https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usSun, 23 Aug 2026 02:40:46 -0400https://pop.com.sg/linearity-of-determinant.htmlhttps://pop.com.sg/linearity-of-determinant.htmlLinearity property of Determinant function: For an $n\times n$ matrix $A$, we can consider det$A$ as a function of the $n$ column vectors in $A$. We will show that if all columns except one are held ...Sun, 23 Aug 2026 06:35:15 -0400https://pop.com.sg/what-is-symmetric-about-set-symmetric-difference.htmlhttps://pop.com.sg/what-is-symmetric-about-set-symmetric-difference.htmlUsing $\Delta$ for set symmetric difference, $A \Delta B$ is all the elements in exactly one of the sets but not all of them. $A \Delta B \Delta C $ is all the elements in exactly one of the sets......Sun, 23 Aug 2026 06:33:52 -0400https://pop.com.sg/how-to-express-the-given-disk-d-in-polar-coordinates.htmlhttps://pop.com.sg/how-to-express-the-given-disk-d-in-polar-coordinates.htmlGiven $D=\{(x,y)|0\le x\le2,0\le y\le\sqrt{4-x^2}\}$ Express $D$ in polar coordinates. I have the answer as $D=\{(r,\theta)|0\le\theta\le\dfrac{\pi}{2},0\le r\le2\}$ One more example is $D=\{(x,y......Sun, 23 Aug 2026 06:15:49 -0400https://pop.com.sg/on-integration-when-solving-differential-equations-specifically-separa.htmlhttps://pop.com.sg/on-integration-when-solving-differential-equations-specifically-separa.htmlSo here is the differential equation and inititial conditions: $$x \frac{\mathrm{d}y}{\mathrm{d}x}=y(3−y) $$ and $$y(2) = 2$$ We have to find the equation $y$ in terms of $x ~~[y(x)]$ that is a so......Sun, 23 Aug 2026 06:05:20 -0400https://pop.com.sg/height-of-an-isosceles-trapezoid.htmlhttps://pop.com.sg/height-of-an-isosceles-trapezoid.htmlSuppose we have an isosceles trapezoid whose length of the associated square is $20$ and the length of the hypotenuse of the triangles is $30$. I want to determine if it is possible to find the hei......Sun, 23 Aug 2026 06:05:04 -0400https://pop.com.sg/learning-mathematics-using-only-audio.htmlhttps://pop.com.sg/learning-mathematics-using-only-audio.htmlAre there any mathematics audio books or other audio sources for learning mathematics, like for example math podcasts which really go into detail. I ask this because I make about 1 hour from my hou......Sun, 23 Aug 2026 05:59:05 -0400https://pop.com.sg/surface-integral-over-graph-of-a-function.htmlhttps://pop.com.sg/surface-integral-over-graph-of-a-function.htmlQ: Evaluate the surface integral $\iint_S \mathbf{F}\cdot d\mathbf{S}$, where $\mathbf{F}(x,y,z)= 3x^2\mathbf{i}-2xy\mathbf{j}+8\mathbf{k}$, and $S$ is the graph of the function $z=f(x,y)=2x-y$ fo......Sun, 23 Aug 2026 05:57:12 -0400https://pop.com.sg/help-solving-2-second-order-differential-equation.htmlhttps://pop.com.sg/help-solving-2-second-order-differential-equation.html$y'' + (y')^2 = y$ and $y*y''= (y')^2 - (y')^3$ I know that both of these equations can be solved by using $p(y) = y'$, but Im getting stuck in the middle, and can't seem to solve the thing aft......Sun, 23 Aug 2026 05:56:42 -0400https://pop.com.sg/convert-expression-to-nand-only.htmlhttps://pop.com.sg/convert-expression-to-nand-only.htmlI have to convert the following to NAND only $$\bar{A}\cdot\bar{B}\cdot\bar{C} + A\cdot\bar{B}\cdot C + A\cdot B\cdot \bar{C} + A \cdot B\cdot C$$ I've looked at the following website: http://en....Sun, 23 Aug 2026 05:48:23 -0400https://pop.com.sg/if-sin-theta-cos-theta-tan-theta-cot-theta-sec-theta-csc-theta-7-then-.htmlhttps://pop.com.sg/if-sin-theta-cos-theta-tan-theta-cot-theta-sec-theta-csc-theta-7-then-.html$$ 0<\theta<\pi/2$$ and $$\sin\theta+\cos\theta+\tan\theta+\cot\theta+\sec\theta+\csc\theta=7$$ then show that $\sin 2\theta$ is a root of the equation $$x^2 -44x +36=0$$ I tried to use ......Sun, 23 Aug 2026 05:41:04 -0400https://pop.com.sg/which-of-the-following-statements-are-true-about-finite-cyclic-groups.htmlhttps://pop.com.sg/which-of-the-following-statements-are-true-about-finite-cyclic-groups.html$1.$ If $G$ is a finite cyclic group, then for every subgroup $H,K$ of $G,$ we have either $H\subset K$ or $K\subset H.$ $2.