OCME · Bilingual mathematics & derivations
Function Invertibility and Finite Probability Moments: Seven-Round Workflow
On this page
- Function mapping
- Quantifiers and explicit domains
- Function composition
- Affine functions and degree-one functions
- Sample space
- Real number
- Probability on finite sample spaces
- Natural-number powers and principal square roots
- Injectivity and distinct inputs
- Surjectivity and codomain coverage
- Bijectivity and unique preimages
- Inverse functions of bijections
- Real random variables on finite sample spaces
- Expectation in finite models
- Variance in finite probability models
Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations.
This is a suggested learning sequence, not a universal logical dependency or a formal proof.
Function mapping
Existing canonical prerequisite for the new objects in this pulse.
Quantifiers and explicit domains
Existing canonical prerequisite for the new objects in this pulse.
Function composition
Existing canonical prerequisite for the new objects in this pulse.
Affine functions and degree-one functions
Existing canonical prerequisite for the new objects in this pulse.
Sample space
Existing canonical prerequisite for the new objects in this pulse.
Real number
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.
Natural-number powers and principal square roots
Existing canonical prerequisite for the new objects in this pulse.
Injectivity and distinct inputs
Injectivity requires distinct permitted inputs to have distinct outputs; it does not require coverage of the whole codomain.
Surjectivity and codomain coverage
Surjectivity requires a permitted preimage for every codomain element; an output may have several preimages.
Bijectivity and unique preimages
A bijection is both injective and surjective; equivalently, every codomain element has exactly one permitted preimage.
Inverse functions of bijections
For a bijection f:A→B, an inverse g:B→A restores values in both directions; exchanging a table is not an unconditional inverse construction.
Real random variables on finite sample spaces
A random variable in a finite model is a total function from elementary outcomes to real numbers, distinct from an outcome or its probability weight.
Expectation in finite models
Finite expectation is the weighted sum of random-variable values under a valid probability law; linearity does not require independence.
Variance in finite probability models
This entry defines variance as E[(X−E[X])²], distinct from sample variance with an n−1 denominator. Zero-weight outliers do not affect it.
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