# Wave6 Roots: Group Actions, Real Bounds, Topological Bases, Markov, Pigeonholes, Euler and Subsequences Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations. This is a suggested learning sequence, not a universal logical dependency or a formal proof. - Stage 1 · required ## [Finite group](/en/mko/mko-finite-group/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-finite-group) [Concept guides](/en/derivations/?q=mko-finite-group) - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-function-mapping) [Concept guides](/en/derivations/?q=mko-function-mapping) - Stage 1 · required ## [Set](/en/mko/mko-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-set) [Concept guides](/en/derivations/?q=mko-set) - Stage 1 · required ## [Real number](/en/mko/mko-real-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-real-number) [Concept guides](/en/derivations/?q=mko-real-number) - Stage 1 · required ## [Usual order on the real numbers](/en/mko/mko-order-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-order-relation) [Concept guides](/en/derivations/?q=mko-order-relation) - Stage 1 · required ## [Quantifiers and explicit domains](/en/mko/mko-quantifier/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-quantifier) [Concept guides](/en/derivations/?q=mko-quantifier) - Stage 1 · required ## [Topological space](/en/mko/mko-topological-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-topological-space) [Concept guides](/en/derivations/?q=mko-topological-space) - Stage 1 · required ## [Open and closed sets](/en/mko/mko-open-closed-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-open-closed-set) [Concept guides](/en/derivations/?q=mko-open-closed-set) - Stage 1 · required ## [Set operations](/en/mko/mko-set-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-set-operations) [Concept guides](/en/derivations/?q=mko-set-operations) - Stage 1 · required ## [Real random variables on finite sample spaces](/en/mko/mko-random-variable/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-random-variable) [Concept guides](/en/derivations/?q=mko-random-variable) - Stage 1 · required ## [Expectation in finite models](/en/mko/mko-expectation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-expectation) [Concept guides](/en/derivations/?q=mko-expectation) - Stage 1 · required ## [Probability on finite sample spaces](/en/mko/mko-probability/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-probability) [Concept guides](/en/derivations/?q=mko-probability) - Stage 1 · required ## [Basic counting principles](/en/mko/mko-counting-principle/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-counting-principle) [Concept guides](/en/derivations/?q=mko-counting-principle) - Stage 1 · required ## [Cardinality comparison of sets](/en/mko/mko-cardinality-comparison/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-cardinality-comparison) [Concept guides](/en/derivations/?q=mko-cardinality-comparison) - Stage 1 · required ## [Euler totient function](/en/mko/mko-euler-totient-function/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-euler-totient-function) [Concept guides](/en/derivations/?q=mko-euler-totient-function) - Stage 1 · required ## [Greatest common divisor and coprimality](/en/mko/mko-gcd-coprime/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-gcd-coprime) [Concept guides](/en/derivations/?q=mko-gcd-coprime) - Stage 1 · required ## [Natural-number congruence modulo a positive modulus](/en/mko/mko-modular-congruence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-modular-congruence) [Concept guides](/en/derivations/?q=mko-modular-congruence) - Stage 1 · required ## [Natural-number powers and principal square roots](/en/mko/mko-power-root/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-power-root) [Concept guides](/en/derivations/?q=mko-power-root) - Stage 1 · required ## [Sequence](/en/mko/mko-sequence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-sequence) [Concept guides](/en/derivations/?q=mko-sequence) - Stage 1 · required ## [Injectivity and distinct inputs](/en/mko/mko-injective/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-injective) [Concept guides](/en/derivations/?q=mko-injective) - Stage 2 · optional ## [Finite group action](/en/mko/mko-finite-group-action/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) An action of a finite group on a finite set is a total action table satisfying the identity and multiplication-compatibility laws. [Related questions](/en/explore/?mko=mko-finite-group-action) [Concept guides](/en/derivations/?q=mko-finite-group-action) - Stage 3 · optional ## [Supremum and infimum](/en/mko/mko-supremum-and-infimum/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) For a nonempty appropriately bounded real set, the supremum is its least upper bound and the infimum its greatest lower bound; either need not belong to the set. [Related questions](/en/explore/?mko=mko-supremum-and-infimum) [Concept guides](/en/derivations/?q=mko-supremum-and-infimum) - Stage 4 · optional ## [Topological basis](/en/mko/mko-topological-basis/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) A topological basis generates a topology through carrier coverage and pointwise local intersection refinement. [Related questions](/en/explore/?mko=mko-topological-basis) [Concept guides](/en/derivations/?q=mko-topological-basis) - Stage 5 · optional ## [Markov inequality](/en/mko/mko-markov-inequality/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) For a nonnegative random variable with finite expectation, its tail probability at a positive threshold is at most E[X] divided by that threshold. [Related questions](/en/explore/?mko=mko-markov-inequality) [Concept guides](/en/derivations/?q=mko-markov-inequality) - Stage 6 · optional ## [Pigeonhole principle](/en/mko/mko-pigeonhole-principle/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) A total map from a larger finite domain into a smaller finite codomain sends two distinct source elements to the same value. [Related questions](/en/explore/?mko=mko-pigeonhole-principle) [Concept guides](/en/derivations/?q=mko-pigeonhole-principle) - Stage 7 · optional ## [Euler's theorem in number theory](/en/mko/mko-euler-theorem-number-theory/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) For n≥2 and gcd(a,n)=1, a^phi(n) is congruent to1 modulo n. [Related questions](/en/explore/?mko=mko-euler-theorem-number-theory) [Concept guides](/en/derivations/?q=mko-euler-theorem-number-theory) - Stage 8 · optional ## [Subsequence](/en/mko/mko-subsequence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261107%2F) A subsequence selects terms by a strictly increasing natural-index map and preserves their original order. [Related questions](/en/explore/?mko=mko-subsequence) [Concept guides](/en/derivations/?q=mko-subsequence) Path source & statuspath-pulse-foundations-20261107 · candidate [Original paths JSON](/data/architecture/learning-paths.json)