$ If for every subgroup $H,K$ of a finite group $G,$ we have either $H\su......Sun, 23 Aug 2026 05:34:00 -0400https://pop.com.sg/da-in-polar-coordinates.htmlhttps://pop.com.sg/da-in-polar-coordinates.htmlI have seen a picture for $dV$ so that $dV = r^{2} \sin(\theta)\,dr\,d\theta\,d\phi$. But how can I deduce things like $dA$ and $dV$? In a simpler coordinate (not sure about the name), $dA = r \,dr......Sun, 23 Aug 2026 05:30:08 -0400https://pop.com.sg/are-the-domain-and-range-of-a-function-and-its-inverse-always-exactly-.htmlhttps://pop.com.sg/are-the-domain-and-range-of-a-function-and-its-inverse-always-exactly-.htmlThe function $ f(x)=\sqrt{x+4} $ has domain $[-4,\infty)$ and range $[0,\infty)$. Its inverse function $ f^{-1}(x)=x^2-4 $ by itself seems to have domain $\Bbb{R}$ and range $[-4,\infty)$. Howeve......Sun, 23 Aug 2026 05:26:53 -0400https://pop.com.sg/proving-the-inverse-of-the-invertible-matrix-theorem.htmlhttps://pop.com.sg/proving-the-inverse-of-the-invertible-matrix-theorem.htmlLet $A$ be an $n \times n$ invertible matrix and $v_1,v_2,...,v_m$ an element of $\mathbb{R}^{n}$. If $\{v_1,v_2,...,v_m\}$ spans $\mathbb{R}^{n}$ then $\{Av_1, Av_2,...,Av_m\}$ also spans $\mathbb......Sun, 23 Aug 2026 05:26:39 -0400https://pop.com.sg/diagonal-terms-of-the-inverse-of-positive-definite-matrix.htmlhttps://pop.com.sg/diagonal-terms-of-the-inverse-of-positive-definite-matrix.htmlI have a matrix $A = I + G$ where $G_{ii} = 0$ for all $i$, and $G_{ij} \in \{0, a\}$ with $a \in [0, 1[$, $G$ is symmetric, and such that $A$ is positive definite. We know that the inverse of $A$,......Sun, 23 Aug 2026 05:23:46 -0400https://pop.com.sg/arc-length-of-y-1-3-sqrt-x-x-3.htmlhttps://pop.com.sg/arc-length-of-y-1-3-sqrt-x-x-3.html$y = 1/3 \sqrt{x} (x-3)$ from 1-9 In my book it is actually $x = 1/3 \sqrt{y} (y-3)$ but I prefer to work with y so I just swap the two variables and I think everything should be the same. The fi......Sun, 23 Aug 2026 05:18:57 -0400https://pop.com.sg/what-is-the-difference-between-l-2-norm-and-ell-2-norm.htmlhttps://pop.com.sg/what-is-the-difference-between-l-2-norm-and-ell-2-norm.htmlI can find no precise definitions on the internet for the $L^2$ and $\ell^2$ norms. Certain websites keep switching between the two. Can someone please help me?...Sun, 23 Aug 2026 05:15:41 -0400https://pop.com.sg/how-to-solve-trigonometric-equations-without-a-calculator-duplicate.htmlhttps://pop.com.sg/how-to-solve-trigonometric-equations-without-a-calculator-duplicate.htmlHow does one solve for sine, cosine, or tangent (and inverse sine, cosine, tangent) without the use of a calculatour? For example: cosine of 131 degrees — how must one calculate this?...Sun, 23 Aug 2026 04:56:01 -0400https://pop.com.sg/f-x-x-1-what-does-x-1-mean.htmlhttps://pop.com.sg/f-x-x-1-what-does-x-1-mean.htmlThe question is, "Find the domain and the range of the real function $f$ defined by $f(x) = |x-1|$. I have no idea what $|x-1|$ means, someone help me?...Sun, 23 Aug 2026 04:51:21 -0400https://pop.com.sg/topological-isomorphism-vs-isometric-isomorphism.htmlhttps://pop.com.sg/topological-isomorphism-vs-isometric-isomorphism.htmlWe say that: $T:(X,\|\cdot\|_X)\rightarrow (Y,\|\cdot\|_Y)$ is a isometric isomorphism if it is a linear isomorphism, and it is an isometry, that is $\|T(x)\|_Y=\|x\|_X\quad \forall x\in X;$ $T:(X......Sun, 23 Aug 2026 04:49:55 -0400https://pop.com.sg/find-the-smallest-positive-integer-for-which-x-mod-3-2-and-x-mod-4-3.htmlhttps://pop.com.sg/find-the-smallest-positive-integer-for-which-x-mod-3-2-and-x-mod-4-3.htmlI came across a question on a piece of homework and my solution yields no possible solutions. Find the smallest positive integer for which x mod 3=2 and x mod 4=3 (and then find the next one) So I ...Sun, 23 Aug 2026 04:47:43 -0400https://pop.com.sg/related-rate-question-water-is-leaking-out-of-an-inverted-conical-tank.htmlhttps://pop.com.sg/related-rate-question-water-is-leaking-out-of-an-inverted-conical-tank.htmlWater is leaking out of an inverted conical tank at a rate of 9,500 cm3/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at......Sun, 23 Aug 2026 04:44:16 -0400https://pop.com.sg/on-provability-of-paris-harrington-theorem.htmlhttps://pop.com.sg/on-provability-of-paris-harrington-theorem.htmlIt is said that the Paris–Harrington theorem is true, but not provable in Peano arithmetic. I want to ask: So how do they know that it is true if it has no proof? I cannot imagine someone knows som......Sun, 23 Aug 2026 04:41:26 -0400https://pop.com.sg/normal-subgroups-of-s-4.htmlhttps://pop.com.sg/normal-subgroups-of-s-4.htmlCan anyone tell me how to find all normal subgroups of the symmetric group $S_4$? In particular are $H=\{e,(1 2)(3 4)\}$ and $K=\{e,(1 2)(3 4), (1 3)(2 4),(1 4)(2 3)\}$ normal subgroups?...Sun, 23 Aug 2026 04:37:03 -0400https://pop.com.sg/solving-heat-equation-using-bender-schmidt-method.htmlhttps://pop.com.sg/solving-heat-equation-using-bender-schmidt-method.htmlI was trying to solve this heat equation using Bender-Schmidt method $$u^{j+1} = A u^{j}+rB$$ The heat equation is $$u_{xx} = 