Concept guide
A complete coloring and a nonoptimal greedy trace
Audit a proper coloring edgewise and show an upper bound need not be optimal.
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Concept guide
Universal closure of a line through zero
U={(x,2x):x in R} is a subspace of R2, whereas U+(0,1) is not.
Concept guide
Closure and interior in a three-point chain topology
Find the closure and interior of A={b}.
Concept guide
Selecting a subcover by original indices
Audit a supplied open cover of K={a,b} and its subcover indices.
Concept guide
Atomwise tails and a null exception
Give a convergence or failure certificate for every atom.
Concept guide
Maximal versus maximum in P4
Audit selected edges and endpoints, then distinguish matching properties.
Concept guide
Counting phi(12) representative by representative
phi(12)=4, phi(1)=1, and here phi(3*4)=phi(3)*phi(4).
Concept guide
Tracing 1011 through a complete DFA
The specified DFA rejects 1011 and accepts epsilon.
Concept guide
Prove a basis before reading dimension
B=((1,1),(1,-1)) is an ordered basis of R2, so dim R2=2.
Concept guide
The full kernel of a two-by-two map
The kernel of T(x,y)=(x+2y,2x+4y) is span{(-2,1)}.
Concept guide
The full image from consistency
im T=span{(1,2)}; (3,6) lies in the image and (0,1) does not.
Concept guide
Exact boundary comparison with the complement
Compute the boundaries of A={b} and its complement in a chain topology.
Concept guide
A universal compactness certificate for a finite topology
Prove every open cover of the three-point chain topology has a finite subcover.
Concept guide
Bayes reversal in a four-atom test model
Compute priors, likelihoods and posteriors from a complete joint law.
Concept guide
A parity invariant for arbitrary word length
All words over {0,1} with an even number of 1s form a regular language.
Concept guide
Full RREF and rank bounds
A=[[1,2,3],[2,4,6],[1,1,1]] has row and column rank 2.
Concept guide
Pointwise tail certificates on a finite domain
Prove pointwise convergence of f_n(p)=1+2/(n+1), f_n(q)=-2-3/(n+1) to f.
Concept guide
Complete tails for geometric and telescoping series
Prove convergence of two series with exact partial sums and remainders.
Concept guide
A complete model with two parameter values and four sample points
Validate every parameter law and distinguish likelihood from posterior.
Concept guide
Excluding every nontrivial clopen set
Decide connectedness of a three-point chain topology.
Concept guide
All eight edge-subset candidates on a three-edge graph
Enumerate all2³ subsets for the maximum matching and all witnesses.
Concept guide
The normalized inverse of 17 modulo 43
The unique inverse residue class of 17 modulo 43 is represented by 38; 17*38=15*43+1.
Concept guide
Rank and nullity for three source coordinates
For T:R3 to R2, A=[[1,2,3],[2,4,6]] has rank 1 and nullity 2, with 3=1+2.
Concept guide
Exact supremum error on a three-point domain
Prove g_n(a)=1/(n+2),g_n(b)=-4/(n+2),g_n(c)=2/(n+2) converges uniformly to zero.