OCME · Bilingual mathematics & derivations
Seven Rounds of Discrete Structure: Bijections, Inverses, Relations, Graphs, Trees, and Counterexamples
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- Injectivity and distinct inputs
- Surjectivity and codomain coverage
- Function composition
- Affine functions and degree-one functions
- Set membership
- Set
- Real number
- Rational number
- Quantifiers and explicit domains
- Mathematical proof
- Bijectivity and unique preimages
- Binary relations and set projections
- Usual order on the real numbers
- Counterexamples and domain eligibility
- Inverse functions of bijections
- Finite simple undirected graphs
- Nonempty finite trees: connectivity, cycles, and unique paths
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.
Injectivity and distinct inputs
Existing canonical prerequisite for this density pulse.
Surjectivity and codomain coverage
Existing canonical prerequisite for this density pulse.
Function composition
Existing canonical prerequisite for this density pulse.
Affine functions and degree-one functions
Existing canonical prerequisite for this density pulse.
Set membership
Existing canonical prerequisite for this density pulse.
Set
Existing canonical prerequisite for this density pulse.
Real number
Existing canonical prerequisite for this density pulse.
Rational number
Existing canonical prerequisite for this density pulse.
Quantifiers and explicit domains
Existing canonical prerequisite for this density pulse.
Mathematical proof
Existing canonical prerequisite for this density pulse.
Bijectivity and unique preimages
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.
Binary relations and set projections
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.
Usual order on the real numbers
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.
Counterexamples and domain eligibility
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.
Inverse functions of bijections
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.
Finite simple undirected graphs
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.
Nonempty finite trees: connectivity, cycles, and unique paths
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.
Path source & status
path-pulse-foundations-20261017 · candidate