Concept guide
The full unit permutation modulo 10
For a=-3,n=10, phi(10)=4 and a^4=1 modulo 10.
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Concept derivation guides
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Concept guide
An all-index subsequence certificate for affine indices
Construct a subsequence of a_n=3n-2 using k(j)=1+2j, then take a further subsequence.
Concept guide
Complete orbits and their partition
The distinct orbits are {0,1,2} and {3}; same-orbit is an equivalence relation.
Concept guide
A bounded monotone tail and the supremum argument
For a_n=3-2/(n+1), prove monotonicity, boundedness, supremum 3 and convergence.
Concept guide
Relative openness differs from ambient openness
Compute the subspace topology and all relative closed sets of Y={b,c}.
Concept guide
Chebyshev from squared deviations
Derive Chebyshev from Markov and compute an exact instance.
Concept guide
All seven intersection terms for three sets
Audit the full alternating sum and each element's multiplicity.
Concept guide
Existence, Euclid's lemma and unique factors
Every positive integer n>1 has a unique prime multiset; 360=2^3*3^2*5, with signs and units separate.
Concept guide
Orientation, all degrees and a walk audit
Audit a fully specified digraph under the declared loop-free simple convention.
Concept guide
All group elements fixing point zero
Stab_G(0)={e,s} is a subgroup; Stab_G(3)=G, distinct from orbit and action-kernel constructions.
Concept guide
Two convergent subsequences of an oscillatory tail
For a_n=(-1)^n+1/(n+1), provide boundedness and two complete convergent subsequences.
Concept guide
Compactness of a finite subspace
Prove K={b,c} compact in its ambient space and audit a relative-open cover.
Concept guide
Total expectation with a nonempty null block
Audit the partition and compute E[X] by two routes.
Concept guide
A full binomial expansion by choosing y factors
Establish coefficient counting and expand n=4 completely.
Concept guide
Complete factorization and mu(30)
mu(30)=-1, mu(12)=0, mu(1)=1; the sum over every divisor of 30 is zero.
Concept guide
A CFG production list and derivation witness
The grammar S to aSb or epsilon has finite derivation witnesses for aaabbb and epsilon.
Concept guide
Fibers as left cosets of the stabilizer
At point 0 the orbit has size 3 and stabilizer size 2; all three fibers are left cosets, giving 6=3*2.
Concept guide
The mean-value point of a quadratic
Directly construct a mean-value point for f(t)=A*t^2+B*t+C on a<b.
Concept guide
Pull back covers and return the same indices
Demonstrate the compact-image cover proof with an explicit finite map.
Concept guide
Exact within-between variance decomposition
Decompose variance in a complete law with a null block.
Concept guide
An oldest-to-newest order-two recurrence trace
Align states, coefficients and all initial values, generate a5 and explain uniqueness.
Concept guide
Divisor-poset inversion of an affine function
For f(k)=2k+1 at n=6, F(6)=28 and inversion returns f(6)=13; n=1 roundtrips too.
Concept guide
A bottom-first PDA run accepting aabb
The specified PDA has a finite run consuming aabb and ending at final f with stack [Z].
Concept guide
Lagrange counting from all left cosets
H={e,s} has three distinct left cosets of size 2, giving 6=[G:H]|H|=3*2.