-solutions-in-this-trigonometric-equation-sin-theta-0.htmlmy question is Why are there 3 solutions in this trigonometric equation $sin\theta = 0$? I thought since the range for the angle is between 0 and 360 it has 2 solutions? I tried and I have got 2 ...Mon, 31 Aug 2026 15:47:55 -0400https://pop.com.sg/the-significance-of-linear-approximation.htmlhttps://pop.com.sg/the-significance-of-linear-approximation.htmlI want to know what makes linear approximation so important (or useful). What I am aware of in my current state of limited understanding is that linear approximation is one of the applications of a ...Mon, 31 Aug 2026 15:47:30 -0400https://pop.com.sg/constant-of-integration-of-the-integrating-factor.htmlhttps://pop.com.sg/constant-of-integration-of-the-integrating-factor.htmlI was reading my notes and at some point I write that in solving a first order linear DE using the integrating factor the final solution should have only one arvitrary constant arising from the ...Mon, 31 Aug 2026 15:44:02 -0400https://pop.com.sg/how-to-compute-dottie-number-accurately.htmlhttps://pop.com.sg/how-to-compute-dottie-number-accurately.htmlDottie number is root of this equation : $cos \alpha = \alpha$, $\alpha \approx 0.73908513321516064165531208767\dots$. I wonder how can I compute it ? I have tried to do it with an approximating f......Mon, 31 Aug 2026 15:43:19 -0400https://pop.com.sg/order-of-elements-in-bbb-z-24.htmlhttps://pop.com.sg/order-of-elements-in-bbb-z-24.htmlWhat elements of $\Bbb Z_{24}$ are order $2$? Order $3$? Order $4$? Order $6$? Order $2: 12$ Order $3: 8,16$ Order $4: 6,18$ Order $6: 4$ I had listed element $20$ as order $6$, but erased it ......Mon, 31 Aug 2026 15:31:33 -0400https://pop.com.sg/finding-domain-of-a-function-with-a-natural-logarithm-at-the-denominat.htmlhttps://pop.com.sg/finding-domain-of-a-function-with-a-natural-logarithm-at-the-denominat.htmlI have the function: $y = f(x) = \frac{x}{\ln x}$ The function is undefined for the conditions: a denominator of a fraction being zero. a logarithm being negative or equal to zero. Hence, is ......Mon, 31 Aug 2026 15:29:37 -0400https://pop.com.sg/cardinality-of-the-set-of-all-prime-factors-of-120-closed.htmlhttps://pop.com.sg/cardinality-of-the-set-of-all-prime-factors-of-120-closed.htmlI need to find the cardinality of the set of all prime factors of $120$. Will it be $16$? Since the set of all prime factors of $120$ will always have $16$ elements i.e $\{ 1, 2, 3, 4, 5, 6, 8, 10......Mon, 31 Aug 2026 15:28:20 -0400https://pop.com.sg/the-wald-test-with-poisson-distribution.htmlhttps://pop.com.sg/the-wald-test-with-poisson-distribution.htmlLet $X_1,\ldots, X_n\sim \operatorname{Poisson}(\lambda)$. Let $\lambda_w>0$ be given, I am trying to find the size $\alpha$ Wald test for $H_0$: $\lambda=\lambda_w$ vs $H_1$: $\lambda\neq \lamb......Mon, 31 Aug 2026 15:11:26 -0400https://pop.com.sg/find-the-residues-of-frac-1-sin-pi-z-closed.htmlhttps://pop.com.sg/find-the-residues-of-frac-1-sin-pi-z-closed.htmlUse Euler's formula $\sin \pi z =\frac{ e^{i\pi z} - e^{-i\pi z}}{2i}$ to find the residue of $\frac{1}{\sin \pi z}$. Show that the complex zeros of $\sin \pi z$ are exactly at the integers, and ......Mon, 31 Aug 2026 15:06:52 -0400https://pop.com.sg/probability-of-winning-a-game-of-craps.htmlhttps://pop.com.sg/probability-of-winning-a-game-of-craps.htmlThe dice game craps is played as follows: The player throws 2 dice, and if the sum is 7 or 11, he/she wins. If the sum is 2, 3, or 12, he/she loses. If the sum is anything else, then he/she continues ...Mon, 31 Aug 2026 15:05:09 -0400https://pop.com.sg/converting-an-equation-from-cartesian-to-polar-form.htmlhttps://pop.com.sg/converting-an-equation-from-cartesian-to-polar-form.htmlI'm trying to convert $x^2 +y^2 =(2-x)^2$ into a polar equation in the form $r=f(\theta)$. The answer is apparently $r=\frac{2}{1+cos(\theta)}$, but I can't seem to get this....Mon, 31 Aug 2026 14:51:09 -0400https://pop.com.sg/how-to-evaluate-the-complex-logarithms-log-i-and-log-3-4i.htmlhttps://pop.com.sg/how-to-evaluate-the-complex-logarithms-log-i-and-log-3-4i.htmlHow to evaluate $$\log(i)?