Discrete mathematical structure complete notes of 3rd semester B.tech.

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In this, you will get the questions and answers to the problems of maths of the 3rd semester of B.tech (CS, IT, CTIS). Topics Covered in this notes are: 1. Mathematical Logic. 2. Relations and Functions. 3. Combinations. 4. Induction, Recursion. 5. Group and Coding Theory.

Engineering

DISCRETE MATHEMATICAL
STRUCTURE - DMS
Hand-Written notes
By

Page 1 of 144
Index
S.NO. Topic Name Page Number
1 Mathematical Logic 3 – 39
2 Relations and Functions 40 – 75
3 Combinators 76 – 99
4 Induction, Recursion and
Inclusion
100 – 118
And half is on 144 – 145
5 Group and Coding Theory 119 - 143
*NOTE: - Important topics, formulas and
questions are marked by green colures pen
whereas most important topics, Formulas and
questions are marked by red pen.

The document discusses methods for solving systems of linear equations. It introduces Gauss elimination and Gauss Jordan methods. Gauss elimination transforms the augmented matrix of the system into row echelon form through elementary row operations, then back-substitutes to solve for the variables. Gauss Jordan additionally transforms the matrix to reduced row echelon form to read solutions directly from the matrix. An example demonstrates applying each method to solve a system of equations.

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The document discusses maxima and minima of functions of two independent variables. It defines a relative minimum or maximum point of a function f(x,y) and explains how to determine the stationary points by simultaneously solving the partial derivatives of f with respect to x and y. The working rule outlined finds the extreme values by checking the signs of the second order partial derivatives at the stationary points. An example of finding the maxima of f(x,y)=x^2+y^2+12x+22y is provided to illustrate the process.

Eigenvectors & Eigenvalues: The Road to Diagonalisation

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The document discusses eigenvectors and eigenvalues. It begins by defining diagonal matrices and provides examples. It then states that the goal is to understand diagonalization using eigenvectors and eigenvalues. Diagonalization involves finding a matrix such that when it transforms another matrix, the result is a diagonal matrix. This requires the eigenvectors of the original matrix to be linearly independent. The document provides examples of calculating eigenvectors and eigenvalues from matrices and shows how this relates to diagonalization. It also gives a brief introduction to linear independence and its implications for diagonalization.

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This document provides an overview of vector differentiation, including gradient, divergence, curl, and related concepts. It begins with definitions of scalar and vector point functions. It then defines the vector differential operator Del and explores using it to calculate the gradient of a scalar function, directional derivatives, and normal derivatives. The document also covers divergence and curl, providing their definitions and formulas. Examples are given for calculating gradient, divergence, curl, and directional derivatives. The document concludes with exercises and references for further reading.

Triangularization method

Kamran Ansari

This document describes the triangularization method for solving systems of linear equations. It involves decomposing the coefficient matrix A into the product of a lower triangular matrix L and an upper triangular matrix U. This decomposition is done by calculating the elements of L and U such that their product equals the original matrix A. The system of linear equations can then be solved through back substitution by first solving the system LUx=B as two separate systems involving L and U. The triangularization method is preferable to Gaussian elimination as it requires fewer operations and can also be used to find the inverse of a matrix. However, it fails if any diagonal elements of L or U are zero.

Linear transformation and application

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This document discusses linear transformations between vector spaces. It begins by defining a linear transformation as a function between vector spaces that satisfies the properties of vector addition and scalar multiplication. It then provides examples of standard linear transformations like the matrix transformation and zero transformation. The document also covers properties of linear transformations such as how they are determined by the images of basis vectors. Finally, it provides applications of linear operators like reflection, rotation, and shear transformations.

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The document describes m-way search trees, B-trees, heaps, and their related operations. An m-way search tree is a tree where each node has at most m child nodes and keys are arranged in ascending order. B-trees are similar but ensure the number of child nodes falls in a range and all leaf nodes are at the same depth. Common operations like searching, insertion, and deletion are explained for each with examples. Heaps store data in a complete binary tree structure where a node's value is greater than its children's values.

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Tensors are mathematical objects that generalize scalars, vectors and matrices to higher dimensions. A tensor is a multidimensional array of numbers that transforms in a predictable way under changes of coordinates. Lower-rank tensors include scalars (rank-0), vectors (rank-1) and matrices (rank-2). Tensors are important in physics for formulating and solving problems involving quantities that have both magnitude and direction, such as mechanics, stress and elasticity.

