Blood Relations – Advanced Reasoning & Theory
(MIT‑Level Study Notes – Reasoning Ability – Chandigarh Police SI)
Goal: Master the formal reasoning of kinship, handle edge cases, and solve complex puzzles rapidly.
1. Foundations – The “First Principles”
| Concept | Formal Definition | Typical Use in Reasoning |
|---|---|---|
| Person (p) | Element of a set (P). | Base unit of any relation. |
| Parent of (p) | A person (q) such that (q) is a biological or legal parent of (p). | Directed “parent → child” edge. |
| Child of (p) | A person (q) such that (p) is a parent of (q). | Reverse of above. |
| Sibling | Two persons share both parents. | Same‑branch relation. |
| Half‑sibling | Share exactly one parent. | Partial sibling relation. |
| Ancestor | Reflexive‑transitive closure of “parent” relation. | (a \rightarrow^* b). |
| Descendant | Reflexive‑transitive closure of “child” relation. | (b \rightarrow^* a). |
| Collateral relative | Two persons with a common ancestor but none is ancestor of the other. | Cousin‐type relations. |
| Degree of kinship | Number of edges in the shortest ancestral path between two persons. | “Nth cousin” etc. |
| In‑law | Relationship obtained through marriage, not blood. | Excluded unless explicitly stated. |
Key Insight: Model the family as a directed acyclic graph (DAG) (G=(P,E)) where (E\subseteq P\times P) and ((q,p)\in E) iff (q) is a parent of (p).
All reasoning about blood relations reduces to graph‑theoretic queries on (G).
2. Graph‑Theoretic Machinery
2.1 Ancestor/Descendant Queries
- Breadth‑first search (BFS) from a node gives all descendants in (O(|V|+|E|)).
- Reverse‑graph BFS gives all ancestors.
- Depth‑first search (DFS) can compute lowest common ancestors (LCAs), useful for cousin relationships.
2.2 Distance / Degree of Separation
- The shortest path length between two nodes (u,v) in the undirected version of (G) equals their degree of kinship.
- For cousins, the path is (u \rightarrow \dots \rightarrow \text{LCA} \leftarrow \dots \leftarrow v).
- If LCA is (k) generations above (u) and (m) generations above (v):
- Degree = (\min(k,m)-1) (cousin count).
- Removal = (|k-m|).
- If LCA is (k) generations above (u) and (m) generations above (v):
2.3 Handling Multiple Parents
- Bipedal families → each child has two parent edges.
- The graph remains a DAG because biological reproduction is acyclic.
- Half‑siblings appear when two children share exactly one parent edge.
2.4 Disjoint Set Union (Union‑Find)
- Useful to answer equivalence queries quickly (e.g., “Are (X) and (Y) related?”).
- Union when adding a parent–child edge.
- Find to determine if two nodes are in the same connected component (i.e., related by blood).
3. Deeper Theory – Algebraic View
3.1 Partial Order & Rank
- Define a partial order (\preceq) where (p \preceq q) iff (p) is an ancestor of (q).
- The rank of a node is its distance from a root (generation number).
- The height of a node (p) is the length of the longest ancestor path ending at (p).
3.2 Equivalence Relations
- Sibling relation is an equivalence relation on the set of children of a given pair of parents.
- Half‑sibling is a partial relation (not transitive).
- Cousin is defined by a shared ancestor but not ancestor/descendant; forms an equivalence class under “share a common ancestor at generation (k)”.
3.3 Lattice of Relations
- The set of all blood relationships forms a lattice under the “is a more distant relative of” partial order.
- Bottom: self (reflexive).
- Top: no relation (dis