$$ Also, how to evaluate $$\log(3+4i)?$$ I am reading complex analysis and I know that logarithm is a multibranched function and is periodic. What I have for the definitio......Mon, 31 Aug 2026 14:47:23 -0400https://pop.com.sg/non-isosceles-triangles-with-vertices-in-a-20-sided-regular-polygon.htmlhttps://pop.com.sg/non-isosceles-triangles-with-vertices-in-a-20-sided-regular-polygon.htmlSuppose $A_1A_2 . . . A_{20}$ is a $20-$sided regular polygon. How many non-isosceles (scalene) triangles can be formed whose vertices are among the vertices of the polygon but whose sides are no......Mon, 31 Aug 2026 14:46:15 -0400https://pop.com.sg/random-sample-vs-simple-random-sample.htmlhttps://pop.com.sg/random-sample-vs-simple-random-sample.htmlI am reading, just for fun, the book Essentials of Statististics of Mario Triola. I am trying to see the differences between Random Sample and Simple Random Sample. In the book I found these ...Mon, 31 Aug 2026 14:42:59 -0400https://pop.com.sg/how-do-find-if-a-relation-is-a-function-algebraically.htmlhttps://pop.com.sg/how-do-find-if-a-relation-is-a-function-algebraically.htmlIs there a way to see if a relation is a function without having to do a "vertical line test" (where you draw a vertical line on the graph and if there line touches two points then it's not a funct......Mon, 31 Aug 2026 14:42:48 -0400https://pop.com.sg/what-does-smooth-curve-mean.htmlhttps://pop.com.sg/what-does-smooth-curve-mean.htmlIn this problem, I know that the hypothesis of Green's theorem must ensure that the simple closed curve is smooth, but what is smooth? Could you give a definition and an intuitive explanation?...Mon, 31 Aug 2026 14:42:02 -0400https://pop.com.sg/linear-algebra-understanding-the-column-picture.htmlhttps://pop.com.sg/linear-algebra-understanding-the-column-picture.htmlI'm proceeding through MIT OCW's 18.06 class in Linear Algebra, and I've reached a sticking point on the first lecture - I was wondering if someone can offer some clarification on a specific point ......Mon, 31 Aug 2026 14:38:45 -0400https://pop.com.sg/what-is-the-angle-of-b.htmlhttps://pop.com.sg/what-is-the-angle-of-b.htmlSo first off, I know how to find the missing length of the leg of the triangle using the pythagorean theorem. $6^2 + b^2 = c^2$ $36 + b^2 = 100$ $100 - 36 = 64$ $\sqrt{64} = 8$. So angle angle $......Mon, 31 Aug 2026 14:31:39 -0400https://pop.com.sg/the-precise-meaning-of-the-limit-that-defines-the-integral.htmlhttps://pop.com.sg/the-precise-meaning-of-the-limit-that-defines-the-integral.htmlI'm having some trouble understanding what the following definition is saying, i think i have a basic understanding but would appreciate some clarification on the technical aspects. The precise mea......Mon, 31 Aug 2026 14:26:13 -0400https://pop.com.sg/what-is-the-contrapositive-of-the-definition-of-onto.htmlhttps://pop.com.sg/what-is-the-contrapositive-of-the-definition-of-onto.htmlI feel that the problem I am working on will be easier to prove by contrapositive, namely instead of showing surjective I want to show the contrapositive of surjective....Mon, 31 Aug 2026 14:04:42 -0400https://pop.com.sg/prove-that-triangle-amc-is-an-isosceles-right-triangle.htmlhttps://pop.com.sg/prove-that-triangle-amc-is-an-isosceles-right-triangle.htmlIn $\triangle ABC$ construct squares $ABDE$ and $BCFG$. Let $M$ be the midpoint of $EF$ then, prove that $\triangle AMC$ is an isosceles right triangle. I was able to prove that $\triangle AMC$ is......Mon, 31 Aug 2026 13:57:38 -0400https://pop.com.sg/limit-as-x-goes-to-infinity-of-absolute-value.htmlhttps://pop.com.sg/limit-as-x-goes-to-infinity-of-absolute-value.htmlI'm looking at $\lim_{x \to \infty} \frac{x}{|x|}$ and my intuition says it's supposed to be 1, which is confirmed by wolframalpha. Yet I'm confused how I'm supposed to prove it's the case when ope......Mon, 31 Aug 2026 13:46:24 -0400https://pop.com.sg/an-example-of-an-operation-in-ordinary-arithmetic-that-is-idempotent.htmlhttps://pop.com.sg/an-example-of-an-operation-in-ordinary-arithmetic-that-is-idempotent.htmlI came across this question in the book Axiomatic set theory by Suppes: Can you give an example of an operation of ordinary arithmetic which is idempotent? So, here I have some things that I ca......Mon, 31 Aug 2026 13:41:19 -0400https://pop.com.sg/solution-of-a-dirichlet-problem-on-the-unit-disk.htmlhttps://pop.com.sg/solution-of-a-dirichlet-problem-on-the-unit-disk.htmlFind the solution of the Dirichlet problem: $$\Delta u(r,\phi)=0, r<1, u(1,\phi)=f(\phi)$$ where $x=r\cos\phi$ and $y=r\sin\phi$ and $$f(\phi)=\sin^3(\phi).$$ I start by doing the following: ...Mon, 31 Aug 2026 13:40:15 -0400https://pop.com.sg/what-is-sin-pi-2-iln2.htmlhttps://pop.com.sg/what-is-sin-pi-2-iln2.htmlI have tried this is a couple of different ways and have different answers. $\sin(\pi/2+i\ln2)=\cos(i\ln2)=\cosh(\ln2)=\frac{e^{\ln2}+e^{-\ln2}}{2}=5/4$ $\sin(\pi/2+i\ln2)={\rm Im}(e^{i(\pi/2+i\ln......Mon, 31 Aug 2026 13:37:13 -0400https://pop.com.sg/understanding-field-extension.htmlhttps://pop.com.sg/understanding-field-extension.htmlI have some questions concerning field extensions, which I hope someone can help me with. 1) Understanding $K(\alpha)$. Let $L,K$ be fields, where $K$ is a subfield of $L$. I know that $K(\alpha)$......Mon, 31 Aug 2026 13:36:15 -0400https://pop.com.sg/pendulum-with-oscillating-fulcrum-equations-of-motion.htmlhttps://pop.com.sg/pendulum-with-oscillating-fulcrum-equations-of-motion.htmlA pendulum with a horizontally oscillating fulcrum can be described using the Lagrange formalism (related to a setting, as it is shown e.g. here) by the following system of nonlinear ordinary ...Mon, 31 Aug 2026 13:32:30 -0400https://pop.com.sg/intuition-for-polynomial-long-division.htmlhttps://pop.com.sg/intuition-for-polynomial-long-division.htmlSo I am trying to learn some basic algebra and go over some precalc material and I am terribly confused on why or how polynomial long division works. I have looked at many sources online and they s......Mon, 31 Aug 2026 13:22:18 -0400https://pop.com.sg/length-of-tangent-from-one-circle-to-another.htmlhttps://pop.com.sg/length-of-tangent-from-one-circle-to-another.htmlShow that length of tangent from any point on the circle$ x^2 + y^2 + 2gx + 2fy +c=0 $ to the circle $x^2 + y^2 + 2gx + 2fy +c_1 =0 $ is $\sqrt {c}-c_1$...Mon, 31 Aug 2026 13:21:03 -0400https://pop.com.sg/proof-of-and-requirements-for-the-reverse-of-jensen-039-s-inequality-f.htmlhttps://pop.com.sg/proof-of-and-requirements-for-the-reverse-of-jensen-039-s-inequality-f.htmlAs I understand it, Jensen's Inequality states $$\int_{U}f_{V}\left(h(u)g(u)\right)du\geq f_{V}\left(\int_{U}h(u)g(u)du\right)$$ For a convex function $f_{V}$, a probability distribution $g(u)$ o......Mon, 31 Aug 2026 13:16:11 -0400https://pop.com.sg/karnaugh-map-and-circuit-of-a-full-adder.htmlhttps://pop.com.sg/karnaugh-map-and-circuit-of-a-full-adder.htmlI have the