32u_{t},0<x<1,u(x,0)=0,u(0,t)=0,u(1,t)=10+t$$taking $$h = 0.25,k......Sun, 23 Aug 2026 04:34:43 -0400https://pop.com.sg/reference-request-for-set-theoretic-foundations-of-geometry.htmlhttps://pop.com.sg/reference-request-for-set-theoretic-foundations-of-geometry.htmlMy question is, Is it possible to define geometrical concepts (say, of Euclidean Geometry) like 'point', 'striaght line' in purely set theoretic terms? So far, I could think of the following ...Sun, 23 Aug 2026 04:29:01 -0400https://pop.com.sg/dimension-of-mathbb-q-otimes-mathbb-z-mathbb-q-as-a-vector-space-over-.htmlhttps://pop.com.sg/dimension-of-mathbb-q-otimes-mathbb-z-mathbb-q-as-a-vector-space-over-.htmlThe following problem was subject of examination that was taken place in June. The document is here. Problem 1 states: The tensor product $\mathbb{Q}\otimes_{\mathbb Z}\mathbb{Q}$ is a vector sp......Sun, 23 Aug 2026 04:25:46 -0400https://pop.com.sg/negation-of-for-every-and-there-exist-statements-duplicate.htmlhttps://pop.com.sg/negation-of-for-every-and-there-exist-statements-duplicate.htmlI need to find the negations of the following statements: For every n (is an element of natural numbers) there is a prime number p such that n < p. There is n (is an element of natural numbers......Sun, 23 Aug 2026 04:15:35 -0400https://pop.com.sg/what-are-the-different-symbols-used-for-denoting-an-angle.htmlhttps://pop.com.sg/what-are-the-different-symbols-used-for-denoting-an-angle.htmlThis is not a question involving numbers and stuff, but suddenly had this basic question in mind: What are the different symbols that different people use for denoting an angle? (I mean, we usually......Sun, 23 Aug 2026 04:08:39 -0400https://pop.com.sg/how-do-you-compute-numerically-the-earth-mover-039-s-distance-emd.htmlhttps://pop.com.sg/how-do-you-compute-numerically-the-earth-mover-039-s-distance-emd.htmlI was trying to compute numerically (write a program) that calculated the EMD for two probability distribution $p_X$ and $q_X$. However, I had a hard time finding an outline of how to exactly compute ...Sun, 23 Aug 2026 03:56:45 -0400https://pop.com.sg/solve-the-inequality-25-x-3000-closed.htmlhttps://pop.com.sg/solve-the-inequality-25-x-3000-closed.htmlSolve the inequality $25^x>3000$. Note: No computer programs or calculators allowed. I tried factoring the two, I ended up with $5^{2z}>2^3\cdot3\cdot5^3$, which doesn't seem to help....Sun, 23 Aug 2026 03:52:27 -0400https://pop.com.sg/conditional-number-growth-of-hilbert-matrix-theoretical-vs-matlab.htmlhttps://pop.com.sg/conditional-number-growth-of-hilbert-matrix-theoretical-vs-matlab.htmlI need to investigate how the condition number of the Hilbert matrix grows with the size N. The Matlab command is: "cond(hilb(N),2)" Compute the condition number of the Hilbert matrices Hn ∈ R, N×......Sun, 23 Aug 2026 03:37:09 -0400https://pop.com.sg/intuition-behind-logarithm-inequality-1-frac1x-leq-log-x-leq-x-1.htmlhttps://pop.com.sg/intuition-behind-logarithm-inequality-1-frac1x-leq-log-x-leq-x-1.htmlOne of fundamental inequalities on logarithm is: $$ 1 - \frac1x \leq \log x \leq x-1 \quad\text{for all $x > 0$},$$ which you may prefer write in the form of $$ \frac{x}{1+x} \leq \log{(1+x)} \l......Sun, 23 Aug 2026 03:32:59 -0400https://pop.com.sg/for-f-x-x-3-2-times-e-0-35x-at-what-x-value-does-the-maximum-occur.htmlhttps://pop.com.sg/for-f-x-x-3-2-times-e-0-35x-at-what-x-value-does-the-maximum-occur.htmlFor $f(x) = x^{3.2}\times \mathrm{e}^{-0.35x}$, at what $x$ value does the maximum occur for minimum or maximum $f'(x)=0$ this implies $3.2 x^{2.2} \mathrm{e}^{-0.35x}-x^{3.2}\mathrm{e}^{-0.35x}\......Sun, 23 Aug 2026 03:32:34 -0400https://pop.com.sg/cube-roots-of-the-complex-numbers-1-i.htmlhttps://pop.com.sg/cube-roots-of-the-complex-numbers-1-i.htmlI cant get any good reference in my books regarding cube of complex numbers. Please help me find cube roots of the Complex number i+1??...Sun, 23 Aug 2026 03:31:39 -0400https://pop.com.sg/partition-of-an-interval-of-mathbb-r.htmlhttps://pop.com.sg/partition-of-an-interval-of-mathbb-r.htmlA partition of an inteval $[a,b]$ of $\mathbb{R}$ is generally defined as a finite sequence of the form: $$a = x_0 < x_1 < x_2 < \dots < x_n = b$$ Then, $[a,b]$ is seen as the following......Sun, 23 Aug 2026 03:26:51 -0400https://pop.com.sg/which-of-the-43-380-possible-nets-for-a-dodecahedron-is-the-narrowest.htmlhttps://pop.com.sg/which-of-the-43-380-possible-nets-for-a-dodecahedron-is-the-narrowest.htmlI want to fit multiple regular dodecahedron nets on to an infinitely long roll of paper. I want this to result in the largest