Discrete math’s /2023
Sunanda koirala(Hnd third semester)
1
2023
DISCRETE
Math
1/5/2023
MATHEMATICS
ASSIGNMENT
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Guidelines and tips
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Discrete Mathematics Assignment: Introduction to Boolean Algebra and Tree Structures, Assignments of Discrete Mathematics
the final assignment of discrete mathematics
Typology: Assignments
2019/2020
1 / 70
This page cannot be seen from the preview
Don't miss anything!
On special offer
Related documents
Partial preview of the text
Download Discrete Mathematics Assignment: Introduction to Boolean Algebra and Tree Structures and more Assignments Discrete Mathematics in PDF only on Docsity!
Sunanda koirala(Hnd third semester) 1
DISCRETE
Math
MATHEMATICS
ASSIGNMENT
Introduction A subset of mathematics known as discrete mathematics deals with discrete (i.e. separate and distinct) things as Oppose to continuous things. Discrete mathematics covers areas such as graph theory, combinatory, number theory and algorithms, among others. It is fundamental to computer science because many ideas and methods from Discrete mathematics are applied to data structures, algorithms, and other disciplines. Discrete mathematics can be used to prove theorems, create new mathematical concepts, and model and solve real-world problem Table of Content: Part 1………………………………………………………………………... Introduction ……………………………………………………………… Set theory……………………………………………………………………. Algebraic Set operations:……………………………………………………. Multisets or Bags:…………………………………. …………………………… Determining cardinal number of given bags:…………………………………….. 14 Functions:………………………………………………………………………. Determination of inverse functions by assuming……………………………… Example 1…. …………………………………………………………………. Example 2………………………………………………………………………. Properties of sets:………………………………………………………………. Proof …………………………………………………………………………… Proof ……………………………………………………………………………. Graph theory……………………………………………………………….......... Sunanda koirala(Hnd third semester)
Application on Group theory Presentation slide:…………………………………. 68 Conclusion:………………………………………………………………………… References…………………………………………………………………………. Part- On the basis of the given scenario we need to prepare a report, which examines the set theory and functions applicable to software engineering and analyze mathematical structures of objects using graph theory. Section 1 Perform algebraic set operations in a formulated mathematical problem. Determine the cardinality of a given bag (multi set). Determine the inverse of a function using appropriate mathematical techniques. Formulate corresponding proof principles to prove properties about defined sets. Section 2 Model contextualized problems using trees, both quantitatively and qualitatively. Use Dijkstra's algorithm to find a shortest path spanning tree in a graph. Assess whether an Euler Ian and Hamiltonian circuit exists in an undirected graph. Construct a proof of the Five Color Theorem. Introduction: Sunanda koirala(Hnd third semester)
We works as a mathematical analyst at Planning division within the Nepal Railways, the department helps the Government of Nepal by providing statistics in planning and developing infrastructures. You are supposed to contribute bIn this section, we will apply algebraic set operations to the presented mathematical problem. It also introduces the idea of the bag and defines the cardinality of the set. Here, we compute the inverse function using appropriate mathematical methods and establish a proof-of-fit principle to describe the properties of a particular set. The use of trees to model contextualized tasks, both statistically and qualitatively, is also discussed in the next section. It also uses Dijkstra's method to simultaneously find the shortest path in the graph's spanning tree. It also goes into detail on some basics of graph theory. It also checks whether there are Hamiltonian and Euler-Jan schemes. Set theory: In any mathematics set plays pivotal part. In simple world a set is the collection of well define unordered rudiments or Item. Individual objects in a set are called member rudiments. The proposition is less precious in direct operation to ordinary experience than as a base for precise and adaptable language for the description of complex and sophisticated fine generalities. A set might be any effects like a collection of number like odd number, indeed number, high number, rudiments, vowels, etc. In general, a set or collection of particulars all ways enclose in a curled braces {} along with set memos. Simple illustration of the set along with the memorandum is demonstrate below: The set notation are listed below: Sunanda koirala(Hnd third semester)
Intersection: Intersection of two sets A and B is the set of all those rudiments which belong to both A and B and is denoted by A ∩B. Difference between two sets: The difference of two sets A and B is a set of all those rudiments which belongs to A but don't belong to B and is denoted by A- B Components of set operations: The Complement of a Set A is a set of all those rudiments of the universal set which don't belong to A and is denoted by Ac. Symmetric difference of set operations. Sunanda koirala(Hnd third semester)
Symmetric difference of set operations: The symmetric difference of two sets A and B is the set containing all the rudiments that are in A or B but not in both and is denoted by A ⨁ B i.e. Disjoint set operations: The disjoint of the two sets A and B are set to be disjoint if the element of both sets aren't common and empty Some common parcels or lows of set operations related to union, crossroad and complements are Sunanda koirala(Hnd third semester)
B = { 2, 4, 6, 8}.
