https://9to5science.com/Step-by-step answers to science and math questions. https://9to5science.com/are-singleton-sets-in-mathbb-r-both-closed-and-openhttps://9to5science.com/are-singleton-sets-in-mathbb-r-both-closed-and-open Mon, 12 Oct 2026 00:00:00 GMTIn ℝ with the usual topology, a singleton {a} is closed but not open. Proofs of both facts, and why only ∅ and ℝ are both open and closed. https://9to5science.com/boolean-algebra-simplification-sum-of-products-from-truth-tablehttps://9to5science.com/boolean-algebra-simplification-sum-of-products-from-truth-table Mon, 12 Oct 2026 00:00:00 GMTWrite the canonical SOP from the rows where the output is 1, then simplify with Boolean algebra or a Karnaugh map. Fully worked 3-variable example. https://9to5science.com/does-standard-deviation-change-if-multiplied-by-a-constant-or-if-a-constant-is-added-to-ithttps://9to5science.com/does-standard-deviation-change-if-multiplied-by-a-constant-or-if-a-constant-is-added-to-it Mon, 12 Oct 2026 00:00:00 GMTAdding a constant leaves the standard deviation unchanged; multiplying by a scales it by |a| and the variance by a². Proof, examples and the general rule for aX + b. https://9to5science.com/drawing-a-figure-without-lifting-the-penhttps://9to5science.com/drawing-a-figure-without-lifting-the-pen Mon, 12 Oct 2026 00:00:00 GMTA figure can be drawn in one stroke exactly when it is connected and has 0 or 2 points where an odd number of lines meet. How to check, and where to start. https://9to5science.com/every-singleton-set-is-openhttps://9to5science.com/every-singleton-set-is-open Mon, 12 Oct 2026 00:00:00 GMTSingleton sets are not open in general. In ℝ, {0} is closed but not open. A singleton {x} is open exactly when x is an isolated point, as in discrete spaces. https://9to5science.com/finding-horizontal-amp-vertical-asymptote-s-using-limitshttps://9to5science.com/finding-horizontal-amp-vertical-asymptote-s-using-limits Mon, 12 Oct 2026 00:00:00 GMTHorizontal asymptotes come from limits as x → ±∞; vertical asymptotes from infinite one-sided limits at a point. Step-by-step method with worked examples. https://9to5science.com/given-mean-and-standard-deviation-find-the-probabilityhttps://9to5science.com/given-mean-and-standard-deviation-find-the-probability Mon, 12 Oct 2026 00:00:00 GMTConvert values to z-scores and use the standard normal table to find P(X < b), P(X > a) and P(a < X < b) from a mean and standard deviation, with examples. https://9to5science.com/how-to-calculate-the-phase-anglehttps://9to5science.com/how-to-calculate-the-phase-angle Mon, 12 Oct 2026 00:00:00 GMTPhase angle formulas for series RLC circuits (φ = arctan((X_L − X_C)/R)), complex phasors, and two sine waves from a time shift, with worked examples. https://9to5science.com/how-to-find-quartiles-using-histogramhttps://9to5science.com/how-to-find-quartiles-using-histogram Mon, 12 Oct 2026 00:00:00 GMTFind Q1, the median and Q3 from a histogram using cumulative frequencies and the grouped-data interpolation formula, with a fully worked example. https://9to5science.com/hydrogen-molecule-potential-energy-graphhttps://9to5science.com/hydrogen-molecule-potential-energy-graph Mon, 12 Oct 2026 00:00:00 GMTThe H₂ potential energy curve explained. The minimum marks the 74 pm bond length, its depth the bond energy, and repulsion dominates at short range. https://9to5science.com/insertion-sort-average-casehttps://9to5science.com/insertion-sort-average-case Mon, 12 Oct 2026 00:00:00 GMTDerivation of insertion sort's average case on a random permutation, about n²/4 comparisons and shifts, with the exact formula (n² + 3n)/4 − Hₙ. https://9to5science.com/inverse-of-gaussian-kernel-matrixhttps://9to5science.com/inverse-of-gaussian-kernel-matrix Mon, 12 Oct 2026 00:00:00 GMTA Gaussian kernel matrix is positive definite for distinct points, so it is invertible, but it is often badly conditioned. Use a ridge term and a Cholesky solve instead. https://9to5science.com/show-that-the-singleton-set-is-open-in-a-finite-metric-spcehttps://9to5science.com/show-that-the-singleton-set-is-open-in-a-finite-metric-spce Mon, 12 Oct 2026 00:00:00 GMTProof that {x} is open in any finite metric space, using the minimum distance to the other points, and why this makes every subset open (the discrete topology). https://9to5science.com/what-is-the-standard-deviation-of-dice-rollinghttps://9to5science.com/what-is-the-standard-deviation-of-dice-rolling Mon, 12 Oct 2026 00:00:00 GMTThe standard deviation of one fair six-sided die is √(35/12) ≈ 1.708. See the derivation, the formula for any die, and how it scales for sums and averages.