possible dodecahedrons, for a roll of a given width. My hunch is that ......Sun, 23 Aug 2026 03:24:08 -0400https://pop.com.sg/if-det-a-2-det-a-then-a-2-in-a-a-1-a-a-1.htmlhttps://pop.com.sg/if-det-a-2-det-a-then-a-2-in-a-a-1-a-a-1.htmlIs the following statement always true? Let $A \in GL_n( \mathbb Z)$. If $\det(A^2)= \det(A)$, then $A^2 \in \{A, A^{-1}, -A, -A^{-1}\}$ . Note: $GL_n(\mathbb Z)$ is defined as the set of all ...Sun, 23 Aug 2026 03:18:42 -0400https://pop.com.sg/find-the-prime-factors-of-3-32-2-32.htmlhttps://pop.com.sg/find-the-prime-factors-of-3-32-2-32.htmlI'm having a go at BMO 2006/7 Q1 which states: "Find four prime numbers less than 100 which are factors of $3^{32}-2^{32}$." My working is as follows (basically just follows difference of two squa......Sun, 23 Aug 2026 03:14:42 -0400https://pop.com.sg/classifying-irreducible-finite-dimensional-representations-of-the-q-we.htmlhttps://pop.com.sg/classifying-irreducible-finite-dimensional-representations-of-the-q-we.htmlI have that $A = \langle x, y : yx = qxy \rangle$ is the $q$-Weyl algebra, with basis $x^iy^j$, $i, j \in \mathbb{Z}$. The base field is $\mathbb{C}$. I need to classify the irreducible finite-...Sun, 23 Aug 2026 03:13:54 -0400https://pop.com.sg/find-primitive-root-mod-17.htmlhttps://pop.com.sg/find-primitive-root-mod-17.htmlI have to list the quadratic residues of $17$ and find a primitive root. I have calculated that: Quadratic residues $mod(17)$ are $1,2,4,8,9,13,15,16$ How am I then meant to use this to obtain a ...Sun, 23 Aug 2026 03:13:36 -0400https://pop.com.sg/degree-sum-formula-closed.htmlhttps://pop.com.sg/degree-sum-formula-closed.htmlSuppose the G = (V,E) is a connected graph with n vertices and n-1 edges. Use the degree-sum formula for vertices to prove that G has a vertex of degree 1. I can't seem to find where to start in t......Sun, 23 Aug 2026 03:10:44 -0400https://pop.com.sg/what-does-the-small-number-on-top-of-the-square-root-symbol-mean.htmlhttps://pop.com.sg/what-does-the-small-number-on-top-of-the-square-root-symbol-mean.htmlI just came across this annotation in my school's maths compendium: The compendium is very brief and doesn't explain what this means....Sun, 23 Aug 2026 03:08:31 -0400https://pop.com.sg/how-to-recognize-the-language-of-a-dfa.htmlhttps://pop.com.sg/how-to-recognize-the-language-of-a-dfa.htmlGiven the following Deterministic Finite Automata, I have to be able to tell what language the Automata accepts.image here] As I can see, it accepts the letters a, b and c. Furthermore, my guess wo......Sun, 23 Aug 2026 02:44:20 -0400https://pop.com.sg/negation-of-some-logic-statement.htmlhttps://pop.com.sg/negation-of-some-logic-statement.htmlI need to negate the following statement : "Some integers are not odd" I have the below, where O(x) is "odd" $$ \exists x (\neg O(x)) $$ Would the negation be $$ \forall x (O(x)) $$ I'm confused, ...Sun, 23 Aug 2026 02:38:02 -0400https://pop.com.sg/linear-algebra-row-picture-and-column-picture-for-system-of-linear-equ.htmlhttps://pop.com.sg/linear-algebra-row-picture-and-column-picture-for-system-of-linear-equ.htmlI am presently going through Strang's linear algebra book in which he explains row picture and column picture for a system of linear equations as such For example: $$\begin{cases} 2x + 3y = 5 \\ ......Sun, 23 Aug 2026 02:26:45 -0400https://pop.com.sg/negative-variance.htmlhttps://pop.com.sg/negative-variance.htmlI have two independent variables $X$ and $Y$. $W=X-Y$ when $X\sim \mbox{Bernoulli}\left(1/2\right)$ and $Y\sim N(0,1)$. This puts $\operatorname{Var}(x)=1/4$ and $\operatorname{Var}(Y)=1$, but I h......Sun, 23 Aug 2026 01:54:56 -0400https://pop.com.sg/pitchfork-bifurcation-subcritical-and-supercritical.htmlhttps://pop.com.sg/pitchfork-bifurcation-subcritical-and-supercritical.htmlI am slightly confused about the terminology of a subcritical and supercritical pitchfork bifurcation. Consider the diagrams below: (both pictures from wikipedia, by Claudio Rocchini, CC by 2.5......Sun, 23 Aug 2026 01:40:24 -0400https://pop.com.sg/lyndon-words-over-the-binary-alphabet-that-begin-with-00.htmlhttps://pop.com.sg/lyndon-words-over-the-binary-alphabet-that-begin-with-00.htmlLydon words over the binary alphabet are strings of $1$s and $0$s such that all rotations of the word are strictly lexicographically later. I've been looking at Lyndon words of length $n$, which are ...Sun, 23 Aug 2026 01:34:06 -0400https://pop.com.sg/value-of-tan-pi-2-duplicate.htmlhttps://pop.com.sg/value-of-tan-pi-2-duplicate.htmlI understand that this is a very stupid question but I'm not getting the