What's the cardinality of B? A ⋃ B, A ⋂ B? The cardinality of B is 4, since there are 4 rudiments in the set. The cardinality of A ⋃ B is 7, since A ⋃ B = { 1, 2, 3, 4, 5, 6, 8}, which contains 7 rudiments. The cardinality of A ⋂ B is 3, since A ⋂ B = { 2, 4, 6}, which contains 3 rudiments. Functions A function is a rule that assigns each input exactly one affair. We call the affair the image of the input. The set of all inputs for a function is called the sphere. The set of all permissible labors is called the codomain.( Levin,n.d.) We'd write fX → YfX → Y to describe a function with name f, f, sphere XX and codomainY.Y. This doesn't tell us which serve ff is however. To define the function, we must describe the rule. This is frequently done by giving a formula to cipher the affair for any input( although this is clearly not the only way to describe the rule). For illustration, consider the function fN → NfN → N defined by f( x) = x2 3. f( x) = x
- Then the sphere and codomain are the same set( the natural figures). The rule is take your input, multiply it by itself and add 3. This works because we can apply this rule to every natural number( every element of the sphere) and the result is always a natural number( an element of the codomain). Notice though that not every natural number is actually an affair( there is no way to get 0, 1, 2, 5,etc.). The set of natural figures that are labors is called the range of the function( in this case, the range is{,},{,}, all the natural figures that are 3 further than a perfect forecourt). Simple structure of functions, The term related to function which we define at over, likewise to know we've pasted then for more easily. There are colorful types of function in mathematics which are describe below Sunanda koirala(Hnd third semester)
- One to one function still, the function is said to be one – one function, If each element in the sphere of a function has a distinct image in theco-domain. In another term one to one function is also called as injective functions. 2. Onto functions A function is called an Onto function if each element in theco-domain has at least one pre
- image in the sphere. It's also called as surjective functions.
- many to one functions still, the function is known as many to one, If there are at least two rudiments in the sphere whose images are same.
- Invertible Function Bijection function are also known as invertible function because they've inverse function property. The antipode of bijection f is denoted as f- 1. It's a function which assigns to b, a unique element a similar that f( a) = b. hence f- 1 ( b) = a.
- Linear functions: All functions in the form of ax + b where a, b\in Rb∈R & a ≠ 0 are called as linear functions. The graph will be a straight line. In other words, a linear polynomial function is a first-degree polynomial where the input needs to be multiplied by m and added to c. It can be expressed by f(x) = mx + c. For example, f(x) = 2x + 1 at x = 1 f(1) = 2.1 + 1 = 3 Sunanda koirala(Hnd third semester)
Example, Steps to find inverse functions: Step 1: You will giving some function like this f(x) = 2x - Step 2: Replace f(x) = y Step 3: Interchange x and y Step 4: Solve for y in terms of x Step 5: Replace y with f-1(x) and the inverse of the function is obtained. Determination of inverse functions by assuming: Example 1: Let’s assume a function f(x) = (3x+2)/(x-1) and find inverse of respective assuming functions. Solutions: First, replace f(x) with y and the function becomes, y = (3x+2)/(x-1) By replacing x with y we get, x = (3y+2)/(y-1) Now, solve y in terms of x: x (y – 1) = 3y + 2 => xy – x = 3y + => xy – 3y = 2 + x => y (x – 3) = 2 + x Sunanda koirala(Hnd third semester)
=> y = (2 + x) / (x – 3) So, y = f-1(x) = (x+2)/(x-3) Therefore, inverse function of respective given functions is f-1(x) = (x+2)/(x-3) Example 2: Let’s assume a function f(x) f(x) = 2x + 3, at x = 4 and find inverse of respective assuming functions. Solutions: We have, f(4) = 2 × 4 + 3 f(4) = 11 Now, let’s apply for reverse on 11. f-1(11) = (11 – 3) / 2 f-1(11) = 4 Magically we get 4 again. Therefore, f-1(f(4)) = 4 So, when we apply function f and its reverse f-1 gives the original value back again, i.e, f- 1(f(x)) = x. Properties of sets: A sets is a well define collections of element or number. There are various types of properties are there in a sets.
- Commutative Laws: For any two finite sets A and B; Sunanda koirala(Hnd third semester)
(ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C) Thus, union and intersection are distributive over intersection and union respectively.
- De Morgan’s Laws: For any two finite sets A and B; (i) A – (B U C) = (A – B) ∩ (A – C) (ii) A - (B ∩ C) = (A – B) U (A – C) De Morgan’s Laws can also we written as: (i) (A U B)’ = A' ∩ B' (ii) (A ∩ B)’ = A' U B' Boolean identities how calculation is performed in table: Sunanda koirala(Hnd third semester)
Proof 1: Associative law proving using logic gate and Boolean truth table. Proof using OR and AND gate: Proof 2: De-Morgan’s law proving using logic gate and truth table. Sunanda koirala(Hnd third semester)
Tree: A connected graph with no cycles. (If we remove the requirement that the graph is connected, the graph is called a forest.) The vertices in a tree with degree 1 are called leaves. Tree have a different part. The edge of the tree is known as the branch, the element of the tree are called nodes and the child nodes of the tree are called leaf nodes. A tree with ‘n’ vertices has n- 1 edges. If it has one more edge extra than n-1, then the extra edge should obviously has to pair up with two vertices which leads to form a cycle. Then, it becomes a cyclic graph which is a violation for the tree graph. Example of tree: In this graph it is tree because it has no cycle and it is connected. It has the number of four vertices and three number of edges. i.e for n vertices n-1 edge as mentioned in here Sunanda koirala(Hnd third semester)
the center of the tree is known as the vertices having minimal eccentricity. The eccentricity of the given vertices X in a tree G having the maximum distance between the two vertices X and other any vertices of the tree. The maximum eccentricity is tree diameter. If a tree has only one center it is known as central tree. And if the tree has more than one center it is known as the Bi- central tree. Some of algorithm to find out the centers and bi-centers of the tree Step1: Remove all the vertices of degree 1 from the given tree and remove their incident edges Step 2: First of all repeat the step one until either the single vertex or the two vertices that are joined by the edge is a left. If the single vertex is left than it is the center of the tree and if more than the two vertices are joined by the edge is left than it is the bi centers for the tree. Problem 1: Find out the central tree of the given tree Solution At first, we will remove all vertices of degree 1 and also remove their incident edges and get the following tree − Sunanda koirala(Hnd third semester)