following task: The addition can be implemented by the rules 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 10. Full addition requires carry-in and carry-out bits. In general, you must b......Mon, 31 Aug 2026 13:15:11 -0400https://pop.com.sg/proving-that-that-f-x-cos-x-sin-tan-x-if-x-neq-pi-2-is-continuous.htmlhttps://pop.com.sg/proving-that-that-f-x-cos-x-sin-tan-x-if-x-neq-pi-2-is-continuous.htmlI'm struggling to solve this one rigurously. I'm given a function $ f(x) $ defined as follows: $$ f(x)= \begin{cases} \cos(x)\sin(\tan(x)),& \text{if } x\neq \frac{\pi}{2}\\ 0, ......Mon, 31 Aug 2026 13:05:23 -0400https://pop.com.sg/linear-algebra-dimension-theorem.htmlhttps://pop.com.sg/linear-algebra-dimension-theorem.htmlSuppose we have a vector space $V$, and $U$, $W$ subspaces of $V$. Dimension theorem states: $$ \dim(U+W)=\dim U+ \dim W - \dim (U\cap W).$$ My question is: Why is $U \cap W$ necessary in this ...Mon, 31 Aug 2026 13:04:35 -0400https://pop.com.sg/if-3-gon-and-5-gon-are-constructible-show-that-15-gon-is-too.htmlhttps://pop.com.sg/if-3-gon-and-5-gon-are-constructible-show-that-15-gon-is-too.htmlUse the fact that the regular $3-$gon and the regular $5-$gon are constructible to show that the regular $15-$gon is constructible. What is the best way to prove this? I have found a theorem that ...Mon, 31 Aug 2026 13:04:20 -0400https://pop.com.sg/inner-product-space-parallelogram-law.htmlhttps://pop.com.sg/inner-product-space-parallelogram-law.htmlKind of confused what is being asked here. Isn't this just obviously true? How would I start this proof?...Mon, 31 Aug 2026 13:00:17 -0400https://pop.com.sg/can-i-get-multivariable-taylor-series-expansion-on-wolfram-alpha-or-ma.htmlhttps://pop.com.sg/can-i-get-multivariable-taylor-series-expansion-on-wolfram-alpha-or-ma.htmlI need something like this : say $a, x, y$ and $t$ are three non-negative real numbers. Now define the complex number, $z = -y + i(a+x-t)$ and consider the function $f(x,y) = z\text{ }tanh (\pi z) ......Mon, 31 Aug 2026 12:46:58 -0400https://pop.com.sg/what-is-a-physical-dimension-in-the-sense-of-dimensional-analysis.htmlhttps://pop.com.sg/what-is-a-physical-dimension-in-the-sense-of-dimensional-analysis.htmlMathematically speaking, what does it mean to say that a physical quantity is some numerical value with a “dimension” associated with it? When we say that the velocity of light is some constant, c ......Mon, 31 Aug 2026 12:44:38 -0400https://pop.com.sg/recursively-defined-set-subset-proof.htmlhttps://pop.com.sg/recursively-defined-set-subset-proof.htmlConsider the subset $S$ of the set of integers recursively defined by BASIS STEP: $3 \in S$. RECURSIVE STEP: If $x \in S$ and $y \in S$, then $x+y \in S$. Q: Show that the set $S$ is the......Mon, 31 Aug 2026 12:38:24 -0400https://pop.com.sg/proving-the-complementation-law-for-sets.htmlhttps://pop.com.sg/proving-the-complementation-law-for-sets.htmlI am new to Discrete Mathematics, and have been asked to prove the Complementation Law for sets, that is: $\overline {(\overline A)} \equiv A$. Our teacher advised us to turn the sets into proposit......Mon, 31 Aug 2026 12:33:09 -0400https://pop.com.sg/the-graph-has-10-vertices-and-47-edges-closed.htmlhttps://pop.com.sg/the-graph-has-10-vertices-and-47-edges-closed.htmlThe graph has $10$ vertices and $47$ edges. Does such a graph actually exist?...Mon, 31 Aug 2026 12:32:24 -0400https://pop.com.sg/logarithmic-differentiation-of-some-functions.htmlhttps://pop.com.sg/logarithmic-differentiation-of-some-functions.htmlGiven $y=f(x)$, where $f(x) $ is a positive function, we can write $\ln y = \ln f(x) $. Now let's say that $f$ takes zero values at certain points in an interval. At