answer. At $x=\pi/2$, what is the value of $tan(x)$? Should it be $-\infty$ or $+\infty$? Text tells it to be $+\infty$. ......Sun, 23 Aug 2026 01:25:42 -0400https://pop.com.sg/area-of-circumscribed-circle.htmlhttps://pop.com.sg/area-of-circumscribed-circle.htmlI'm looking at the solutions for a 1989 ARML contest, and it has this question: If the area of a rectangle is 1/2 of the area of its circumscribed circle, compute the acute angle between the dia......Sun, 23 Aug 2026 01:22:25 -0400https://pop.com.sg/solve-equation-3-x-2-3-x.htmlhttps://pop.com.sg/solve-equation-3-x-2-3-x.htmlUsing the substitution $u=3^x$, or otherwise, solve, correct to 3 significant figures, the equation $3^x=2+3^{-x}$ My attempt: $u=2+u^{-1}$ $u-u^{-1}=2$ $ {{u^2-1}\over {u}}=2$ $\log ({{u^2......Sun, 23 Aug 2026 01:21:24 -0400https://pop.com.sg/show-that-limit-does-not-exist-two-variables.htmlhttps://pop.com.sg/show-that-limit-does-not-exist-two-variables.htmlI just started looking into multiple variable calculus and limits involving them. I'm not amazing at limits either. I want to answer this question: Show that the following limit does not exist $$\......Sun, 23 Aug 2026 01:18:48 -0400https://pop.com.sg/summation-of-n-squared-cubed-etc-duplicate.htmlhttps://pop.com.sg/summation-of-n-squared-cubed-etc-duplicate.htmlHow do you in general derive a formula for summation of n-squared, n-cubed, etc...? Clear explanation with reference would be great....Sun, 23 Aug 2026 01:15:14 -0400https://pop.com.sg/finding-if-a-relation-implicit-function-is-an-even-or-odd-function.htmlhttps://pop.com.sg/finding-if-a-relation-implicit-function-is-an-even-or-odd-function.htmlI have this relation, and if graphed on Desmos.com looks very symmetrical. $$x^{\frac{2}{3}} + y^{\frac{2}{3}} = 1 $$ It has cusps(vertical tangents on the y-axis). How can one determine if this re......Sun, 23 Aug 2026 01:07:42 -0400https://pop.com.sg/how-to-find-a-point-between-two-points-with-given-distance.htmlhttps://pop.com.sg/how-to-find-a-point-between-two-points-with-given-distance.htmlLets say I have point P1(10,10) and P2(20,20). I want to find a P3 which is on between this two points and 3 units away from P1. What is the formula to find P3 ? Known values: X1, X2, Y1 , Y2, ...Sun, 23 Aug 2026 01:04:13 -0400https://pop.com.sg/why-is-the-definition-of-cardinal-number-as-the-set-of-all-sets-equiva.htmlhttps://pop.com.sg/why-is-the-definition-of-cardinal-number-as-the-set-of-all-sets-equiva.htmlIn Fundamentals of Mathematics, Volume 1 Foundations of Mathematics: The Real Number System and Algebra, after defining set equivalence as the ability to put the elements of the related sets in one......Sun, 23 Aug 2026 01:02:13 -0400https://pop.com.sg/linear-dependence-of-functions.htmlhttps://pop.com.sg/linear-dependence-of-functions.htmlI'm in an intro to DE class. On one of the quizzes, the professor asked us to determine whether the set of functions $\ x, x-1,$ and$\ x+3$ are linearly independent or linearly dependent on $\ (- \......Sun, 23 Aug 2026 01:02:05 -0400https://pop.com.sg/definition-of-a-divergent-sequence.htmlhttps://pop.com.sg/definition-of-a-divergent-sequence.htmlI have that the definition of a convergent sequence is: ($\exists$ L $\in$ $\mathbb{R}$)($\forall$ $\epsilon$ > 0)($\exists$ N $\in$ $\mathbb{N}$)($\forall$ n $\in$ $\mathbb{N}$)[n $\ge$ N $\Right......Sun, 23 Aug 2026 01:01:07 -0400https://pop.com.sg/solving-equations-that-contain-summations.htmlhttps://pop.com.sg/solving-equations-that-contain-summations.htmlMy previous algebra course did not go over summation at all, and now that I'm in my new course, Discrete Math for Information Technology, we have been introduced to summation. I understand summation ...Sun, 23 Aug 2026 01:00:06 -0400https://pop.com.sg/4-regular-planar-graph-is-unique.htmlhttps://pop.com.sg/4-regular-planar-graph-is-unique.htmlI'm working on a project for a class and as part of that project I (previously) decided to do the following problem from our textbook, Combinatorics and Graph Theory 2nd ed. by Harris, Hirst, & ...Sun, 23 Aug 2026 01:00:05 -0400https://pop.com.sg/derivation-of-fibonacci-log-n-time-sequence.htmlhttps://pop.com.sg/derivation-of-fibonacci-log-n-time-sequence.htmlI was trying to derive following equation to compute the nth fibonacci number in O(log(n)) time. F(2n) = (2*F(n-1) + F(n)) * F(n) which i found on wiki form the fibonacci matrix equation stated th......Sun, 23 Aug 2026 00:49:10 -0400https://pop.com.sg/books-on-complex-analysis-ahlfors-conway-and-lang.htmlhttps://pop.com.sg/books-on-complex-analysis-ahlfors-conway-and-lang.htmlTo