OCME · Bilingual mathematics & derivations
Daily Seven-Round Cycle I: Quantifiers, Discrete Structures, Linear Systems, and Conditional Models
On this page
- Proposition
- Set
- Mathematical proof
- Set membership
- Real number
- Arithmetic operations
- Coordinate plane
- Linear equation in one variable
- Probability on finite sample spaces
- Set operations
- Quantifiers and explicit domains
- Counterexamples and domain eligibility
- Binary relations and set projections
- Finite simple undirected graphs
- Plane vectors in a fixed basis
- Linear systems in two variables and degenerate constraints
- Conditional probability in finite 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.
Proposition
Existing canonical prerequisite for the new objects in this pulse.
Set
Existing canonical prerequisite for the new objects in this pulse.
Mathematical proof
Existing canonical prerequisite for the new objects in this pulse.
Set membership
Existing canonical prerequisite for the new objects in this pulse.
Real number
Existing canonical prerequisite for the new objects in this pulse.
Arithmetic operations
Existing canonical prerequisite for the new objects in this pulse.
Coordinate plane
Existing canonical prerequisite for the new objects in this pulse.
Linear equation in one variable
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.
Set operations
Existing canonical prerequisite for the new objects in this pulse.
Quantifiers and explicit domains
Universal and existential quantifiers must bind a domain. Under classical truth semantics, a universal statement over the empty domain is true and an existential statement is false.
Counterexamples and domain eligibility
A counterexample must belong to the claimed domain and falsify the universal predicate. An outside value or failure to find a counterexample does not by itself prove the original statement.
Binary relations and set projections
A binary relation is a set of ordered pairs in a specified A×B. An input need not have a unique output or any output.
Finite simple undirected graphs
This entry covers finite simple undirected graphs. An edge is an unordered set of two distinct vertices, and degree counts distinct neighbors.
Plane vectors in a fixed basis
This entry represents plane vectors as ordered pairs in R² in a fixed basis, defining addition and scalar multiplication componentwise.
Linear systems in two variables and degenerate constraints
A solution must satisfy two real linear constraints simultaneously. A nonzero determinant gives a unique solution; singular systems and zero rows require separate consistency checks.
Conditional probability in finite models
The conditioning event must have positive probability. In a finite model, P(A|B)=P(A∩B)/P(B), and the direction cannot be freely exchanged.
Path source & status
path-pulse-foundations-20261004 · candidate