these points, the natural logar......Mon, 31 Aug 2026 12:32:22 -0400https://pop.com.sg/nested-summation-and-product-notation-of-two-expressions.htmlhttps://pop.com.sg/nested-summation-and-product-notation-of-two-expressions.htmlCould someone please tell me how I should evaluate these two expressions together? I understand how each notation should be evaluated separately but not together. Also, would the way you evaluate c......Mon, 31 Aug 2026 11:59:15 -0400https://pop.com.sg/proving-rolle-039-s-theorem.htmlhttps://pop.com.sg/proving-rolle-039-s-theorem.htmlTrying to prove Rolle's Theorem, which says that for a function $f$ continuous over $[a,b]$ and differentiable over $(a,b)$ (no idea why the endpoints aren't included here), such that $f(a) = f(b)$......Mon, 31 Aug 2026 11:59:04 -0400https://pop.com.sg/proof-of-an-abstract-bayes-039-theorem.htmlhttps://pop.com.sg/proof-of-an-abstract-bayes-039-theorem.htmlIn Björk (2009) a Bayes' theorem is given by Assume that $X$ is a random variable on $(\Omega, \mathcal F, P)$ and let $Q$ be another probability measure on $(\Omega, \mathcal F)$ with Radon-Ni......Mon, 31 Aug 2026 11:53:41 -0400https://pop.com.sg/solving-a-integro-differential-equation.htmlhttps://pop.com.sg/solving-a-integro-differential-equation.htmlI'm trying to solve the following problem, but my solution doesn't seem to be correct. Could I be rewriting the integral incorrectly? $$ \frac{dy}{dt} + 25 \int_0^t{y(t-w)e^{-10w}dw} = L[4]; y(0)=......Mon, 31 Aug 2026 11:49:33 -0400https://pop.com.sg/an-easy-example-of-a-non-constructive-proof-without-an-obvious-fix.htmlhttps://pop.com.sg/an-easy-example-of-a-non-constructive-proof-without-an-obvious-fix.htmlI wanted to give an easy example of a non-constructive proof, or, more precisely, of a proof which states that an object exists, but gives no obvious recipe to create/find it. Euclid's proof of the ...Mon, 31 Aug 2026 11:44:25 -0400https://pop.com.sg/understanding-euclid-039-s-proof-that-the-number-of-primes-is-infinite.htmlhttps://pop.com.sg/understanding-euclid-039-s-proof-that-the-number-of-primes-is-infinite.htmlIn Euclid's proof, if $p_1, p_2, \dots, p_n$ are the only primes then $p_1 \times p_2 \times \dots \times p_n + 1$ is not divisible by any of $p_1, p_2, \dots, p_n$ (because of some algebraic facts), ...Mon, 31 Aug 2026 11:39:48 -0400https://pop.com.sg/is-there-any-symbol-for-undefined-duplicate.htmlhttps://pop.com.sg/is-there-any-symbol-for-undefined-duplicate.htmlFor example we have $\frac{0}{0}$ which is undefined or we have a multiplication of $2\times2$ and $3\times3$ matrices which is also undefined. Is there any symbol for representing it?...Mon, 31 Aug 2026 11:21:51 -0400https://pop.com.sg/write-the-following-in-the-form-of-ax-b.htmlhttps://pop.com.sg/write-the-following-in-the-form-of-ax-b.htmlWrite the following system of equations in the form $AX = B$, and calculate the solution using the equation $X = A^{-1}B$. $$2x - 3y = - 1$$ $$-5x +5y = 20$$ I'm not the strongest at linear alge......Mon, 31 Aug 2026 11:20:47 -0400https://pop.com.sg/what-does-long-term-relative-frequency-mean.htmlhttps://pop.com.sg/what-does-long-term-relative-frequency-mean.htmlI'm reading an introductory statistics textbook and it mentioned this: "The probability of any outcome is the long-term relative frequency of that outcome. Probabilities are between zero and one, ...Mon, 31 Aug 2026 11:11:07 -0400https://pop.com.sg/solving-the-general-cubic-by-completing-the-cube.htmlhttps://pop.com.sg/solving-the-general-cubic-by-completing-the-cube.htmlCompleting the Square: In a quadratic of the form $x^2+ax+b=0$, we can complete the square by moving $b$ to the right hand side and finding