make my question slightly different from others, I would like to know how would you rate on the complex analysis books by Ahlfors, Conway and Lang? I had a basic course on complex analysis during ...Sun, 23 Aug 2026 00:47:24 -0400https://pop.com.sg/regular-k-gons-that-can-tessellate-a-plane.htmlhttps://pop.com.sg/regular-k-gons-that-can-tessellate-a-plane.htmlWhat is the sum of all possible values of $k$ for which regular $k-$gons can tessellate a plane? This is one of the NAT problems. While I am familiar with tessellations, I don't quite get what the......Sun, 23 Aug 2026 00:41:48 -0400https://pop.com.sg/real-and-imaginary-part-of-1-i-sqrt-3-6.htmlhttps://pop.com.sg/real-and-imaginary-part-of-1-i-sqrt-3-6.htmli am a bit stuck here. As the title says i try to find out how to write complex numbers like for example$$ (1-i\sqrt{3})^6$$ in the normal representation$$ z = x + i*y$$ I already found out that ......Sun, 23 Aug 2026 00:34:48 -0400https://pop.com.sg/how-to-bisect-an-isosceles-trapezoid-into-two-equal-area-parts.htmlhttps://pop.com.sg/how-to-bisect-an-isosceles-trapezoid-into-two-equal-area-parts.htmlGiven an isosceles trapezoid: I want to draw a line parallel to the bases (that is, parallel to and in between AD and BC) such that the top half and the bottom half both have equal area. Specific......Sun, 23 Aug 2026 00:34:14 -0400https://pop.com.sg/show-that-n-kl-is-a-perfect-square.htmlhttps://pop.com.sg/show-that-n-kl-is-a-perfect-square.htmlI faced a doubt in this question while solving some maths problem. Please, solve it. A natural number $n$ is chosen strictly between two consecutive perfect squares. The smaller of these two square ...Sun, 23 Aug 2026 00:31:46 -0400https://pop.com.sg/what-is-an-intermediate-definition-for-a-tangent-to-a-curve.htmlhttps://pop.com.sg/what-is-an-intermediate-definition-for-a-tangent-to-a-curve.htmlMost students come to calculus with an intuitive sense of what a tangent line should be for a curve. It is easy enough to give a definition of a tangent to a circle that is both elementary and rigo......Sun, 23 Aug 2026 00:28:07 -0400https://pop.com.sg/different-definitions-of-p-points-ultrafilters.htmlhttps://pop.com.sg/different-definitions-of-p-points-ultrafilters.htmlRecently I have been exposed to the concept of P-Points. I have three definitions: A non-principal ultrafilter $u$ is a P-Point if: For every sequence $\left < A_n \right >_{n\in \omega}$ of ...Sun, 23 Aug 2026 00:25:48 -0400https://pop.com.sg/clarifying-the-definition-of-the-dual-map.htmlhttps://pop.com.sg/clarifying-the-definition-of-the-dual-map.htmlI would appreciate help understanding the definition of the dual map $T'$ as presented in Axler's "Linear Algebra Done Right" 3rd ed. on page 103. If $T \in \cal {L}$$(V,W)$ then the dua......Sun, 23 Aug 2026 00:22:53 -0400https://pop.com.sg/geometric-interpretation-of-mixed-partial-derivatives.htmlhttps://pop.com.sg/geometric-interpretation-of-mixed-partial-derivatives.htmlI'm looking for a geometric interpretation of this theorem: My book doesn't give any kind of explanation of it. Again, I'm not looking for a proof - I'm looking for a geometric interpretation. Th......Sun, 23 Aug 2026 00:18:52 -0400https://pop.com.sg/geometric-understanding-of-differential-forms.htmlhttps://pop.com.sg/geometric-understanding-of-differential-forms.htmlI would like to understand differential forms more intuitively. I have yet to find a book which explains how the use of the exterior product in differential forms ties into the geometrical signific......Sun, 23 Aug 2026 00:18:37 -0400https://pop.com.sg/given-two-adjacent-sides-of-a-rectangle-are-equivalent-prove-that-the-.htmlhttps://pop.com.sg/given-two-adjacent-sides-of-a-rectangle-are-equivalent-prove-that-the-.htmlIn Geometry class today, we were talking about quadrilaterals and the types of them. I was wondering that if, given a rectangle with two adjacent equal congruent sides, if that was enough to prove ......Sun, 23 Aug 2026 00:12:11 -0400https://pop.com.sg/when-is-a-power-series-a-geometric-series.htmlhttps://pop.com.sg/when-is-a-power-series-a-geometric-series.htmlA geometric series is one having a common ratio, right? Something like: $$\sum_{n=0}^{\infty} ar^n$$ And a power series is one with the form: $$\sum_{n=0}^{\infty} c_n(x-a)^n$$ Initially, I tho......Sun, 23 Aug 2026 00:04:58 -0400https://pop.com.sg/rolling-standard-deviations.htmlhttps://pop.com.sg/rolling-standard-deviations.htmlI am trying to calculate standard deviations on an array of numbers. My psuedo code looks like this: deviation = getStandardDeviation(array(32, 47, 