Relations

• How a relation from set A to set B is a subset of A × B, and how to represent it as a set of ordered pairs or a relation matrix.
• The definitions of reflexive, symmetric, transitive, and antisymmetric relations — tested heavily in CBSE and JEE.
• To identify equivalence relations and partition a set into equivalence classes.
• To find domain, range, and co-domain of a relation and interpret relations on real numbers (≤, <, divides).

Level 1 — Cartesian product and relation basics

Verbal: A relation R from A to B picks out certain ordered pairs (a, b) where a ∈ A and b ∈ B. Every relation is a subset of the Cartesian product A × B.

Symbolic: A × B = {(a, b) : a ∈ A, b ∈ B}. If A = {1, 2} and B = {a, b}, then A × B has 2 × 2 = 4 pairs. A relation R ⊆ A × B might be R = {(1, a), (2, b)}.

Domain: {a : (a, b) ∈ R}. Range: {b : (a, b) ∈ R}. Co-domain: B (the target set).

Number of relations: From A (m elements) to B (n elements), there are 2^(mn) possible relations (each subset of A × B).

Level 2 — Types of relations on a set A

A relation R on A (i.e., R ⊆ A × A) may have these properties:

Equivalence relation: Reflexive + symmetric + transitive. Example: "has same remainder on division by 5" on ℤ partitions integers into 5 classes: [0], [1], …, [4].

Partial order: Reflexive + antisymmetric + transitive. Example: set inclusion ⊆ on P(A).

NCERT spotlight — Relations on finite sets

For a finite set A with n elements, the number of reflexive relations on A is 2^(n^2 - n) because all (a,a) pairs must be included. A relation matrix uses 1 if (i,j) belongs to R. Reflexive means diagonal entries are all 1.

Functions as relations: A relation from A to B is a function iff every a in A appears in exactly one ordered pair. Number of functions from A to B equals |B|^|A|, which is far smaller than 2^(|A||B|) total relations.

Equivalence classes: Integers modulo 3 partition Z into three disjoint classes. Each class contains all integers with the same remainder when divided by 3.

Let A = {1, 2, 3, 4}. Define R = {(a, b) : a, b ∈ A, a divides b}. List R and state which properties R satisfies.

Step 1 — List pairs where a|b:
         (1,1),(1,2),(1,3),(1,4),(2,2),(2,4),(3,3),(4,4).
Step 2 — Reflexive: a|a for all a → yes.
Step 3 — Symmetric? 1|2 but 2∤1 → not symmetric.
Step 4 — Transitive: if a|b and b|c then a|c → yes.
Step 5 — Antisymmetric: if a|b and b|a with a,b ∈ A, then a = b → yes.
         R is a partial order (not equivalence, since not symmetric).

Applications — divisibility and modular arithmetic

On set {1,2,...,12}, define a R b if a divides b. This relation is reflexive and antisymmetric but not symmetric (2|4 but 4 does not divide 2). Transitive because a|b and b|c implies a|c. Such concrete relations clarify abstract definitions better than arbitrary letter pairs alone.

Review and practice drill

Review checklist: (1) State reflexive, symmetric, transitive with counterexample for each failure mode. (2) Count relations: total 2^(n^2) on set of n elements. (3) Equivalence classes partition the set — disjoint and cover all elements. (4) Function is relation with unique output per input. Practice: On {1,2,3}, relation R = {(1,1),(2,2),(3,3),(1,2),(2,1)} — reflexive and symmetric; is it transitive? Yes, (1,2) and (2,1) imply (1,1) in R already; check all triples systematically.

• How many relations exist from A = {1, 2} to B = {x}? (List |A × B| first.)
• On ℤ, is "a − b is even" an equivalence relation? Justify.
• Find domain and range of R = {(1,2), (3,2), (4,5)} if co-domain is {1,2,3,4,5}.

Open the Practice tab for graded questions on Relations.

• Use the platform-native live simulation or PhET-style tool for this topic (number line, Venn, physics playground, molecule builder, sensor dashboard, etc.).
• Mirror / body / home activity: physically do the concept (count objects, measure, role-play) and photograph or describe for portfolio.
• Voice or text reflection with AI Mentor: explain the concept to a younger student or family member.

• "Explain this concept to a Class 6 student using one real example from an Indian home, school, market, or festival."
• "What is one common mistake students make here, and how would you catch yourself making it?"
• Stretch: "How does this connect to coding, robotics, money, health, environment, or a future career?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.