OCME · Bilingual mathematics & derivations
Wave6 Roots: Group Actions, Real Bounds, Topological Bases, Markov, Pigeonholes, Euler and Subsequences
On this page
- Finite group
- Function mapping
- Set
- Real number
- Usual order on the real numbers
- Quantifiers and explicit domains
- Topological space
- Open and closed sets
- Set operations
- Real random variables on finite sample spaces
- Expectation in finite models
- Probability on finite sample spaces
- Basic counting principles
- Cardinality comparison of sets
- Euler totient function
- Greatest common divisor and coprimality
- Natural-number congruence modulo a positive modulus
- Natural-number powers and principal square roots
- Sequence
- Injectivity and distinct inputs
- Finite group action
- Supremum and infimum
- Topological basis
- Markov inequality
- Pigeonhole principle
- Euler's theorem in number theory
- Subsequence
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.
Finite group
Existing canonical prerequisite for the new objects in this pulse.
Function mapping
Existing canonical prerequisite for the new objects in this pulse.
Set
Existing canonical prerequisite for the new objects in this pulse.
Real number
Existing canonical prerequisite for the new objects in this pulse.
Usual order on the real numbers
Existing canonical prerequisite for the new objects in this pulse.
Quantifiers and explicit domains
Existing canonical prerequisite for the new objects in this pulse.
Topological space
Existing canonical prerequisite for the new objects in this pulse.
Open and closed sets
Existing canonical prerequisite for the new objects in this pulse.
Set operations
Existing canonical prerequisite for the new objects in this pulse.
Real random variables on finite sample spaces
Existing canonical prerequisite for the new objects in this pulse.
Expectation in finite models
Existing canonical prerequisite for the new objects in this pulse.
Probability on finite sample spaces
Existing canonical prerequisite for the new objects in this pulse.
Basic counting principles
Existing canonical prerequisite for the new objects in this pulse.
Cardinality comparison of sets
Existing canonical prerequisite for the new objects in this pulse.
Euler totient function
Existing canonical prerequisite for the new objects in this pulse.
Greatest common divisor and coprimality
Existing canonical prerequisite for the new objects in this pulse.
Natural-number congruence modulo a positive modulus
Existing canonical prerequisite for the new objects in this pulse.
Natural-number powers and principal square roots
Existing canonical prerequisite for the new objects in this pulse.
Sequence
Existing canonical prerequisite for the new objects in this pulse.
Injectivity and distinct inputs
Existing canonical prerequisite for the new objects in this pulse.
Finite group action
An action of a finite group on a finite set is a total action table satisfying the identity and multiplication-compatibility laws.
Supremum and infimum
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.
Topological basis
A topological basis generates a topology through carrier coverage and pointwise local intersection refinement.
Markov inequality
For a nonnegative random variable with finite expectation, its tail probability at a positive threshold is at most E[X] divided by that threshold.
Pigeonhole principle
A total map from a larger finite domain into a smaller finite codomain sends two distinct source elements to the same value.
Euler's theorem in number theory
For n≥2 and gcd(a,n)=1, a^phi(n) is congruent to1 modulo n.
Subsequence
A subsequence selects terms by a strictly increasing natural-index map and preserves their original order.
Path source & status
path-pulse-foundations-20261107 · candidate