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Discrete maths assignment solution in short method
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Qualification HND in Computing (RQF) Center Name Center Registration No Student name Assessor name Unit Number/Name 18: Discrete Maths Assignment number and title Submitted on Abstract Algebra Click here to enter a date. Student Declaration Student signature: Date: Click here to enter a date.
Assignment discrete maths Part 1: A=(1,2,3,4,5,6,7,8,9,10) F=(1,3,5,7,9) A n F =(1,3,5,7,9) A n F = F A union F A u F =(1,2,3,4,5,6,7,8,9,10) A U F A f A F
Au(BnC)=(CuB)nA solution: If: Au(BnC)=(CuB)nA (BnC) included in A u(BnC) so (BnC) included in (CuB)nA (CuB)nA included in A so (BnC) included in A so Au(BnC)=A A included in Au(BnC) so A included in (CuB)nA (CuB)nA included in (CuB) so A included in (CuB) so (CuB)nA=A [Au(BnC)=(CuB)nA ]=>[ An(BnC)=A and (CuB)nA=A] b)[ An(BnC)=A and (CuB)nA=A]=>[Au(BnC)=(CuB)nA ] therefore c) [Au(BnC)=(CuB)nA] <=>[Au(BnC)=A and (CuB)nA=A] <=>[ BnC included in A and A included in CuB] L02: Introduction of Graphs In simple words, graphs are data structures that are used to depict connections amidst a couple of elements where these elements are called nodes (or vertex) that generally real-time objects, persons or entities and connections amid nodes are termed as edges. Also, two nodes only get connected if there is an edge between them "A graph is essentially an interrelationship of nodes/vertices connected by edges." Graphs can be divided into two parts;
Undirected Graphs: For every couple of associated nodes, if an individual could move from one node to another in both directions, then the graph is termed as an undirected graph. Directed Graphs: For every couple of associated graphs, if an individual could move from one node to another in a specific (single) direction, then the graph is known as the directed graph. In this case, arrows are implemented rather than simple lines in order to represent directed edges.
The weight graphs are the graphs where edges of the graph have “a weight” or “cost” and also where weight could reflect distance, time, money or anything that displays the “association” amid a couple of nodes it links. These weights are an essential element under Dijkstra's Algorithm. Unweighted graphs An unweighted undirected graph is defined by vertices and edges. The vertices are also often called the nodes in the graph. Vertices may be connected by edges. A common definition of a graph is via its adjacency matrix What is Dijkstra’s Algorithm?
Path refers to the sequence of nodes along the edges of a tree. Root, The node at the top of the tree is called root. There is only one root per tree and one path from the root node to any node. Parent, Any node except the root node has one edge upward to a node called parent Example1 : we need to find shortest path to arrive from Constantine to biskra
Constantine
Setif batna Biskra
Example : eulerian path Below is an example diagram of euler path
2. Eulerian circuit: An Eulerian circuit is an Eulerian trail that starts and ends on the same vertex, i.e., the path is a cycle. An undirected graph has an Eulerian cycle if and only if 1. Every vertex has an even degree, and 2. All of its vertices with a non-zero degree belong to a single connected component.
For example, the following graph has an Eulerian cycle since every vertex has an even degree: Hamiltonian Path If there exists a Path in the connected graph that contains all the vertices of the graph then such a path is called as a Hamiltonian path Note In Hamiltonian path all the edges may or may not be covered but edges must not repeat Hamiltonian Path Examples
Z Z=temp< 4 c ( 0 ) C X
Properties of Binary Operation Closure Property: A binary operation * on a non-empty set P has closure property, if a ∈ P, b ∈ P ⇒ a * b ∈ P. For example, addition is a binary operation that is closed on natural numbers, integers, and rational numbers. Associative Property: The associative property of binary operations holds if, for a non-empty set S, we can write (a * b) c = a(b * c), where {a, b, c} ∈ S. Suppose Z be the set of integers and multiplication be the binary operation. Let, a = -3, b = 5, and c = -16. We can write (a × b) × c = 240 = a × (b × c). Please note that all binary operations are not associative, for example, subtraction denoted by '-'. Commutative Property: A binary operation * on a non-empty set S is commutative, if a * b = b * a, for all (a, b) ∈ S. Suppose addition be the binary operation and N be the set of natural numbers. Let, a = 4 and b = 5, a + b = 9 = b + a. Distributive Property: Let * and # be two binary operations defined on a non-empty set S. The binary operations are distributive if, a* (b # c) = (a * b) # (a * c), for all {a, b, c} ∈ S. Suppose * is the multiplication operation and # is the subtraction operation defined on Z (set of integers). Let, a = 3, b = 4, and c = 7. Then, a*(b # c) = a × (b − c) = 3 × (4 − 7) = -9. And, (a * b) # (a
Example if we are considering multiplication operation on real numbers, then, a e = e a = a here e becomes 1 which is also a part of the set. If all the above three conditions are met, then the set is a group under that binary operation