42, 45, 80, 90)); In the above example, deviat......Sat, 22 Aug 2026 23:58:17 -0400https://pop.com.sg/how-does-one-interpret-the-meaning-of-a-stochastic-derivative.htmlhttps://pop.com.sg/how-does-one-interpret-the-meaning-of-a-stochastic-derivative.htmlMy understanding of derivatives is in the difference quotient limit sense... How does one interpret the meaning of a stochastic derivative? How can one possibly differentiate a random variable? W......Sat, 22 Aug 2026 23:47:19 -0400https://pop.com.sg/how-to-calculate-the-result-of-a-recursive-equation.htmlhttps://pop.com.sg/how-to-calculate-the-result-of-a-recursive-equation.htmlI'm looking for a tool (like Wolfram Alpha) that can calculate the result of $ B(P,N) $ Where $B(P,N)$ is a recursive function defined as follows: $ \left\{\begin{matrix} B(P,N)=\frac{-(-1)^{\fra......Sat, 22 Aug 2026 23:36:56 -0400https://pop.com.sg/property-of-the-2-3-4-triangle-and-its-extensions.htmlhttps://pop.com.sg/property-of-the-2-3-4-triangle-and-its-extensions.htmlI'm trying to answer this question but I'm fully stuck. I've tried to draw the diagram but it still is a mystery to me. Here is the problem: a) Prove that the $2$-$3$-$4$ triangle has the property......Sat, 22 Aug 2026 23:36:14 -0400https://pop.com.sg/why-is-determinant-a-multilinear-function.htmlhttps://pop.com.sg/why-is-determinant-a-multilinear-function.htmlI am trying to understand (intuitive explanation will be fine) why determinant is a multilinear function and therefore to learn how elementary row operation affect the determinant. I understand th......Sat, 22 Aug 2026 23:27:50 -0400https://pop.com.sg/closure-of-the-union-union-of-closures.htmlhttps://pop.com.sg/closure-of-the-union-union-of-closures.htmlI have seen that $\text{cl}(A\cup B)=\text{cl}(A)\cup \text{cl}(B)$. However I don't see why this is true. I can see the the right to left inclusion, but I can't see the inclusion from left to righ......Sat, 22 Aug 2026 23:13:55 -0400https://pop.com.sg/the-sum-of-two-odd-numbers-is-even.htmlhttps://pop.com.sg/the-sum-of-two-odd-numbers-is-even.htmlI know formal ways of proving this, but I am doing research on how middle grade students create arguments to support conjectures. I would like to anticipate many more solutions before I enact the t......Sat, 22 Aug 2026 23:10:31 -0400https://pop.com.sg/calculate-flux-through-a-surface.htmlhttps://pop.com.sg/calculate-flux-through-a-surface.htmlPart of the surface, S, is: $z=x^2+y^2$ above the disk $ \ x^2+y^2 = 1 \ $ oriented in the $\vec k$ direction. I need to set up an integrated integral to calculate the flux of $\vec F = yz\vec i+x......Sat, 22 Aug 2026 22:55:46 -0400https://pop.com.sg/need-help-visualizing-the-combinatorics-sum-product-rule.htmlhttps://pop.com.sg/need-help-visualizing-the-combinatorics-sum-product-rule.htmlI'm working on some assignments in my discrete math class for the sum & product combinatorics rules and I'm having trouble understanding their definitions and how they work. I'm hoping someone ......Sat, 22 Aug 2026 22:52:48 -0400https://pop.com.sg/is-the-point-at-0-0-of-the-graph-z-x-3-y-3-considered-a-saddle-point.htmlhttps://pop.com.sg/is-the-point-at-0-0-of-the-graph-z-x-3-y-3-considered-a-saddle-point.htmlIs the point at $(0,0)$ of the graph $z=x^3 + y^3$ considered a saddle point? I was given that function, and I used the second-order derivative test only to find that the $Hf(0,0) = 0$, with the ...Sat, 22 Aug 2026 22:28:54 -0400https://pop.com.sg/monic-polynomials-clarification.htmlhttps://pop.com.sg/monic-polynomials-clarification.htmlSo I know a monic polynomial is one where the coefficient of the leading term is 1, but does that mean the constant 1 is a monic polynomial too? Thanks....Sat, 22 Aug 2026 22:28:45 -0400https://pop.com.sg/drawing-vector-fields-beginner.htmlhttps://pop.com.sg/drawing-vector-fields-beginner.htmlI have $f(x,y) = xy$ which has the Gradient Vector Field F$(x,y) = (y,x)$ I believe the correct way to draw individual vectors is to map points into the formula, for example at point $(1,-3)$ we w......Sat, 22 Aug 2026 22:27:03 -0400https://pop.com.sg/difference-in-meaning-between-converges-against-and-converge-to.htmlhttps://pop.com.sg/difference-in-meaning-between-converges-against-and-converge-to.htmlMight be a silly question, but is there any difference between saying that a something (limit, series and so on) converges to a value and converges against a value?...Sat, 22 Aug 2026 22:24:23 -0400https://pop.com.sg/what-is-the-notation-for-the-empty-matrix.htmlhttps://pop.com.sg/what-is-the-notation-for-the-empty-matrix.htmlMy question: