OCME · Bilingual mathematics & derivations
Cross-Domain Depth: Algebra, Sets, Distances, Finite Probability, and Quantifiers
On this page
- Variables and algebraic expressions
- Natural-number powers and principal square roots
- Rational number
- Set membership
- Coordinate plane
- Euclidean length
- Probability on finite sample spaces
- Real random variables on finite sample spaces
- Proposition
- Set
- Univariate polynomials over the rationals
- Set operations
- Euclidean distance in an equal-unit orthonormal frame
- Expectation in finite models
- Quantifiers and explicit domains
- Conditional probability in finite models
- Variance in finite probability models
Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices.
This is a suggested learning sequence, not a universal logical dependency or a formal proof.
Variables and algebraic expressions
Existing canonical prerequisite for this density pulse.
Natural-number powers and principal square roots
Existing canonical prerequisite for this density pulse.
Rational number
Existing canonical prerequisite for this density pulse.
Set membership
Existing canonical prerequisite for this density pulse.
Coordinate plane
Existing canonical prerequisite for this density pulse.
Euclidean length
Existing canonical prerequisite for this density pulse.
Probability on finite sample spaces
Existing canonical prerequisite for this density pulse.
Real random variables on finite sample spaces
Existing canonical prerequisite for this density pulse.
Proposition
Existing canonical prerequisite for this density pulse.
Set
Existing canonical prerequisite for this density pulse.
Univariate polynomials over the rationals
Use the integer-coefficient slice of Q[x] through input degree 3: complete normalized sum/product coefficients (products through degree 6), ordered evaluation at two points, and complete coefficients of p(x+c) by exact expansion. Equality at one point is not polynomial identity.
Set operations
Normalize finite explicit integer sets; compute union/intersection cardinalities, symmetric differences, both relative De Morgan sets, and ordered three-part partitions. Require A,B⊆the declared U before complements; finite instances do not create general set-theory formal Evidence.
Euclidean distance in an equal-unit orthonormal frame
Vectors are integer coordinate differences from a common origin in one equal-unit orthogonal real plane. Check ordered squares, candidate sign and square defect, indexed three-distance tie groups, and sums of two integer principal distances. Exclude nonorthogonal/unequal-scale and default product-metric interpretations; a two-segment sum is not another segment’s length.
Expectation in finite models
On a shared normalized finite integer-valued model, compute wX contributions, E[X−Y], ordered (E[XY],E[X]E[Y]), and an integer value at an explicit null slot of positive mass from a known mean. No independence, sample-estimation, or general integrability claim; equal numerical expressions do not establish independence.
Quantifiers and explicit domains
Enumerate finite explicit integer domains with order, equality, or nonzero-divisibility predicates: true/false partitions, (¬∀P,¬∃P), typed counterexample masks, and ordered nested existence/universality counts on U/V. Empty-domain universal is True and existential False. Do not exchange quantifiers or extrapolate to all integers or general first-order proofs.
Conditional probability in finite models
Use explicit finite S, its full powerset, nonnegative integer masses, and positive D with total mass D. Require positive B mass, and positive A mass for both directions. Compute probability pairs, directional conditionals, every conditional singleton weight, and signed candidate defects. Nonempty zero-mass events cannot condition; no Bayes or independence proof is claimed.
Variance in finite probability models
Compute w(X−μ)², exact (E[X²],μ²), variance after forming the shared-model difference, and the first positive-support mismatch relative to the first positive-weight value. Preserve zero-weight outliers; do not use n−1 sample variance, split Var(X−Y) into two variances, or claim complete covariance theory.
Path source & status
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