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Relations – JEE Mathematics Notes

1.2.1 Definition

Let A and B be two non-empty sets. Every subset of A × B defines a relation from A to B, and every relation from A to B is a subset of A × B. If R ⊆ A × B and (a, b) ∈ R, then we say that a is related to b by the relation R and write it as aRb.

Example: Let A = {1, 2, 5, 8, 9}, B = {1, 3}. Define a relation R from A to B as: a R b if a ≤ b. Then R = {(1,1), (1,3), (2,3)} ⊆ A × B.

(1) Total Number of Relations

If A and B are finite sets with |A| = m and |B| = n, then A × B has m × n elements and total relations from A to B = 2^(m × n).

Example: If |A| = 2, |B| = 3, then A × B has 2×3 = 6 elements. Total relations = 2^6 = 64.

(2) Domain and Range

For R ⊆ A × B: Domain(R) = {a ∈ A : (a,b) ∈ R for some b ∈ B} and Range(R) = {b ∈ B : (a,b) ∈ R for some a ∈ A}. Domain is always a subset of A, and Range is always a subset of B.

(3) Relation on a Set

If A = B, then any subset of A × A is called a relation on A.

Example: If A = {1, 2, 3}, then total relations on A = 2^(3×3) = 2^9 = 512.

1.2.2 Inverse Relation

If R ⊆ A × B, then the inverse relation R⁻¹ is defined as R⁻¹ = {(b,a) : (a,b) ∈ R} ⊆ B × A.

Properties: Domain(R⁻¹) = Range(R) and Range(R⁻¹) = Domain(R).

Example: Let A = {a,b,c}, B = {1,2,3}, R = {(a,1), (a,3), (b,3), (c,3)}. Then R⁻¹ = {(1,a), (3,a), (3,b), (3,c)}. Domain(R) = {a,b,c}, Range(R) = {1,3}.

1.2.3 Types of Relations

Reflexive: R on A is reflexive if (a,a) ∈ R for all a ∈ A. Example: On A={1,2,3}, R={(1,1),(2,2),(3,3)} is reflexive.

Symmetric: R is symmetric if (a,b) ∈ R ⇒ (b,a) ∈ R for all a,b ∈ A. Example: R = {(1,2), (2,1)} is symmetric on A={1,2}.

Anti-Symmetric: R is anti-symmetric if (a,b) ∈ R and (b,a) ∈ R ⇒ a=b. Example: “≤” on real numbers is anti-symmetric.

Transitive: R is transitive if (a,b) ∈ R and (b,c) ∈ R ⇒ (a,c) ∈ R. Example: “≤” on real numbers is transitive.

Identity Relation: I_A = {(a,a) : a ∈ A}. Always reflexive, symmetric, transitive.

Equivalence Relation: R is equivalence if it is Reflexive, Symmetric, and Transitive. Example: Congruence modulo m on integers is an equivalence relation.

1.2.4 Equivalence Classes

If R is an equivalence relation on A and a ∈ A, then the equivalence class of a is [a] = {x ∈ A : x R a}.

Properties: If b ∈ [a], then [a] = [b] and two equivalence classes are either identical or disjoint.

Example: For n=5, congruence modulo 5 gives equivalence classes: [0] = {…, -10, -5, 0, 5, 10, …} and [1] = {…, -9, -4, 1, 6, 11, …}.