a constant $m$ such that the left hand side becomes a pe......Mon, 31 Aug 2026 11:09:52 -0400https://pop.com.sg/what-is-the-history-behind-the-development-of-the-term-coefficient-clo.htmlhttps://pop.com.sg/what-is-the-history-behind-the-development-of-the-term-coefficient-clo.htmlWhy are coefficients called "coefficients"? For example I learned that squaring a number is called "squaring" because it actually refers to "making a square". That's how it was developed. |<-......Mon, 31 Aug 2026 10:59:07 -0400https://pop.com.sg/simplifying-nested-square-roots-sqrt-6-4-sqrt-2-sqrt-2.htmlhttps://pop.com.sg/simplifying-nested-square-roots-sqrt-6-4-sqrt-2-sqrt-2.htmlI guess I learned it many years ago at school, but I must have forgotten it. From a geometry puzzle I got to the solution $\sqrt{6-4\sqrt{2}} + \sqrt{2}$ My calculator tells me that (within its ...Mon, 31 Aug 2026 10:54:06 -0400https://pop.com.sg/using-integration-by-parts-to-evaluate-an-integrals.htmlhttps://pop.com.sg/using-integration-by-parts-to-evaluate-an-integrals.htmlCan't understand how to solve this math: use integration by parts to evaluate this integrals: $$\int x\sin(2x + 1) \,dx$$ can any one solve this so i can understand how to do this! Thanks :)...Mon, 31 Aug 2026 10:50:29 -0400https://pop.com.sg/the-square-of-an-integer-is-congruent-to-0-or-1-mod-4.htmlhttps://pop.com.sg/the-square-of-an-integer-is-congruent-to-0-or-1-mod-4.htmlThis is a question from the free Harvard online abstract algebra lectures. I'm posting my solutions here to get some feedback on them. For a fuller explanation, see this post. This problem is from ...Mon, 31 Aug 2026 10:49:13 -0400https://pop.com.sg/expectation-of-the-function-of-multiple-random-variables.htmlhttps://pop.com.sg/expectation-of-the-function-of-multiple-random-variables.htmlI am working through some basic probability stuff and have a question regarding functions of multiple variables. If I have two random variables $X,Y$ which have some joint probability distribution ......Mon, 31 Aug 2026 10:45:04 -0400https://pop.com.sg/whats-the-difference-between-logical-consequence-entailment-and-simple.htmlhttps://pop.com.sg/whats-the-difference-between-logical-consequence-entailment-and-simple.htmlAccording to Wikipedia: A logical consequence is the relationship between statements that holds true when one logically "follows from" one or more others. So, A ⊨ B B is a logical consequen......Mon, 31 Aug 2026 10:42:54 -0400https://pop.com.sg/definition-of-a-complete-metric-space.htmlhttps://pop.com.sg/definition-of-a-complete-metric-space.htmlI am having a little trouble understanding the definition of a complete set, which is the following : a metric space is said to be complete if every fundamental sequence converges in the space X. ......Mon, 31 Aug 2026 10:40:16 -0400https://pop.com.sg/what-is-the-difference-between-a-hermitian-inner-product-and-an-inner-.htmlhttps://pop.com.sg/what-is-the-difference-between-a-hermitian-inner-product-and-an-inner-.htmlAre there any difference? Or is the Hermitian inner product just a special case of an inner product on a complex space?...Mon, 31 Aug 2026 10:30:52 -0400https://pop.com.sg/can-a-directional-derivative-be-negative.htmlhttps://pop.com.sg/can-a-directional-derivative-be-negative.htmlI just calculated the directional derivative of a function and ended up with $- \frac{1}{3}$ which seems counterintuitive. Can this be possible?...Mon, 31 Aug 2026 10:28:02 -0400https://pop.com.sg/what-is-the-proper-notation-for-the-indefinite-integral-of-an-area.htmlhttps://pop.com.sg/what-is-the-proper-notation-for-the-indefinite-integral-of-an-area.htmlIm not exactly experienced with Calculus and math notation but