# Seven Rounds of Discrete Structure: Bijections, Inverses, Relations, Graphs, Trees, and Counterexamples 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. - Stage 1 · required ## [Injectivity and distinct inputs](/en/mko/mko-injective/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261017%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-injective) [Concept guides](/en/derivations/?q=mko-injective) - Stage 1 · required ## [Surjectivity and codomain coverage](/en/mko/mko-surjective/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261017%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-surjective) [Concept guides](/en/derivations/?q=mko-surjective) - Stage 1 · required ## [Function composition](/en/mko/mko-composition/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261017%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-composition) [Concept guides](/en/derivations/?q=mko-composition) - Stage 1 · required ## [Affine functions and degree-one functions](/en/mko/mko-linear-function/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261017%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-linear-function) [Concept guides](/en/derivations/?q=mko-linear-function) - Stage 1 · required ## [Set membership](/en/mko/mko-set-membership/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261017%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-set-membership) [Concept guides](/en/derivations/?q=mko-set-membership) - Stage 1 · required ## [Set](/en/mko/mko-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261017%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-set) [Concept guides](/en/derivations/?q=mko-set) - Stage 1 · required ## [Real number](/en/mko/mko-real-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261017%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-real-number) [Concept guides](/en/derivations/?q=mko-real-number) - Stage 1 · required ## [Rational number](/en/mko/mko-rational-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261017%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-rational-number) [Concept guides](/en/derivations/?q=mko-rational-number) - Stage 1 · required ## [Quantifiers and explicit domains](/en/mko/mko-quantifier/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261017%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-quantifier) [Concept guides](/en/derivations/?q=mko-quantifier) - Stage 1 · required ## [Mathematical proof](/en/mko/mko-proof/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261017%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-proof) [Concept guides](/en/derivations/?q=mko-proof) - Stage 2 · optional ## [Bijectivity and unique preimages](/en/mko/mko-bijective/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261017%2F) Validate complete finite total maps before separating injectivity, surjectivity and bijectivity. Enumerate fibers, collisions and missing targets, qualify the converse relation, transpose output labels and distinguish cardinality equality from the given map. Empty-to-empty is bijective. [Related questions](/en/explore/?mko=mko-bijective) [Concept guides](/en/derivations/?q=mko-bijective) - Stage 2 · optional ## [Binary relations and set projections](/en/mko/mko-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261017%2F) Use ordered-pair sets with explicit finite source and target: projections and missing sources, restricted images and fibers, double converse, single-valuedness and totality. Normalize repeated pairs; a relation may be multivalued or partial and its converse is not automatically an inverse function. [Related questions](/en/explore/?mko=mko-relation) [Concept guides](/en/derivations/?q=mko-relation) - Stage 2 · optional ## [Usual order on the real numbers](/en/mko/mko-order-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261017%2F) Use the usual real total order on rational inputs with positive original denominators. Compute stable three-value sorts, inclusive interval intersections and membership slacks, and actual scaled values with positive/negative/zero direction guards. Do not extrapolate to complex numbers or arbitrary partial orders. [Related questions](/en/explore/?mko=mko-order-relation) [Concept guides](/en/derivations/?q=mko-order-relation) - Stage 2 · optional ## [Counterexamples and domain eligibility](/en/mko/mko-counterexample/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261017%2F) Use a declared finite domain and complete Boolean predicate table. Enumerate failures and least witnesses; guard candidates by the domain, recheck witnesses under restriction, and explicitly provide any new predicate value on extension. Outside inputs are not counterexamples; empty universal quantification is true and finite enumeration proves no infinite-domain claim. [Related questions](/en/explore/?mko=mko-counterexample) [Concept guides](/en/derivations/?q=mko-counterexample) - Stage 3 · optional ## [Inverse functions of bijections](/en/mko/mko-inverse-function/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261017%2F) Construct inverse tables only after bijectivity; type every candidate inverse cell before checking both identities independently. Guard queries by declared codomain. An inverse is not a reciprocal; affine recovery on R requires nonzero slope and exact normalized rational substitution. [Related questions](/en/explore/?mko=mko-inverse-function) [Concept guides](/en/derivations/?q=mko-inverse-function) - Stage 3 · optional ## [Finite simple undirected graphs](/en/mko/mko-graph/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261017%2F) Use complete simple undirected finite graphs: adjacency matrices and degree sums, neighbor/nonneighbor partitions, valid edge toggles and induced subsets with original labels. Reversed or repeated edges count once; exclude loops and do not infer adjacency from drawings or claim tree qualification here. [Related questions](/en/explore/?mko=mko-graph) [Concept guides](/en/derivations/?q=mko-graph) - Stage 4 · optional ## [Nonempty finite trees: connectivity, cycles, and unique paths](/en/mko/mko-tree/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261017%2F) Require a nonempty connected acyclic simple undirected graph. Check components/cycles, original-tree and endpoint guards before paths, degree-one leaf deletion without relabeling, and a new nonedge’s unique cycle. Empty graphs are not trees; a singleton is a tree but its degree-zero vertex is not a leaf. Edge count alone does not qualify a tree. [Related questions](/en/explore/?mko=mko-tree) [Concept guides](/en/derivations/?q=mko-tree) Path source & statuspath-pulse-foundations-20261017 · candidate [Original paths JSON](/data/architecture/learning-paths.json)