AJAY PATHAK (MATH)

🚆 Mathematics Behind "Where is My Train" App In today’s digital world, apps like “Where is My Train” have made railway travel much easier. With just one click, we can know the exact location of a train, its delay, and expected arrival time. But have you ever wondered — what mathematics works behind this smart system? Let’s explore the beautiful mathematics and logic behind this application in a simple and interesting way. 📍 1. Position Estimation (Basic Idea) The main goal of the app is to find the current position of the train . Mathematically, this depends on: Position = f(Time, Speed, Distance) This means the location of the train depends on how fast it is moving and how much distance it has covered over time. 🌍 2. GPS and Distance Calculation The app uses GPS coordinates (latitude φ and longitude λ) to track the train in real time. Each location on Earth is represented as a point (φ₁, λ₁) and (φ₂, λ₂). To find the distance between these...

Q.3 (a) Question: Define a complete graph and find the order and size of the graph K 2025 . Definition: Complete Graph A complete graph is a simple graph in which every pair of distinct vertices is connected by exactly one edge . A complete graph with n vertices is denoted by K n . Order of the Graph: The order of a graph is the number of vertices in the graph. Order of K 2025 = 2025 Size of the Graph: The size of a graph is the number of edges. A complete graph with n vertices has: n(n − 1)/2 edges For K 2025 : = 2025 × 2024 / 2 = 2025 × 1012 = 2049300 edges Final Answer: Order of K 2025 = 2025 Size of K 2025 = 2049300 Q.3 (b) Question: Show that the proposition [(p → q) ∧ (q → r)] → (p → r) is a tautology . Method: Trut...

Q.1 (a) Prove that: (A ∩ B) ∪ (A − B) = A A ∪ (B − A) = A ∪ B Solution: (i) To prove: (A ∩ B) ∪ (A − B) = A Let x ∈ (A ∩ B) ∪ (A − B) . Then either x ∈ A ∩ B or x ∈ A − B . In both cases, x ∈ A . Hence, (A ∩ B) ∪ (A − B) ⊆ A . Now let x ∈ A . Then either x ∈ B or x ∉ B . If x ∈ B , then x ∈ A ∩ B . If x ∉ B , then x ∈ A − B . Hence, x ∈ (A ∩ B) ∪ (A − B) . Therefore, A ⊆ (A ∩ B) ∪ (A − B) . Thus, (A ∩ B) ∪ (A − B) = A. (ii) To prove: A ∪ (B − A) = A ∪ B Let x ∈ A ∪ (B − A) . Then x ∈ A or x ∈ B − A . In both cases, x ∈ A ∪ B . Hence, A ∪ (B − A) ⊆ A ∪ B . Conversely, let x ∈ A ∪ B . If x ∈ A , then x ∈ A ∪ (B − A) . If x ∈ B and x ∉ A , then x ∈ B − A . Hence, A ∪ B ⊆ A ∪ (B − A) . Therefore, A ∪ (B − A) = A ∪ B. ✔ Proved ...

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 ...