Blood relations: Reasoning Ability Notes for Coast Guard

Blood‑Relations – Advanced Study Notes

Coast Guard – Reasoning Ability
(MIT‑level, first principles, rigorous, edge cases & a challenging problem)

1. Introduction

Blood‑relation questions test the ability to infer relationships from a set of facts.
Typical exam items:

“If A is B’s mother, and C is B’s sister, who is D relative to A?”
“Given a chain of relationships, find the degree of kinship between two persons.”

These questions are essentially graph‑theoretic or logical problems.
We formalise them and then discuss algorithmic and theoretical aspects.

2. Fundamental Definitions

Symbol	Definition	Example
`P(x, y)`	x is the parent of y	`P(A,B)`
`C(x, y)`	x is the child of y (inverse of parent)	`C(B,A)`
`S(x, y)`	x is the sibling of y (share at least one parent)	`S(B,C)`
`M(x, y)`	x is the married to y	`M(A,D)`
`E(x, y, n)`	x is the n‑th cousin of y	`E(B,C,1)` (first cousins)
`G⁺(x, y)`	x is any ancestor of y (transitive closure of `P`)	`G⁺(A,D)` if A→B→C→D
`D(x, y)`	x is any descendant of y (inverse of `G⁺`)	`D(D,A)`

2.1 Basic Rules

Parent → Child: P(x, y) ↔ C(y, x)
Sibling: S(x, y) ⇔ ∃p (P(p,x) ∧ P(p,y) ∧ x≠y)
Cousins: E(x, y, n) ⇔
∃a,b (G⁺(a,x) ∧ G⁺(b,y) ∧ a≠b ∧ depth(a)=depth(b)+n+1 ∧ depth(b)=depth(a)+n+1)
Marriage does not affect blood relation (except for adoptive rules).
Adoption: treat adoptive parent as “parent” for legal purposes, but not for genetic kinship in reasoning‑ability problems unless stated.

3. Graph Representation

3.1 Directed Graph

Vertices: individuals.
Edges: directed from parent to child (P).

The graph is a Directed Acyclic Graph (DAG) if we ignore marriages and adoptions (no cycles).

3.2 Undirected Graph

Edges: sibling relationships (S).
Useful for exploring degree of kinship (distance in the undirected graph).

3.3 Mixed Graph

Combines directed (parent‑child) and undirected (siblings, cousins) edges.

4. From First Principles to Reasoning

4.1 First‑Order Logic (FOL) Encoding

Predicates: Parent(x, y), Sibling(x, y), Cousin(x, y, n).

Axioms:

∀x ∀y (Parent(x, y) → ¬Parent(y, x))          // no cycles
∀x ∀y (Parent(x, y) → ∃z (Sibling(x, z) ∧ Sibling(y, z)))  // siblings share a parent
∀x ∀y (Sibling(x, y) → Sibling(y, x))
∀x ∀y ∀z (Sibling(x, y) ∧ Sibling(y, z) ∧ x≠z → Sibling(x, z))  // transitivity through shared parent

Inference: apply resolution or tableau methods to deduce new relations.

4.2 Set‑Theoretic View

Define for each person p:

Anc(p) = set of all ancestors (transitive closure of parents).
Desc(p) = set of all descendants.

Kinship degree between x and y is the minimal k such that Anc(x) ∩ Desc(y) ≠ ∅ or vice‑versa, plus sibling adjustments.

4.3 Algebraic Notation

Let R be the relationship matrix where R[i][j] encodes the shortest path between person i and j in the family graph.

For parent/child: R[i][j] = 1 if P(i,j) or P(j,i).
For siblings: R[i][j] = 2 (parent → child → sibling).
For cousins: R[i][j] = 2n + 2 where n = cousin degree.

The matrix exponentiation trick can compute all