# Concept derivation guides 299 concept teaching guides, separate from the 36,000 individual question derivations. [Search all questions & derivations →](/en/explore/) [← Previous](/en/derivations/?page=9#results)[1](/en/derivations/?page=1#results) … [9](/en/derivations/?page=9#results) 10 [11](/en/derivations/?page=11#results) … [13](/en/derivations/?page=13#results)[Next →](/en/derivations/?page=11#results) 299 results · Page 10 of 13 Concept guide ## [The full unit permutation modulo 10](/en/derivations/derivation-mko-euler-theorem-number-theory-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D10) For a=-3,n=10, phi(10)=4 and a^4=1 modulo 10. Euler's theorem in number theory derivation-mko-euler-theorem-number-theory-01 Concept guide ## [An all-index subsequence certificate for affine indices](/en/derivations/derivation-mko-subsequence-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D10) Construct a subsequence of a_n=3n-2 using k(j)=1+2j, then take a further subsequence. Subsequence derivation-mko-subsequence-01 Concept guide ## [Complete orbits and their partition](/en/derivations/derivation-mko-finite-group-orbit-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D10) The distinct orbits are {0,1,2} and {3}; same-orbit is an equivalence relation. Orbit of a finite group action derivation-mko-finite-group-orbit-01 Concept guide ## [A bounded monotone tail and the supremum argument](/en/derivations/derivation-mko-monotone-sequence-convergence-theorem-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D10) For a_n=3-2/(n+1), prove monotonicity, boundedness, supremum 3 and convergence. Monotone bounded sequence convergence theorem derivation-mko-monotone-sequence-convergence-theorem-01 Concept guide ## [Relative openness differs from ambient openness](/en/derivations/derivation-mko-subspace-topology-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D10) Compute the subspace topology and all relative closed sets of Y={b,c}. Subspace topology derivation-mko-subspace-topology-01 Concept guide ## [Chebyshev from squared deviations](/en/derivations/derivation-mko-chebyshev-inequality-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D10) Derive Chebyshev from Markov and compute an exact instance. Chebyshev inequality derivation-mko-chebyshev-inequality-01 Concept guide ## [All seven intersection terms for three sets](/en/derivations/derivation-mko-inclusion-exclusion-principle-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D10) Audit the full alternating sum and each element's multiplicity. Inclusion-exclusion principle derivation-mko-inclusion-exclusion-principle-01 Concept guide ## [Existence, Euclid's lemma and unique factors](/en/derivations/derivation-mko-unique-prime-factorization-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D10) Every positive integer n>1 has a unique prime multiset; 360=2^3*3^2*5, with signs and units separate. Unique prime factorization of integers derivation-mko-unique-prime-factorization-01 Concept guide ## [Orientation, all degrees and a walk audit](/en/derivations/derivation-mko-finite-directed-graph-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D10) Audit a fully specified digraph under the declared loop-free simple convention. Finite directed graph derivation-mko-finite-directed-graph-01 Concept guide ## [All group elements fixing point zero](/en/derivations/derivation-mko-finite-group-stabilizer-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D10) Stab_G(0)={e,s} is a subgroup; Stab_G(3)=G, distinct from orbit and action-kernel constructions. Stabilizer of a finite group action derivation-mko-finite-group-stabilizer-01 Concept guide ## [Two convergent subsequences of an oscillatory tail](/en/derivations/derivation-mko-bolzano-weierstrass-theorem-real-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D10) For a_n=(-1)^n+1/(n+1), provide boundedness and two complete convergent subsequences. Bolzano-Weierstrass theorem for real sequences derivation-mko-bolzano-weierstrass-theorem-real-01 Concept guide ## [Compactness of a finite subspace](/en/derivations/derivation-mko-compact-subset-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D10) Prove K={b,c} compact in its ambient space and audit a relative-open cover. Compact subset derivation-mko-compact-subset-01 Concept guide ## [Total expectation with a nonempty null block](/en/derivations/derivation-mko-law-of-total-expectation-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D10) Audit the partition and compute E[X] by two routes. Law of total expectation derivation-mko-law-of-total-expectation-01 Concept guide ## [A full binomial expansion by choosing y factors](/en/derivations/derivation-mko-binomial-theorem-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D10) Establish coefficient counting and expand n=4 completely. Binomial theorem derivation-mko-binomial-theorem-01 Concept guide ## [Complete factorization and mu(30)](/en/derivations/derivation-mko-mobius-function-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D10) mu(30)=-1, mu(12)=0, mu(1)=1; the sum over every divisor of 30 is zero. Mobius function derivation-mko-mobius-function-01 Concept guide ## [A CFG production list and derivation witness](/en/derivations/derivation-mko-context-free-grammar-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D10) The grammar S to aSb or epsilon has finite derivation witnesses for aaabbb and epsilon. Context-free grammar derivation-mko-context-free-grammar-01 Concept guide ## [Fibers as left cosets of the stabilizer](/en/derivations/derivation-mko-orbit-stabilizer-theorem-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D10) At point 0 the orbit has size 3 and stabilizer size 2; all three fibers are left cosets, giving 6=3*2. Orbit-stabilizer theorem derivation-mko-orbit-stabilizer-theorem-01 Concept guide ## [The mean-value point of a quadratic](/en/derivations/derivation-mko-mean-value-theorem-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D10) Directly construct a mean-value point for f(t)=A*t^2+B*t+C on a