Blood Relations – A Rigorous Study Guide
For the IBPS Clerk Reasoning Ability Exam (Reasoning – “Blood Relations”)
Goal – Understand the formal structure of kinship, model it mathematically, and solve advanced puzzles with confidence.
Audience – Students at MIT‑level who want a solid theoretical foundation and sharp problem‑solving skills.
1. Core Definitions (From First Principles)
| Term | Formal Definition | Example |
|---|---|---|
| Individual | An element of a set (U) (the universe of all persons). | (a,b,c,\dots \in U) |
| Parent–Child | Binary relation (P \subseteq U \times U) where ((x,y)\in P) means x is a parent of y. | ((\text{Alice}, \text{Bob})\in P) |
| Sibling | Two distinct individuals (x\neq y) share at least one common parent. | (x \text{ and } y) are siblings if (\exists z: (z,x)\in P \land (z,y)\in P) |
| Consanguineous relation | Any relation that can be derived from (P) by following a chain of parent–child links. | Cousins, great‑aunts, etc. |
| Affinity | Relations formed by marriage (not considered “blood” in this guide). | Husband, wife, step‑sibling |
Key point – Every blood relation is a path in the directed graph induced by (P).
2. Graph‑Theoretic Model
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Graph (G=(U, E))
- Vertex set (U) (people).
- Edge set (E={(x,y): (x,y)\in P}) (directed from parent to child).
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Acyclicity
- (G) is a forest (collection of trees) because a person cannot be an ancestor of themselves.
- Theorem: (G) is a directed acyclic graph (DAG).
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Depth & Height
- Depth of vertex (v): length of longest path from a root (someone with no parents) to (v).
- Height of a tree: maximum depth among its leaves.
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Distance
- Ancestors: (d_{\text{anc}}(x,y)=) length of shortest directed path from (x) to (y).
- Descendants: (d_{\text{desc}}(x,y)=) length of shortest directed path from (y) to (x).
- Kin distance: (d_{\text{kin}}(x,y)=\min{d_{\text{anc}}(x,y), d_{\text{desc}}(x,y),) path through a common ancestor(}).
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Transitive Closure – The relation “is an ancestor of” is transitive.
[ (x,y)\in \text{Anc} \Leftrightarrow \exists n\ge1:\ (x,u_1),(u_1,u_2),\dots,(u_{n-1},y)\in P ]
3. Taxonomy of Blood Relations
| Category | Formal Criterion | Example |
|---|---|---|
| Direct Line | Ancestor–descendant chain | Grandparent, Great‑grandparent |
| Collateral | Share a common ancestor but not ancestor–descendant | Siblings, Cousins |
| Degree of Cousinship | Number of generations between you and the common ancestor minus 1 | 1st cousins share a grandparent |
| Removal | Difference in generations between two collateral relatives | 1st cousin once‑removed (one generation apart) |
3.1 Calculating Cousinship
Let (x) and (y) be two individuals.
Let (g) be the lowest common ancestor (LCA) of (x) and (y).
Let (d_x) = distance from (g) to (x).
Let (d_y) = distance from (g) to (y).
- Degree (k = \min(d_x,d_y)-1)
- Removal (r = |d_x-d_y|)
Example:
(d_x=3) (great‑grandchild), (d_y=4) (great‑great‑grandchild).
Then (k=\min(3,4)-1 = 2) → 2nd cousins,
(r=|3-4| = 1) → once‑removed → 2nd cousins once‑removed.
4. Deeper Theory
4.1 Consanguinity as an Equivalence Relation
- Define relation (C) on (U): (x C y) iff (x) and (y) share a common ancestor.
- Theorem: (C) is an equivalence relation.
- Reflexive: (x) shares an ancestor with itself (trivial).
- Symmetric: If (x) shares an ancestor (a\