# Question batches 36,000 questions have bilingual step-by-step derivations. Browse original batches or search and filter in the question explorer. [Search all questions & derivations →](/en/explore/) [← Previous](/en/practice/?page=7#results)[1](/en/practice/?page=1#results) … [7](/en/practice/?page=7#results) 8 [9](/en/practice/?page=9#results) … [15](/en/practice/?page=15#results)[Next →](/en/practice/?page=9#results) 360 results · Page 8 of 15 Question batch · 100 questions ## [Complete Reflexivity, Symmetry, Transitivity and Qualified Classes](/en/practice/batch-20261013-advance-20261026-r07-equivalence-relation-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Equivalence relation batch-20261013-advance-20261026-r07-equivalence-relation-foundations Question batch · 100 questions ## [Complete Vertex Bijections, Adjacency Iff and Invariant Boundaries](/en/practice/batch-20261013-advance-20261026-r06-graph-isomorphism-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Graph isomorphism batch-20261013-advance-20261026-r06-graph-isomorphism-foundations Question batch · 100 questions ## [One-Law Covariance, Symmetry, Affine Changes and Zero-Covariance Dependence](/en/practice/batch-20261013-advance-20261026-r05-covariance-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Covariance batch-20261013-advance-20261026-r05-covariance-foundations Question batch · 100 questions ## [Exact Joint Qualification, Marginals, Rectangle Events and Pair Masses](/en/practice/batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Finite joint distribution batch-20261013-advance-20261026-r04-finite-joint-distribution-foundations Question batch · 100 questions ## [Finite Tail Audits, Infinite Cauchy Witnesses and Alternating Counterexamples](/en/practice/batch-20261013-advance-20261026-r03-cauchy-sequence-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Cauchy sequence batch-20261013-advance-20261026-r03-cauchy-sequence-foundations Question batch · 100 questions ## [Exact Named Norms, Universal Certificates and Induced Distances](/en/practice/batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Real normed vector space batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations Question batch · 100 questions ## [Qualified Finite Groups, Commutativity, Orders and Cyclic Subgroups](/en/practice/batch-20261013-advance-20261026-r01-finite-abelian-group-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Finite abelian group batch-20261013-advance-20261026-r01-finite-abelian-group-foundations Question batch · 100 questions ## [Whole Partial Polynomials, Fixed-x Slices and Symbolic Quotient Limits](/en/practice/batch-20261013-advance-20261025-r07-partial-derivative-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Partial derivative batch-20261013-advance-20261025-r07-partial-derivative-foundations Question batch · 100 questions ## [Exact Finite-Step Markov Powers, Endpoint Joints and Conditional Paths](/en/practice/batch-20261013-advance-20261025-r06-finite-markov-chain-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Finite-state Markov chain batch-20261013-advance-20261025-r06-finite-markov-chain-foundations Question batch · 100 questions ## [Finite Preimage Laws, Constant Maps and Source-Sigma Refinement](/en/practice/batch-20261013-advance-20261025-r05-measurable-function-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Measurable function batch-20261013-advance-20261025-r05-measurable-function-foundations Question batch · 100 questions ## [Explicit Span, Dependence and Ordered-Basis Coordinate Witnesses](/en/practice/batch-20261013-advance-20261025-r04-linear-independence-basis-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Linear independence and bases batch-20261013-advance-20261025-r04-linear-independence-basis-foundations Question batch · 100 questions ## [Convex-Problem Qualification, Feasibility, Evaluation and Pairwise Comparison](/en/practice/batch-20261013-advance-20261025-r03-convex-optimization-problem-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Convex optimization problem batch-20261013-advance-20261025-r03-convex-optimization-problem-foundations Question batch · 100 questions ## [Universal Affine/Quadratic Convexity Certificates versus Jensen Instances](/en/practice/batch-20261013-advance-20261025-r02-convex-function-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Convex function batch-20261013-advance-20261025-r02-convex-function-foundations Question batch · 100 questions ## [Separate Interior PDE and Complete Rectangle-Boundary Identities](/en/practice/batch-20261013-advance-20261025-r01-boundary-value-problem-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Boundary-value problem batch-20261013-advance-20261025-r01-boundary-value-problem-foundations Question batch · 100 questions ## [Universal Interval, Box and Halfspace Certificates versus Midpoint Counterexamples](/en/practice/batch-20261013-advance-20261024-r07-convex-set-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Convex set batch-20261013-advance-20261024-r07-convex-set-foundations Question batch · 100 questions ## [Explicit PDE Conventions: Point Residuals and Symbolic Identities](/en/practice/batch-20261013-advance-20261024-r06-partial-differential-equation-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Partial differential equation batch-20261013-advance-20261024-r06-partial-differential-equation-foundations Question batch · 100 questions ## [Exact Convex Weights, Weighted Points and Witnesses](/en/practice/batch-20261013-advance-20261024-r05-finite-convex-combination-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Finite convex combination batch-20261013-advance-20261024-r05-finite-convex-combination-foundations Question batch · 100 questions ## [Fixed-Coordinate Polynomial Partials and Ordered Gradients](/en/practice/batch-20261013-advance-20261024-r04-partial-derivative-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Partial derivative batch-20261013-advance-20261024-r04-partial-derivative-foundations Question batch · 100 questions ## [Row-Stochastic Markov Kernels, Finite-Step Laws and Paths](/en/practice/batch-20261013-advance-20261024-r03-finite-markov-chain-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Finite-state Markov chain batch-20261013-advance-20261024-r03-finite-markov-chain-foundations Question batch · 100 questions ## [Complete Finite Measurability through Target Preimages](/en/practice/batch-20261013-advance-20261024-r02-measurable-function-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Measurable function batch-20261013-advance-20261024-r02-measurable-function-foundations Question batch · 100 questions ## [Exact Independence, Spanning and Ordered Plane-Basis Coordinates](/en/practice/batch-20261013-advance-20261024-r01-linear-independence-basis-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Linear independence and bases batch-20261013-advance-20261024-r01-linear-independence-basis-foundations Question batch · 100 questions ## [Declared Bivariate Polynomial Coordinates and Domain Slices](/en/practice/batch-20261013-advance-20261023-r07-multivariable-real-function-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Multivariable real-valued function batch-20261013-advance-20261023-r07-multivariable-real-function-foundations Question batch · 100 questions ## [Complete Finite-Horizon Joint Path Laws and Marginals](/en/practice/batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Discrete-time stochastic process batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations Question batch · 100 questions ## [Finite Sigma-Algebra Closure and Exact Measure Reduction](/en/practice/batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D8) Sigma-algebras and measure spaces batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations [← Previous](/en/practice/?page=7#results)[1](/en/practice/?page=1#results) … [7](/en/practice/?page=7#results) 8 [9](/en/practice/?page=9#results) … [15](/en/practice/?page=15#results)[Next →](/en/practice/?page=9#results) 360 results · Page 8 of 15