AJAY PATHAK (MATH)

Recurrence Relations A recurrence relation defines a sequence in terms of its previous terms. Such relations occur frequently in mathematics, computer science, and discrete structures. Step Rule 1 Write the characteristic equation for the given recurrence relation. 2 Solve the characteristic equation and find all roots. 3 The order of the recurrence = highest index − lowest index. Nature of Roots Complementary Solution a n (h) Real and distinct λ₁ ≠ λ₂ ≠ … ≠ λₖ a n = C₁λ₁ n + C₂λ₂ n + … + Cₖλₖ n Real and equal λ₁ = λ₂ = … = λₖ a n = (C₁ + C₂n + C₃n² + …) λ₁ n Complex conjugate roots λ = α ± iβ a n = r n [ C₁ cos(nθ) + C₂ sin(nθ) ] where r = √(α² + β²),   θ = tan −1 (β/α) f(n) Assumed Particular Solution a n (p) Constant (C) P₀ Polynomial a + bn + cn² + … P₀ + P₁n + P₂n² + … a·b n , b ≠ λ P₀·b n a·b n , b = λ (with multiplicity m) P₀·n m b n Total ...

Matrix Representation of a Relation in Discrete Mathematics In Discrete Mathematics, relations can be represented in different ways such as ordered pairs, digraphs, and matrices. Among these, the matrix representation of a relation is very useful for performing operations like union, intersection, complement, and composition of relations. Matrix Representation of a Relation Let $A = \{a_1, a_2, a_3, \dots, a_n\}$ $B = \{b_1, b_2, b_3, \dots, b_m\}$ be two finite sets containing $n$ and $m$ elements respectively. Then the Cartesian product $A \times B$ contains $n \times m$ ordered pairs. Let $R$ be a relation from set $A$ to set $B$. Then, $$ R \subseteq A \times B $$ The matrix of relation $R$ , denoted by $M_R$, is an $n \times m$ matrix defined as follows: $$ M_R = [m_{ij}] $$ where $m_{ij} = 1$ if $(a_i, b_j) \in R$ $m_{ij} = 0$ if $(a_i, b_j) \notin R$ Question 1 Let $A = \{1,2,3,4\}$ $B = \{1,4,6,8,9,16\}$ A rel...

Types of Relations in Discrete Mathematics – Definitions, Examples & Solved Problems Relations are a fundamental concept in Discrete Mathematics and Graph Theory . They are widely used in equivalence relations, partial order relations, posets, lattices, and many real-life applications. Questions based on relations are frequently asked in GTU and engineering mathematics examinations . In this article, we explain the types of relations in a clear and exam-oriented manner with suitable examples and fully solved problems. Question 1 Explain types of a relation with a suitable example. Solution Definition of Relation Let A and B be two non-empty sets. A relation R from set A to set B is defined as a subset of the Cartesian product A × B . Mathematically, $$ R \subseteq A \times B $$ Types of Relations 1. Reflexive Relation A relation R on a set A is said to be reflexive if every element of A is related to itself. $$ (a,a) \in R \quad \text{for ...