What is a notation for an empty 0x0 matrix (i.e. the matrix for the only linear map $f:\{0\}\to\{0\}$)? Is it written $()$? How can I distinguish the 0x0 matrix with for example the 0x3 ...Sat, 22 Aug 2026 22:23:39 -0400https://pop.com.sg/how-to-find-tangent-line-at-a-given-point-without-equation.htmlhttps://pop.com.sg/how-to-find-tangent-line-at-a-given-point-without-equation.htmlFind the equation of the line that is tangent to the curve at the point $(0,\sqrt{\frac{\pi}{2}})$. Given your answer in slope-intercept form. I don't know how can I get the tangent line, without a ...Sat, 22 Aug 2026 22:21:39 -0400https://pop.com.sg/special-galois-connections.htmlhttps://pop.com.sg/special-galois-connections.htmlLet $(f;g)$ is a Galois connection between two posets. Consider the special case $g\circ f = \operatorname{id}_{\operatorname{dom} f}$. What can be said about this special case of Galois connecti......Sat, 22 Aug 2026 22:19:19 -0400https://pop.com.sg/how-to-find-the-average-rate-of-change-velocity.htmlhttps://pop.com.sg/how-to-find-the-average-rate-of-change-velocity.htmlA problem asked this: The rocket club wants to model rockets straight up into the air. The height of rocket A can being modeled by the function $y=-16x^2+88x$. The function that models the heigh......Sat, 22 Aug 2026 22:16:51 -0400https://pop.com.sg/the-comparison-test-or-the-limit-comparison-test.htmlhttps://pop.com.sg/the-comparison-test-or-the-limit-comparison-test.htmlI have checked many of this site's pages yet I could not find a clear answer about how to choose between the "comparison test" OR the "limit comparison test". Because the difference between the tw......Sat, 22 Aug 2026 22:16:46 -0400https://pop.com.sg/what-is-a-bounty-how-can-i-start-one.htmlhttps://pop.com.sg/what-is-a-bounty-how-can-i-start-one.htmlQ&A for people studying math at any level and professionals in related fields...Sat, 22 Aug 2026 22:16:34 -0400https://pop.com.sg/solution-of-cubic-equations-in-terms-of-quadratic-equations.htmlhttps://pop.com.sg/solution-of-cubic-equations-in-terms-of-quadratic-equations.htmlIs there a general way in which a cubic equation of $3$rd degree can be represented by a quadratic equation of $2$nd degree such that $2$ solutions of a cubic equation is equal to $2$ solutions of ...Sat, 22 Aug 2026 22:15:57 -0400https://pop.com.sg/finding-the-inverse-distribution-for-a-standard-cauchy-distribution-is.htmlhttps://pop.com.sg/finding-the-inverse-distribution-for-a-standard-cauchy-distribution-is.htmlThe standard cauchy distribution was given as follows $f_X(x) = \frac{1}{\pi} \frac{1}{1+x^2} \;\;\;\;\; x\in \mathbb{R}$ I need to find the distribution for $Y= \frac{1}{X}$ and show it is the sam......Sat, 22 Aug 2026 22:14:43 -0400https://pop.com.sg/trig-identity-pythagorean-theorem-confusion.htmlhttps://pop.com.sg/trig-identity-pythagorean-theorem-confusion.htmlI run into a problem when I'm trying to prove how $\tan^2x+1 = \sec^2x$, and $1+\cot^2x=\csc^2x$ I understand that $\sin^2x+\cos^2x = 1$. (To my understanding 1 is the Hypotenuse, please correct m......Sat, 22 Aug 2026 22:14:13 -0400https://pop.com.sg/definitions-of-direct-product-and-of-direct-sum.htmlhttps://pop.com.sg/definitions-of-direct-product-and-of-direct-sum.htmlI was wondering if there are some general definitions for direct product and for direct sum, for example in category theory or in set theory, so that the concepts for vector spaces, Abelian groups,......Sat, 22 Aug 2026 21:45:43 -0400https://pop.com.sg/can-i-factorize-x-4-27x-without-using-the-factor-theorem.htmlhttps://pop.com.sg/can-i-factorize-x-4-27x-without-using-the-factor-theorem.htmlI am doing some self-assigned homework questions. One question asks me to factorize $x^4 + 27x$. I used the factor theorem to solve it. I observed that $f(0) = 0$ and $f(-3) = 0$, so $x$ and $(x + ......Sat, 22 Aug 2026 21:39:40 -0400https://pop.com.sg/find-general-solution-of-first-order-de.htmlhttps://pop.com.sg/find-general-solution-of-first-order-de.htmlFind general solution of $$y'+\frac{2y}{x}=\frac{3}{x^2}$$ and find solution for which $$y(2)=-1$$ I know $$xy'+\frac{r(x)y}{x}=3r(x)x$$ $$(ry')=ry'+r'y$$$$r'=\frac{r}{x},\frac{r'}{r}=\......Sat, 22 Aug 2026 21:39:02 -0400https://pop.com.sg/endomorphisms-of-a-finite-dimensional-vector-space.htmlhttps://pop.com.sg/endomorphisms-of-a-finite-dimensional-vector-space.htmlFrom Humphreys' Introduction to Lie Algebras and Representation Theory: If $V$ is a finite dimensional vector space over $F$, denote by $\text{End }V$ the set of linear transformations $V\righta......Sat, 22 Aug 2026 21:35:55 -0400