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This document discusses numerical methods for solving linear systems of equations. It begins by introducing linear systems in general and matrix forms, and classifying them as homogeneous or non-homogeneous. It then discusses checking for consistency. The main methods covered for obtaining solutions are: Gauss elimination, Gauss-Jordan elimination, using the inverse matrix if it exists, and iterative techniques. Specific examples are provided to demonstrate how to apply Gauss elimination, Gauss-Jordan elimination and using the inverse matrix to solve sample systems.

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This document discusses linear data structures, specifically stacks and queues. It defines a stack as a linear data structure that follows the LIFO principle and can be implemented using arrays or linked lists. Queue is defined as another linear data structure that follows the FIFO principle and supports enqueue and dequeue operations. Various queue implementations like array-based queues, linked list queues, circular queues, and priority queues are described along with their operations. Common applications of stacks and queues in areas like function calls, CPU scheduling, and page replacement algorithms are also mentioned.

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Given weights and values of n items, put these items in a knapsack of capacity W to get the maximum total value in the knapsack. In other words, given two integer arrays val[0..n-1] and wt[0..n-1] which represent values and weights associated with n items respectively. Also given an integer W which represents knapsack capacity, find out the maximum value subset of val[] such that sum of the weights of this subset is smaller than or equal to W. You cannot break an item, either pick the complete item or don’t pick it (0-1 property). Method 1: Recursion by Brute-Force algorithm OR Exhaustive Search. Approach: A simple solution is to consider all subsets of items and calculate the total weight and value of all subsets. Consider the only subsets whose total weight is smaller than W. From all such subsets, pick the maximum value subset. Optimal Sub-structure: To consider all subsets of items, there can be two cases for every item. Case 1: The item is included in the optimal subset. Case 2: The item is not included in the optimal set. Therefore, the maximum value that can be obtained from ‘n’ items is the max of the following two values. Maximum value obtained by n-1 items and W weight (excluding nth item). Value of nth item plus maximum value obtained by n-1 items and W minus the weight of the nth item (including nth item). If the weight of the ‘nth’ item is greater than ‘W’, then the nth item cannot be included and Case 1 is the only possibility.

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This document discusses the prevalence and importance of mathematics in everyday life. It provides examples of how mathematics is used in areas like health, weather, transportation, society, and more. While some applications are directly observable, others involve more complex systems that are still being understood mathematically, like DNA. The document also discusses the historical foundations of mathematics over centuries, with concepts building upon each other like a pyramid, and provides a brief biography of the mathematician Aryabhata, who made important contributions in astronomy and mathematics.

MATRICES

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Ramanujan number

mathematics20152017

1729 is known as the Hardy–Ramanujan number because of a famous anecdote involving the mathematicians G. H. Hardy and Srinivasa Ramanujan. When Hardy remarked that the taxi cab number 1729 seemed dull, Ramanujan revealed that 1729 can be expressed as the sum of two cubes in two different ways: 1729 = 13 + 123 = 93 + 103. This makes 1729 the smallest number that can be expressed in two distinct ways as the sum of cubes, earning it the description of a "taxicab number."

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kishor pokar

The document defines row echelon form and reduced row echelon form for matrices. Row echelon form requires that leading 1's occur farther to the right in lower rows. Reduced row echelon form further requires that all entries above leading 1's are zero. The document also discusses Gauss elimination method and elementary row operations for transforming a matrix into row echelon or reduced row echelon form.

07.03 cartesian product

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The document discusses three relational algebra operations: rename, cartesian product, and selection with a join. The rename operation changes the name of a relation without altering the tuples. The cartesian product pairs all tuples of one relation with all tuples of another relation, resulting in a cross join. The example operation selects tuples from two joined relations where the high school is over 1000 and the major is computer science with a rejection decision.

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Discrete mathematical structure complete notes of 3rd semester B.tech.

1. DISCRETE MATHEMATICAL STRUCTURE - DMS Hand-Written notes By

2. Page 1 of 144 Index S.NO. Topic Name Page Number 1 Mathematical Logic 3 – 39 2 Relations and Functions 40 – 75 3 Combinators 76 – 99 4 Induction, Recursion and Inclusion 100 – 118 And half is on 144 – 145 5 Group and Coding Theory 119 - 143 *NOTE: - Important topics, formulas and questions are marked by green colures pen whereas most important topics, Formulas and questions are marked by red pen.

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