I had question on Leibniz's notation: If we have an equation $dy=2dx$, and we wanted to find $y$, we would take the integral of both s......Mon, 31 Aug 2026 10:27:09 -0400https://pop.com.sg/uniqueness-of-a-complement-of-a-subgroup.htmlhttps://pop.com.sg/uniqueness-of-a-complement-of-a-subgroup.htmllet $A$ and $B$ be two subgroups of $G$. we say that $B$ is a complement of $A$ if : $G=AB$ $A\cap B=\{1\}$ Given a subgroup $A$ of $G$ i don't see how the complement $B$ of $A$ in $G$ is not uni......Mon, 31 Aug 2026 10:25:51 -0400https://pop.com.sg/solution-of-the-equation-below.htmlhttps://pop.com.sg/solution-of-the-equation-below.htmlThe solution of the equation $\sin 7x + \cos 2x = -2$ is/are? My Approach: For the above equation to hold true, Both $\sin 7x$ and $\cos 2x$ have to be -1. $$\sin 7x = -1$$ $$x = \frac{n\pi}{7} +......Mon, 31 Aug 2026 10:23:25 -0400https://pop.com.sg/when-does-a-linear-equation-have-infinitely-many-solutions.htmlhttps://pop.com.sg/when-does-a-linear-equation-have-infinitely-many-solutions.htmlFor which values of $k$ the equation $(k+8)x-3k = 3k(1-x)-6$ has infinitely many solutions? Since this is not a set of linear equations you can't use the augmented matrix and Gaussian elimination,......Mon, 31 Aug 2026 10:20:41 -0400https://pop.com.sg/how-can-i-find-values-that-make-a-matrix-linearly-dependent.htmlhttps://pop.com.sg/how-can-i-find-values-that-make-a-matrix-linearly-dependent.htmlLet $$A=\left(\begin{smallmatrix}2&-1&0&1\\0&0&a&3\\0&0&0&b\end{smallmatrix}\right)$$ For what choices (if any) of real numbers a and b are the rows of A linearly ...Mon, 31 Aug 2026 10:20:07 -0400https://pop.com.sg/arc-length-with-vector-valued-functions.htmlhttps://pop.com.sg/arc-length-with-vector-valued-functions.html"Consider the path of a particle in a conservative force field represented by the vector-valued function $r(t) = \langle 4(\sin t - t \cos t), 4(\sin t + t \sin t), (\frac{3}{2})t^2 \rangle$." "A)......Mon, 31 Aug 2026 10:05:44 -0400https://pop.com.sg/039-does-not-necessarily-equal-039-symbol.htmlhttps://pop.com.sg/039-does-not-necessarily-equal-039-symbol.htmlWhat symbol would I use if I wanted to express that, in the context of some binary relation $P$ implied from context, that $\exists (a,b)\in P: a\ne b$, but not to the extent that $\forall (a,b) \i......Mon, 31 Aug 2026 09:55:33 -0400https://pop.com.sg/is-the-integral-of-the-sum-really-the-sum-of-the-integrals.htmlhttps://pop.com.sg/is-the-integral-of-the-sum-really-the-sum-of-the-integrals.htmlI was asked to find the mclaurin series of $\int_0^x\frac{\arctan (t)}{t}dt$ using the known mclaurin for arctan: $\arctan(t)=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}t^{2n-1}}{2n-1}$ Ok, so what I di......Mon, 31 Aug 2026 09:51:06 -0400https://pop.com.sg/how-to-solve-30-37-mod-77-without-calculator.htmlhttps://pop.com.sg/how-to-solve-30-37-mod-77-without-calculator.htmlI tried doing something like this: $$(30^2)^{15}(30^7)\mod 77$$ but it is not effective, maybe someone knows some tips and tricks to solve this ?...Mon, 31 Aug 2026 09:48:36 -0400https://pop.com.sg/solve-for-f-x-if-f-f-x-6x-f-x.htmlhttps://pop.com.sg/solve-for-f-x-if-f-f-x-6x-f-x.htmlIf $f: [0,\infty) \rightarrow [0,\infty)$ and $f(f(x))=6x-f(x)$ $f(x)>0$ $ \forall x \in (0,\infty) $ Find f(x)...Mon, 31 Aug 2026 09:32:04 -0400https://pop.com.sg/levi-civita-connection-and-riemannian-metrics.htmlhttps://pop.com.sg/levi-civita-connection-and-riemannian-metrics.htmlI'm trying to solve the following problem in preparation for an exam : (i) Let $g_0$ be a Riemannian metric and define $g=cg_0$ where $c$ is a positive constant. Prove that $g_0$ and $g$ have the ......Mon, 31 Aug 2026 09:09:06 -0400