# Daily Seven-Round Cycle I: Quantifiers, Discrete Structures, Linear Systems, and Conditional Models Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations. [← Return to results](/en/paths/) This is a suggested learning sequence, not a universal logical dependency or a formal proof. - Stage 1 · required ## [Proposition](/en/mko/mko-proposition/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261004%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-proposition) [Concept guides](/en/derivations/?q=mko-proposition) - Stage 1 · required ## [Set](/en/mko/mko-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261004%2F%3Ffrom%3D%252Fen%252Fpaths%252F) 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 ## [Mathematical proof](/en/mko/mko-proof/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261004%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-proof) [Concept guides](/en/derivations/?q=mko-proof) - Stage 1 · required ## [Set membership](/en/mko/mko-set-membership/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261004%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-set-membership) [Concept guides](/en/derivations/?q=mko-set-membership) - Stage 1 · required ## [Real number](/en/mko/mko-real-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261004%2F%3Ffrom%3D%252Fen%252Fpaths%252F) 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 ## [Arithmetic operations](/en/mko/mko-arithmetic-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261004%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-arithmetic-operations) [Concept guides](/en/derivations/?q=mko-arithmetic-operations) - Stage 1 · required ## [Coordinate plane](/en/mko/mko-coordinate-plane/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261004%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-coordinate-plane) [Concept guides](/en/derivations/?q=mko-coordinate-plane) - Stage 1 · required ## [Linear equation in one variable](/en/mko/mko-linear-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261004%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-linear-equation) [Concept guides](/en/derivations/?q=mko-linear-equation) - Stage 1 · required ## [Probability on finite sample spaces](/en/mko/mko-probability/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261004%2F%3Ffrom%3D%252Fen%252Fpaths%252F) 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 ## [Set operations](/en/mko/mko-set-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261004%2F%3Ffrom%3D%252Fen%252Fpaths%252F) 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 2 · optional ## [Quantifiers and explicit domains](/en/mko/mko-quantifier/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261004%2F%3Ffrom%3D%252Fen%252Fpaths%252F) 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. [Related questions](/en/explore/?mko=mko-quantifier) [Concept guides](/en/derivations/?q=mko-quantifier) - Stage 3 · optional ## [Counterexamples and domain eligibility](/en/mko/mko-counterexample/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261004%2F%3Ffrom%3D%252Fen%252Fpaths%252F) 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. [Related questions](/en/explore/?mko=mko-counterexample) [Concept guides](/en/derivations/?q=mko-counterexample) - Stage 4 · optional ## [Binary relations and set projections](/en/mko/mko-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261004%2F%3Ffrom%3D%252Fen%252Fpaths%252F) 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. [Related questions](/en/explore/?mko=mko-relation) [Concept guides](/en/derivations/?q=mko-relation) - Stage 5 · optional ## [Finite simple undirected graphs](/en/mko/mko-graph/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261004%2F%3Ffrom%3D%252Fen%252Fpaths%252F) This entry covers finite simple undirected graphs. An edge is an unordered set of two distinct vertices, and degree counts distinct neighbors. [Related questions](/en/explore/?mko=mko-graph) [Concept guides](/en/derivations/?q=mko-graph) - Stage 6 · optional ## [Plane vectors in a fixed basis](/en/mko/mko-vector/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261004%2F%3Ffrom%3D%252Fen%252Fpaths%252F) This entry represents plane vectors as ordered pairs in R² in a fixed basis, defining addition and scalar multiplication componentwise. [Related questions](/en/explore/?mko=mko-vector) [Concept guides](/en/derivations/?q=mko-vector) - Stage 7 · optional ## [Linear systems in two variables and degenerate constraints](/en/mko/mko-system-linear-equations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261004%2F%3Ffrom%3D%252Fen%252Fpaths%252F) 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. [Related questions](/en/explore/?mko=mko-system-linear-equations) [Concept guides](/en/derivations/?q=mko-system-linear-equations) - Stage 8 · optional ## [Conditional probability in finite models](/en/mko/mko-conditional-probability/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261004%2F%3Ffrom%3D%252Fen%252Fpaths%252F) 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. [Related questions](/en/explore/?mko=mko-conditional-probability) [Concept guides](/en/derivations/?q=mko-conditional-probability) Path source & statuspath-pulse-foundations-20261004 · candidate [Original paths JSON](/data/architecture/learning-paths.json)